PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre...
Transcript of PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre...
![Page 1: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/1.jpg)
PHYS 405 - Fundamentals of Quantum Theory I
Term: Fall 2016Meetings: Monday & Wednesday 11:25-12:40Location: 212 Stuart Building
Instructor: Carlo SegreOffice: 106A Life SciencesPhone: 312.567.3498email: [email protected]
Book: Introduction to Quantum Mechanics, 2nd ed.,D. Griffiths (Pearson, 2005)
Web Site: http://phys.iit.edu/∼segre/phys405/16F
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24
![Page 2: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/2.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 3: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/3.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 4: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/4.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 5: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/5.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 6: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/6.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 7: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/7.jpg)
Course Objectives
1 Understand the interpretation of the Schrodinger equation and thewave function.
2 Understand the solution of the time-independent Schrodingerequation for one-dimensional potentials.
3 Understand quantum formalism including operators and the Diracnotation.
4 Understand the solution of three-dimensional potentials.
5 Understand how systems of identical particles are solved.
6 Be able to solve quantum mechanics problems in one and threedimensions and with identical particles.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 2 / 24
![Page 8: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/8.jpg)
Course Grading
15% – Homework assignments
Weekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 9: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/9.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weekly
Due at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 10: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/10.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weeklyDue at beginning of class
May be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 11: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/11.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 12: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/12.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 13: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/13.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 14: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/14.jpg)
Course Grading
15% – Homework assignmentsWeekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard
50% – Two mid-term exams
35% – Final examination (TBA)
Grading scaleA – 88% to 100%B – 75% to 88%C – 62% to 75%D – 50% to 62%E – 0% to 50%
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 3 / 24
![Page 15: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/15.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 16: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/16.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 17: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/17.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 18: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/18.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 19: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/19.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 20: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/20.jpg)
Topics to be Covered (Chapter titles)
1 The wave function
2 Time-independent Schrodinger equation
3 Quantum formalism
4 Three dimensional quantum mechanics
5 Identical particles
6 Other topics as appropriate
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 4 / 24
![Page 21: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/21.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook.
TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 22: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/22.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook.
TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 23: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/23.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 24: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/24.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 25: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/25.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 26: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/26.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort.
Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 27: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/27.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort. Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 28: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/28.jpg)
Tips for success
1 Do the reading assignments before lecture, you willunderstand them better.
2 Attend class or really view the lectures completely, thereare things discussed which are not on the slides or thebook. TAKE NOTES!
3 Ask questions in class, it’s likely that others have thesame ones.
4 Go through the derivations yourself, kill some trees!
5 Do the homework the “right” way, only use the solutionsmanual as a last resort. Struggling is good and helps youlearn!
6 Come to office hours with questions, I’ll be less lonelyand it will help you too!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 5 / 24
![Page 29: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/29.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 30: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/30.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 31: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/31.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 32: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/32.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 33: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/33.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 34: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/34.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 35: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/35.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 36: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/36.jpg)
Course Schedule
Focus on “mechanics” but will bring in some originalarticles if there is time available.
Exam #1 – Monday, October 03, 2016Exam #2 – Wednesday, November 09, 2016
Up-to-date schedule athttp://phys.iit.edu/∼segre/phys405/16F/schedule.html
We have 25 class sessions,
2 mid-term exams,
250 pages to cover,
and we’re online.
Let’s start!
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 6 / 24
![Page 37: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/37.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 38: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/38.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 39: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/39.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 40: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/40.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 41: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/41.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 42: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/42.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 43: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/43.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 44: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/44.jpg)
Today’s Outline - August 22, 2016
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Davisson-Germer experiment
• The 1-D Schrodinger equation
Reading Assignment: Chapter 1.1–1.6
Homework Assignment #01:Chapter 1: 1, 3, 8, 11, 15, 17due Wednesday, August 31, 2016
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 7 / 24
![Page 45: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/45.jpg)
Black Body Radiation
The radiation spectrum of ablack-body depends on thetemperature of the object.
For example, T=5000 K.
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 46: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/46.jpg)
Black Body Radiation
The radiation spectrum of ablack-body depends on thetemperature of the object.
For example, T=5000 K.
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 47: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/47.jpg)
Black Body Radiation
The maximum wavelengthλm is seen to scale inverselywith temperature such that
λmT = 2.898× 10−3m · K3
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 48: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/48.jpg)
Black Body Radiation
The maximum wavelengthλm is seen to scale inverselywith temperature such that
λmT = 2.898× 10−3m · K3
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
4000 K
3000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 49: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/49.jpg)
Black Body Radiation
The maximum wavelengthλm is seen to scale inverselywith temperature such that
λmT = 2.898× 10−3m · K3
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
4000 K
3000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 50: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/50.jpg)
Black Body Radiation
This proves to be a universalcurve.
