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Physics 451
Quantum mechanics I
Fall 2012
Oct 17, 2012
Karine Chesnel
Phys 451
Announcements
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
Practice test 2 M Oct 22
Sign for a problem!
Test 2 : Tu Oct 23 – Fri Oct 26
Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
T a  a
a
is transformed into a scalar multiple of itself
a
is an eigenvector of T
 is an eigenvalue of T
Quantum mechanics
Eigenvectors & eigenvalues
To find the eigenvalues:
T   I  a
 0
det T   I   0
We get a Nth polynomial in : characteristic equation
Find the N roots
 1, 2 ,...N 
Spectrum
Pb A18, A25, A 26
Quantum mechanics
Discrete spectra
Degenerate states
More than one eigenstate for the same eigenvalue
Gram-Schmidt
Orthogonalization procedure
See problem A4, application A26
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
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
Quantum mechanics
Continuous spectra of eigenvalues
Momentum operator:
d
f p  x   pf p  x 
i dx
For real eigenvalue p:
- Dirac orthonormality
f p f p '   ( p  p ')
- Eigenfunctions are complete
f  x 

 c  p  f  x  dp
p

Wave length – momentum: de Broglie formulae
2

p
Quantum mechanics
Continuous spectra of eigenvalues
Position operator:
xf  x    f  x 
- Eigenvalue must be real
- Dirac orthonormality
- Eigenfunctions are complete
Quantum mechanics
Continuous spectra of eigenvalues
Eigenfunctions are not normalizable
Do NOT belong to Hilbert space
Do not represent physical states
but
If eigenvalues are real:
- Dirac orthonormality
- Eigenfunctions are complete
Phys 451
Generalized statistical interpretation

• Particle in a given state
• We measure an observable
• Operator’s eigenstates:
Q
(Hermitian operator)
Q  n  qn  n
eigenvalue
eigenvector
Eigenvectors are complete:
Discrete spectrum

   cn n
n 1
Continuous spectrum
 

 c(q)

q
( x)dq
Phys 451
Generalized statistical interpretation

Particle in a given state
   cn n
n 1
Operator’s eigenstates:
Q  n  qn  n

• Normalization:
    cn
2
n 1

• Expectation value
2
Q   Q   cn qn
n 1
orthonormal
Phys 451
Quiz 18
If you measure an observable Q
on a particle in a certain state    cn  n ,
n
what result will you get?
A. the expectation value
Q
B. one of the eigenvalues of Q
C. the average of all eigenvalues
D. A combination of eigenvalues
with their respective probabilities

q
n 1
n

c
n 1
n
2
qn
Phys 451
Generalized statistical interpretation
Operator ‘position’:
ˆ y  x   yf y  x 
xf

c( y ) 
  ( x  y) ( x, t )dx  ( y, t )

c( y )   ( y, t )
2
Probability of finding the particle at x=y:
2
Phys 451
Generalized statistical interpretation
Operator ‘momentum’:
d
f p  x   pf p  x 
i dx
  x 

 c  p  f  x  dp
p

1
c( p) 
2

 ipx /
e
 ( x, t )dx    p, t 


c ( p )   ( p, t )
2
Probability of measuring momentum p:
Example Harmonic ocillator Pb 3.11
2
Phys 451
The Dirac notation
Different notations to express the wave function:
• Projection on the position eigenstates
• Projection on the momentum eigenstates
• Projection on the energy eigenstates
  x, t     y, t    x  y  dy
eipx /
    p, t 
dp
2
  cn n ( x)eiEnt /
n
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