Quantum Mechanics Review and Test Preparation

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

2. Test 2 preparation Review: Friday Oct 19 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

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

4. Discrete spectrum of eigenvalues: Eigenfunctions in Hilbert space • Continuous spectrum of eigenvalues: Eigenfunctions NOT in Hilbert space Quantum mechanics Eigenvalues of a Hermitian operator Infinite space Two cases

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

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

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