However, the classical the-oretical model (Rayleigh–Jeans), is unable to describethe low wavelength cutoffobserved.∫ ∞0
u(λ)dλ ∝∫ ∞0
λ−4dλ→∞
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
4000 K
3000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 51: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/51.jpg)
Black Body Radiation
This proves to be a universalcurve.
However, the classical the-oretical model (Rayleigh–Jeans), is unable to describethe low wavelength cutoffobserved.
∫ ∞0
u(λ)dλ ∝∫ ∞0
λ−4dλ→∞
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
4000 K
3000 K
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 52: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/52.jpg)
Black Body Radiation
This proves to be a universalcurve.
However, the classical the-oretical model (Rayleigh–Jeans), is unable to describethe low wavelength cutoffobserved.∫ ∞0
u(λ)dλ ∝∫ ∞0
λ−4dλ→∞
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
Inte
nsity (
arb
.)
Wavelength (µm)
5000 K
4000 K
3000 K
Rayleigh-Jeans(5000 K)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 8 / 24
![Page 53: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/53.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 54: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/54.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 55: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/55.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 56: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/56.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 57: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/57.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 58: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/58.jpg)
Planck’s Solution
By assuming that the modesof oscillation in the black-body cavity were quantized.
The resulting function forthe energy distribution is
which cuts off properly asλ→ 0.
Em = mhν, m = 0, 1, 2, 3, · · ·
u(λ) ∝ λ−5
ehc/λkT − 1
limλ→0
u(λ) =e−hc/λkT
λ5
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 9 / 24
![Page 59: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/59.jpg)
Photoelectric Effect
In the photoelectric effect, the emission of electrons depends on the colorof the incident light rather than its intensity.
Einstein (1905) explained this by reasoning that light must be quantizedaccording to its frequency, thereby acting as a particle.
1
2mv2
max = hν − φ
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 10 / 24
![Page 60: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/60.jpg)
Photoelectric Effect
In the photoelectric effect, the emission of electrons depends on the colorof the incident light rather than its intensity.
Einstein (1905) explained this by reasoning that light must be quantizedaccording to its frequency, thereby acting as a particle.
1
2mv2
max = hν − φ
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 10 / 24
![Page 61: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/61.jpg)
Photoelectric Effect
In the photoelectric effect, the emission of electrons depends on the colorof the incident light rather than its intensity.
Einstein (1905) explained this by reasoning that light must be quantizedaccording to its frequency, thereby acting as a particle.
1
2mv2
max = hν − φ
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 10 / 24
![Page 62: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/62.jpg)
Compton Scattering Experiment
In 1923, Arthur Compton mea-sured the scattering of x-raysfrom a carbon foil.
He observed x-rays at lower en-ergies than the incident energyand that the energy dependedon the observation angle.
This could be explained bytreating the x-rays as particlescolliding with the electrons inthe carbon atoms of the foil.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 11 / 24
![Page 63: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/63.jpg)
Compton Scattering Experiment
In 1923, Arthur Compton mea-sured the scattering of x-raysfrom a carbon foil.
He observed x-rays at lower en-ergies than the incident energyand that the energy dependedon the observation angle.
This could be explained bytreating the x-rays as particlescolliding with the electrons inthe carbon atoms of the foil.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 11 / 24
![Page 64: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/64.jpg)
Compton Scattering Experiment
In 1923, Arthur Compton mea-sured the scattering of x-raysfrom a carbon foil.
He observed x-rays at lower en-ergies than the incident energyand that the energy dependedon the observation angle.
