Physics 451
Quantum mechanics I
Fall 2012
Oct 15, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW # 13 due Tuesday Oct 16
Pb 3.3, 3.5, A18, A19, A23, A25
• HW #14 due Thursday Oct 18
Pb 3.7, 3.9, 3.10, 3.11, A26
Test 2 preparation
Review: Friday Oct 19
Practice test: Monday Oct 22
Quantum mechanics
Eigenvalues of an Hermitian operator
Finite space
Q q
Generalization of
Determinate state:
operator
Hermitian operator:
eigenvalue
eigenstate
Q†  Q
1. The eigenvalues are real
2. The eigenvectors corresponding to distinct eigenvalues are orthogonal
3. The eigenvectors span the space
Quantum mechanics
Eigenvalues of a Hermitian operator
Infinite space
Two cases
• Discrete spectrum of eigenvalues:
Eigenfunctions in Hilbert space
• Continuous spectrum of eigenvalues:
Eigenfunctions NOT in Hilbert space
Quantum mechanics
Quiz 17
In which categories fall the following potentials?
1. Harmonic oscillator
2. Free particle
A. Discrete spectrum
B. Continuous spectrum
3. Infinite square well
4. Finite square well
C. Could have both
Quantum mechanics
Discrete spectra of eigenvalues
Theorem 1: the eigenvalues are real
Theorem 2: the eigenfunctions of distinct
eigenvalues are orthogonal
Axiom 3: the eigenvectors of a Hermitian operator
are complete
Quantum mechanics
Degenerate states
More than one eigenstate for the same eigenvalue
Gram-Schmidt
Orthogonalization procedure
See problem A4
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
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