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

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

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

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

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

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

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

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

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

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