This could be explained bytreating the x-rays as particlescolliding with the electrons inthe carbon atoms of the foil.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 11 / 24
![Page 65: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/65.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 66: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/66.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 67: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/67.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λ
p′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 68: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/68.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 69: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/69.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 70: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/70.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 71: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/71.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 72: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/72.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 73: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/73.jpg)
Compton Scattering Phenomenon
A photon-electron collision
ϕ
θ
λ
v
λ
p = ~k = 2π~/λp′ = ~k′ = 2π~/λ′
|k| 6=∣∣k′∣∣
Treat the electron relativistically and conserve energy and momentum
mc2 +hc
λ=
hc
λ′+ γmc2 (energy)
h
λ=
h
λ′cosφ+ γmv cos θ (x-axis)
0 =h
λ′sinφ+ γmv sin θ (y-axis)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 12 / 24
![Page 74: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/74.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 75: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/75.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ
(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 76: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/76.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 77: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/77.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together,
substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 78: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/78.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,
and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 79: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/79.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange
and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 80: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/80.jpg)
Compton Scattering Derivation
squaring the momentumequations
(h
λ− h
λ′cosφ
)2
= γ2m2v2 cos2 θ(− h
λ′sinφ
)2
= γ2m2v2 sin2 θ
now add them together, substitute sin2 θ + cos2 θ = 1, expand the squares,and sin2 φ+ cos2 φ = 1, then rearrange and substitute v = βc
γ2m2v2(sin2 θ + cos2 θ
)=
(h
λ− h
λ′cosφ
)2
+
(− h
λ′sinφ
)2
γ2m2v2 =h2
λ2− 2h2
λλ′cosφ+
h2
λ′2sin2 φ+
h2
λ′2cos2 φ
m2c2β2
1− β2=
m2v2
1− β2=
h2
λ2− 2h2
λλ′cosφ+
h2
λ′2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 13 / 24
![Page 81: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/81.jpg)
Compton Scattering Derivation
Now take the energy equation and square it,
then solve it for β2 which issubstituted into the equation from the momentum.
(mc2 +
hc
λ− hc
λ′
)2
= γ2m2c4 =m2c4
1− β2
β2 = 1− m2c4(mc2 + hc
λ −hcλ′
)2h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
m2c2β2
1− β2
=1
c2
(mc2 +
hc
λ− hc
λ′
)2
−m2c2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 14 / 24
![Page 82: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/82.jpg)
Compton Scattering Derivation
Now take the energy equation and square it, then solve it for β2
which issubstituted into the equation from the momentum.
(mc2 +
hc
λ− hc
λ′
)2
= γ2m2c4 =m2c4
1− β2
β2 = 1− m2c4(mc2 + hc
λ −hcλ′
)2
h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
m2c2β2
1− β2
=1
c2
(mc2 +
hc
λ− hc
λ′
)2
−m2c2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 14 / 24
![Page 83: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/83.jpg)
Compton Scattering Derivation
Now take the energy equation and square it, then solve it for β2 which issubstituted into the equation from the momentum.(
mc2 +hc
λ− hc
λ′
)2
= γ2m2c4 =m2c4
1− β2
β2 = 1− m2c4(mc2 + hc
λ −hcλ′
)2h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
m2c2β2
1− β2
=1
c2
(mc2 +
hc
λ− hc
λ′
)2
−m2c2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 14 / 24
![Page 84: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/84.jpg)
Compton Scattering Derivation
Now take the energy equation and square it, then solve it for β2 which issubstituted into the equation from the momentum.(
mc2 +hc
λ− hc
λ′
)2
= γ2m2c4 =m2c4
1− β2
β2 = 1− m2c4(mc2 + hc
λ −hcλ′
)2h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
m2c2β2
1− β2
=1
c2
(mc2 +
hc
λ− hc
λ′
)2
−m2c2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 14 / 24
![Page 85: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/85.jpg)
Compton Scattering Derivation
After expansion, cancellation, and rearrangement, we obtain
h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
��λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 86: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/86.jpg)
Compton Scattering Derivation
After expansion,
cancellation, and rearrangement, we obtain
h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
��λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 87: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/87.jpg)
Compton Scattering Derivation
After expansion, cancellation,
and rearrangement, we obtain
h2
λ2+
h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
h2
λ2+
h2
λ′2− 2h2
λλ′
2h2
��λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 88: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/88.jpg)
Compton Scattering Derivation
After expansion, cancellation,
and rearrangement, we obtain
���h2
λ2+���h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
��λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 89: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/89.jpg)
Compton Scattering Derivation
After expansion, cancellation, and rearrangement, we obtain
���h2
λ2+���h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)
= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 90: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/90.jpg)
Compton Scattering Derivation
After expansion, cancellation,
and rearrangement, we obtain
���h2
λ2+���h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)
=2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 91: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/91.jpg)
Compton Scattering Derivation
After expansion, cancellation,
and rearrangement, we obtain
���h2
λ2+���h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 92: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/92.jpg)
Compton Scattering Derivation
After expansion, cancellation,
and rearrangement, we obtain
���h2
λ2+���h2
λ′2− 2h2
λλ′cosφ =
(mc +
h
λ− h
λ′
)2
−m2c2
= ���m2c2 +h2
λ2+
h2
λ′2− 2mch
λ− 2mch
λ′+
2h2
λλ′−���m2c2
= 2m
(hc
λ− hc
λ′
)+
���h2
λ2+���h2
λ′2− 2h2
λλ′
2h2
��λλ′(1− cosφ) = 2m
(hc
λ− hc
λ′
)= 2mhc
(λ′ − λλλ′
)=
2mhc∆λ
��λλ′
∆λ =h
mc(1− cosφ)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 15 / 24
![Page 93: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/93.jpg)
Compton Scattering Equation
∆λ =h
mc(1− cosφ)
This explains the change in en-ergy of the broader peak withincreasing angle
It also provides the basis for un-derstanding why the Comptonpeak is so broad . . .
