ADVANCED QUANTUM MECHANICS

Time: Fri BCD 9:00-12:00 Location: ED 201

Instructor: Oleksandr Voskoboynikov 霍斯科

Phone: 5712121 ext. 54174 Office: 646 ED bld.4 E-mail: vam@faculty.nctu.edu.tw Office hours: by appointment Pre-requisite courses: Engineering Mathematics, Linear Algebra, Modern Physic, Electromagnetics Text book: J. J. Sakurai , Jim J. Napolitano, Modern Quantum Mechanics (2nd Edition), Addison-Wesley, 2011 (or any Edition) Reference books: 1. Daniel R. Bes, Quantum Mechanics (Second, Revised Edition), Springer, 2007 2. Franz Schwabl, Quantum Mechanics (Fourth Edition), Springer, 2007 3. Yehuda B. Band and Yshai Avishai, Quantum Mechanics with applications to nanotechnology and

information science, Elsevier, 2013 4. Class Notes. Credits: 3 (hours for weekly study –3) Grade: Home Works and Quiz 15% Midterm: 40% Final: 45%

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Complex Variables

Web: http://web.it.nctu.edu.tw/~vam/

Course Description:

The course treats non-relativistic quantum mechanics. It

introduces foundations, principles and basic approaches of

quantum mechanics to experimental and theoretical exploration of

the nature. The course introduces techniques to solve quantum

mechanical problems. Some of the most important central force

problems including the theory of angular momentum and spin are

considered. Perturbation theory as well as other approximation

methods are discussed. Basic problems on atomic systems, multi-

particle systems, and quantum theory of scattering are considered.

In addition an introductory consideration is being given to the

quantum information processing.

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Introduction

Basic concepts of quantum mechanics

Wave functions, operators,

observables, and quantum measurement

1. Linear vector spaces. Hilbert space

2. Operators and representations

3. Wave function and density operator

4. Measurement, observables, uncertainty

relations

Evolutions of quantum systems

1.Time evolution and the Schrodinger equation

2.Schrodinger and Heisenberg pictures

3. Examples of solvable cases for the Schrodinger

equation

4. Propagators

5.Potentials and Gauge Transformations

Theory of Angular Momentum 1.Rotation and angular momentum

2.Spin

3.Eigenvalues and eigenstates of angular

momentum operators

4. Schrodinger equation for central potentials

5. Addition of Angular Momenta

Approximation Methods 1. Time independent perturbation theory. Non-

degenerate and degenerate cases

2.Variational Methods

3.Hydrogenlike atoms. Fine structure and

Zeeman effect

4. The Wentzel-Kramers-Brillouin (WKB)

approximation

5.Time dependent perturbation theory

6.Interaction with the classical radiation fields.

Light emission and absorption

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Scattering Theory 1.The Lippmann-Schwinger Equation

2.The Born approximation

3.Optical theorem

4.Method of partial waves

5.Resonance scattering

Multi-practical Systems

1. Basic quantum statistics

2. Two-electron systems

3. The helium atom

4. The Hartree–Fock method

Elements of Quantum Information 1.Qubits’ concept

2.Quantum gates and elementary qubit operations

3. Quantum information processing. Physical

implementations

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“…trying to find a computer simulation of physics, seems

to me to be an excellent program to follow out … and I’m not happy with all the analyses that go with just

the classical theory, because

NATURE IST’T CLASSICAL,

dammit, and if you want to make a simulation of nature, you’d better

MAKE IT QUANTUM MECHANICAL,

and golly it’s a wonderful problem because it doesn’t look so easy.”

Richard Feynman (International Journal of Theoretical Physics, Vol.21,

p. 486, 1981)

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A FEW OF WELL KNOWN EXAMPLES:

#1. How does a magnet work?

People in ancient China discovered that natural lodestone magnets attracted iron. The

Chinese also found that a piece of lodestone would point in a north-south direction if it was

allowed to rotate freely. They used this characteristic of lodestone to tell fortunes and as a

guide for building. By A.D. 1200, sailors used magnetic compasses to steer their ships.

John H. Van Vleck of the United States and Louis E. F. Neel of France applied quantum

mechanics to understand the magnetic properties of atoms and molecules.

All magnets are macro-quantum objects

and

ħ 0 M, 0

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#2. How does the light work?

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#3. How does a superconductor work?

High Temperature

Ordinary Conductivity

At high temperature one observes

a state of ordinary conductivity

due to disorderly dynamics of the electrons

and a corresponding inner friction.

Low Temperature

Super-conductivity

At low temperature there is a unique

state of superconductivity

due to the coherent quantum

dynamics of the electrons with a

characteristic frictionless flow of the

electrons

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#4. How does a transistor work? (“To the electron -- may it never be of any use to anybody." JJ. Thomson's favorite toast)

What is a quasi-electron? Why has it “an effective mass”

What is a crystal ? What is a

semiconductor? The First Transistor (1947) The workbench of

John Bardeen and Walter Brattain at Bell Laboratories

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#5. What is a quantum computer?

Two-state bit can not compete a multi-state quantum bit

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.

.

.

n

Quantum bit Classical bit

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#6. How is designed the Universe?

The distribution of galaxies may help to confirm theories of quantum cosmology (NASA Picture )

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History

The electromagnetic radiation processed both a wave and a corpuscular

character. It’s energy is absorbed and emitted in separate portions – quanta-

photons

The photons momentum is determined by the vector

Where

1900 M. Plank 190 1905 A. Einstien

E2/h

Plank’s constant = 6.62x10-34Js

kp

ck

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1912 J. Frank and H. Hertz

The atomic energy states have a discrete character

(from the ionization potentials of gases)

1913 N. Bohr

The first successful attempt to explain the properties of the

hydrogen atom

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1924 L.de Broglie

A hypothesis of the wave properties of all particles of small mass

1926 E. Schrödinger

The Schrödinger's wave equation

vppk m ,

),,(),()(),(2

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tt

itVtm

rrrr

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1927 C. J. Davison and G.P. Thomson

The experimental discovery of the diffraction of electrons by crystals

1926-1930 W. Heisenberg, P. Dirac, and W. Pauli

The discovery of new productive forms of the quantum theory

Quantum Mechanics

is now the basis of many new branches of the modern science

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From: Y. B. Band and Y. Avishai, Quantum Mechanics with applications to nanotechnology and information science.

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