Physics 451
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Transcript of Physics 451
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Physics 451Quantum mechanics I
Fall 2012
Oct 17, 2012
Karine Chesnel
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Next homework assignments:
•HW # 14 due Thursday Oct 18 by 7pm
Pb 3.7, 3.9, 3.10, 3.11, A26
• HW #15 due Tuesday Oct 23
AnnouncementsPhys 451
Practice test 2 M Oct 22 Sign for a problem!
Test 2 : Tu Oct 23 – Fri Oct 26
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Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
T a a
is transformed into a scalar multiple of itselfa
a is an eigenvector of T
is an eigenvalue of T
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Quantum mechanics
Eigenvectors & eigenvalues
0T I a
det 0T I
To find the eigenvalues:
We get a Nth polynomial in : characteristic equation
Find the N roots 1 2, ,... N
Spectrum Pb A18, A25, A 26
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Quantum mechanics
Gram-Schmidt Orthogonalization procedure
Discrete spectraDegenerate states
More than one eigenstate for the same eigenvalue
See problem A4, application A26
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Quantum mechanics
Discrete spectra of eigenvalues
1. Theorem: the eigenvalues are real
2. Theorem: the eigenfunctions of distinct eigenvalues are orthogonal
3. Axiom: the eigenvectors of a Hermitian operator are complete
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Quantum mechanics
Continuous spectra of eigenvalues
Q̂f x f x
No proof of theorem 1 and 2… but intuition for:
- Eigenvalues being real- Orthogonality between eigenstates- Compliteness of the eigenstates
Orthogonalization Pb 3.7
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Quantum mechanics
Continuous spectra of eigenvalues
p pd f x pf x
i dx
Momentum operator:
For real eigenvalue p: - Dirac orthonormality
- Eigenfunctions are complete
Wave length – momentum: de Broglie formulae2p
' ( ')p pf f p p
pf x c p f x dp
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Quantum mechanics
Continuous spectra of eigenvalues
Position operator:
xf x f x
- Eigenvalue must be real
- Dirac orthonormality
- Eigenfunctions are complete
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Quantum mechanics
Continuous spectra of eigenvalues
Eigenfunctions are not normalizableDo NOT belong to Hilbert spaceDo not represent physical states
If eigenvalues are real:- Dirac orthonormality- Eigenfunctions are complete
but
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Generalized statistical interpretation
• Operator’s eigenstates: n n nQ q
eigenvectoreigenvalue
• Particle in a given state
• We measure an observable Q (Hermitian operator)
Eigenvectors are complete:
Discrete spectrum
1n n
n
c
Continuous spectrum
( ) ( )qc q x dq
Phys 451
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Generalized statistical interpretation
Particle in a given state
• Normalization:
1n n
n
c
2
1n
n
c
• Expectation value
2
1n n
n
Q Q c q
Operator’s eigenstates: n n nQ q orthonormal
Phys 451
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Quiz 18
A. the expectation value
B. one of the eigenvalues of Q
C. the average of all eigenvalues
D. A combination of eigenvalues
with their respective probabilities
If you measure an observable Q on a particle in a certain state ,
what result will you get?
Q
2
1n n
n
c q
1n
n
q
n nn
c
Phys 451
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Operator ‘position’:
ˆy yxf x yf x
Generalized statistical interpretation
( ) ( ) ( , ) ( , )c y x y x t dx y t
Probability of finding the particle at x=y:2 2( ) ( , )c y y t
Phys 451
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p pd f x pf x
i dx
Operator ‘momentum’:
Generalized statistical interpretation
/1( ) ( , ) ,2
ipxc p e x t dx p t
Probability of measuring momentum p:2 2( ) ( , )c p p t
px c p f x dp
Phys 451
Example Harmonic ocillator Pb 3.11
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The Dirac notation
Different notations to express the wave function:
• Projection on the energy eigenstates
• Projection on the position eigenstates
• Projection on the momentum eigenstates
, ,x t y t x y dy
/
,2
ipxep t dp
/( ) niE tn n
n
c x e
Phys 451