. . . since the electrons are notreally “stationary”, there will bea spread in energy and momen-tum of the outgoing photon.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 16 / 24
![Page 94: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/94.jpg)
Compton Scattering Equation
∆λ =h
mc(1− cosφ)
This explains the change in en-ergy of the broader peak withincreasing angle
It also provides the basis for un-derstanding why the Comptonpeak is so broad . . .
. . . since the electrons are notreally “stationary”, there will bea spread in energy and momen-tum of the outgoing photon.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 16 / 24
![Page 95: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/95.jpg)
Compton Scattering Equation
∆λ =h
mc(1− cosφ)
This explains the change in en-ergy of the broader peak withincreasing angle
It also provides the basis for un-derstanding why the Comptonpeak is so broad . . .
. . . since the electrons are notreally “stationary”, there will bea spread in energy and momen-tum of the outgoing photon.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 16 / 24
![Page 96: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/96.jpg)
Compton Scattering Equation
∆λ =h
mc(1− cosφ)
This explains the change in en-ergy of the broader peak withincreasing angle
It also provides the basis for un-derstanding why the Comptonpeak is so broad . . .
. . . since the electrons are notreally “stationary”, there will bea spread in energy and momen-tum of the outgoing photon.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 16 / 24
![Page 97: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/97.jpg)
Davisson-Germer Experiment
In 1928, Davisson & Germershowed that DeBroglie’s hy-pothesis of the wave nature ofparticles was correct.
By measuring the electronsscattered at various energiesfrom a metal foil, the observa-tion of Bragg’s Law for elec-trons was made.
This could only be explained byinterference between electrons.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 17 / 24
![Page 98: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/98.jpg)
Davisson-Germer Experiment
In 1928, Davisson & Germershowed that DeBroglie’s hy-pothesis of the wave nature ofparticles was correct.
By measuring the electronsscattered at various energiesfrom a metal foil, the observa-tion of Bragg’s Law for elec-trons was made.
This could only be explained byinterference between electrons.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 17 / 24
![Page 99: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/99.jpg)
Davisson-Germer Experiment
In 1928, Davisson & Germershowed that DeBroglie’s hy-pothesis of the wave nature ofparticles was correct.
By measuring the electronsscattered at various energiesfrom a metal foil, the observa-tion of Bragg’s Law for elec-trons was made.
This could only be explained byinterference between electrons.
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 17 / 24
![Page 100: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/100.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 101: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/101.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 102: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/102.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 103: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/103.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 104: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/104.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 105: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/105.jpg)
1D Schrodinger equation
i~∂Ψ
∂t= − ~2
2m
∂2Ψ
∂x2+ V Ψ
i~∂Ψ
∂t= Total Energy
− ~2
2m
∂2Ψ
∂x2= Kinetic Energy
V Ψ = Potential Energy
where the wave function,Ψ(x , t) is a function of bothtime and space
this equation can be viewed asan expression of conservation ofenergy
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 18 / 24
![Page 106: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/106.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 107: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/107.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation
with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 108: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/108.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation
with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 109: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/109.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 110: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/110.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting
results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 111: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/111.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting
results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 112: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/112.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 113: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/113.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c
−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 114: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/114.jpg)
Deriving the Schrodinger equation
“When Schrodinger first wrote it down, he gave a kind of derivation basedon some heuristic arguments and some brilliant intuitive guesses. Some ofthe arguments he used were even false, but that does not matter; the onlyimportant thing is that the ultimate equation gives a correct description ofnature.” - Richard Feynman
Inspired by wave optics,Schrodinger started with thewave equation for electromagneticradiation with solution
E (x , t) = E0e i(kx−ωt)
taking the derivatives and substi-tuting results in the dispersion re-lation for photons
0 =∂2E
∂x2− 1
c2
∂2E
∂t2
0 =
(−k2 +
ω2
c2
)E0e i(kx−ωt)
k =ω
c−→ E = pc
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 19 / 24
![Page 115: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/115.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 116: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/116.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy
andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 117: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/117.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy
andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 118: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/118.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 119: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/119.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k
−→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 120: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/120.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 121: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/121.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 122: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/122.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t)
+ V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 123: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/123.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t)
+ V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 124: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/124.jpg)
Deriving the Schrodinger equation (cont.)
