1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Oct 15, 2012
Karine Chesnel

2. Practice test: Monday Oct 22
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
Review: Friday Oct 19
Practice test: Monday Oct 22
Test 2 preparation

3. Eigenvalues of an Hermitian operator
Quantum mechanics
Eigenvalues of an Hermitian operator
Finite space
Generalization of
Determinate state:
operator
eigenstate
eigenvalue
Hermitian operator:
1. The eigenvalues are real
2. The eigenvectors corresponding to distinct eigenvalues are orthogonal
3. The eigenvectors span the space

4. Eigenvalues of a Hermitian operator
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

5. In which categories fall the following potentials?
Quantum mechanics
Quiz 17
In which categories fall the following potentials?
1. Harmonic oscillator
Discrete spectrum
Continuous spectrum
Could have both
2. Free particle
3. Infinite square well
4. Finite square well

6. Discrete spectra of eigenvalues
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

7. Orthogonalization procedure
Quantum mechanics
Degenerate states
More than one eigenstate for the same eigenvalue
Gram-Schmidt
Orthogonalization procedure
See problem A4

8. Continuous spectra of eigenvalues
Quantum mechanics
Continuous spectra of eigenvalues
No proof of theorem 1 and 2… but intuition for:
Eigenvalues being real
Orthogonality between eigenstates
Compliteness of the eigenstates

9. Continuous spectra of eigenvalues
Quantum mechanics
Continuous spectra of eigenvalues
Momentum operator:
For real eigenvalue p:
Dirac orthonormality
Eigenfunctions are complete
Wave length – momentum: de Broglie formulae

10. Continuous spectra of eigenvalues
Quantum mechanics
Continuous spectra of eigenvalues
Position operator:
- Eigenvalue must be real
Dirac orthonormality
Eigenfunctions are complete