This works for a relativistic massless particle like the photon, but for anon-relativistic particle with mass, we have to start with a differentdispersion relation.
The dispersion relation for a non-relativistic particle must be consis-tent with the classical energy andDeBroglie’s relation
E =p2
2m= ~ω
p = ~k −→ ~2k2
2m= ~ω
Therefore, we need a wave equation which gives this dispersion relationwhen applied to a traveling matter plane wave, Ψ(x , t) = ψ0e i(kx−ωt)
i~∂
∂tΨ(x , t) = − ~2
2m
∂2
∂x2Ψ(x , t) + V (x)Ψ(x , t)
when including the possibility of a potential, V (x)
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 20 / 24
![Page 125: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/125.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 126: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/126.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 127: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/127.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 128: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/128.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 129: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/129.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 130: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/130.jpg)
Meaning of the wave function
The wave function, Ψ(x , t) de-scribes everything about a par-ticle (system)
a complex quantity but itsphase is meaningless
spatial integral gives probabilityof the particle being found inthe interval from a to b
Copenhagen interpretation hasproven to be correct one – col-lapse of the wave function aftermeasurement!
∫ b
a|Ψ(x , t)|2 dx
|Ψ|2
xa b
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 21 / 24
![Page 131: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/131.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j .
Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 132: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/132.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 133: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/133.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 134: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/134.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 135: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/135.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 136: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/136.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 137: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/137.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 138: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/138.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 139: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/139.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 140: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/140.jpg)
Probability review
N =∞∑j=0
N(j)
P(j) =N(j)
N
1 =∞∑j=0
P(j)
〈j〉 =
∑jN(j)
N
=∞∑j=0
jP(j)
Suppose we have a distribution of discrete quantitiessuch as ages of people in a sports stadium, where N(j)is the number of individuals with the age j . Thetotal number of people, N, is
The probability of an individual chosen at randomfrom the crowd having the age j is
The sum of all the probabilities must be 1
The average value of the age (not the most probable)is given by
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 22 / 24
![Page 141: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/141.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as
and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 142: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/142.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 143: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/143.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 144: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/144.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩
σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 145: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/145.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩
σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 146: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/146.jpg)
Expectation values
In general, the average value of any quan-tity, f (j) which depends on this distribu-tion may be calculated as and given thename, expectation value
One particular quantity, the variance, de-scribes the “width” of the distributionand is given by
Where σ is called the standard deviationof the distribution
〈f (j)〉 =∞∑j=0
f (j)P(j)
σ2 ≡⟨(∆j)2
⟩σ =
√〈j2〉 − 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 23 / 24
![Page 147: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/147.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩
=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 148: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/148.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 149: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/149.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 150: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/150.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 151: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/151.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 152: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/152.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 153: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/153.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2
=⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 154: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/154.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24
![Page 155: PHYS 405 - Fundamentals of Quantum Theory Icsrri.iit.edu/~segre/phys405/16F/lecture_01.pdfC. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 1 / 24 Course Objectives 1 Understand](https://reader030.fdocuments.us/reader030/viewer/2022021501/5aa2466c7f8b9a436d8cbf56/html5/thumbnails/155.jpg)
Computing the variance
σ2 =⟨(∆j)2
⟩=∑
(∆j)2 P(j)
=∑
(j − 〈j〉)2 P(j)
=∑(
j2 − 2j 〈j〉+ 〈j〉2)
P(j)
=∑
j2P(j) +∑
2j 〈j〉P(j) +∑〈j〉2 P(j)
=⟨j2⟩− 2 〈j〉 〈j〉+ 〈j〉2 =
⟨j2⟩− 〈j〉2
∆j = (j − 〈j〉)
expanding the square
dividing into threesums
Since σ2 ≥ 0,⟨j2⟩≥ 〈j〉2
C. Segre (IIT) PHYS 405 - Fall 2016 August 22, 2016 24 / 24