Physics 451 Quantum mechanics I Fall 2012 Oct 17, 2012 Karine Chesnel

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 Announcements Phys 451 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: is transformed into a scalar multiple of itself is an eigenvector of T is an eigenvalue of T

Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a N th polynomial in : characteristic equation Find the N roots Spectrum Pb A18, A25, A 26

Quantum mechanics Gram-Schmidt Orthogonalization procedure Discrete spectra Degenerate states More than one eigenstate for the same eigenvalue 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 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 : For real eigenvalue p : - Dirac orthonormality - Eigenfunctions are complete Wave length – momentum: de Broglie formulae

Quantum mechanics Continuous spectra of eigenvalues Position operator : - 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 If eigenvalues are real: - Dirac orthonormality - Eigenfunctions are complete but

Generalized statistical interpretation Operator’s eigenstates: eigenvectoreigenvalue Particle in a given state We measure an observable(Hermitian operator) Eigenvectors are complete: Discrete spectrumContinuous spectrum Phys 451

Generalized statistical interpretation Particle in a given state Normalization: Expectation value Operator’s eigenstates:orthonormal Phys 451

Quiz 18 A.the expectation value B.one of the eigenvalues of Q C.the average of all eigenvalues D. A combination of eigenvalues with their respective probabilities If you measure an observable Q on a particle in a certain state, what result will you get? Phys 451

Operator ‘position’ : Generalized statistical interpretation Probability of finding the particle at x=y: Phys 451

Operator ‘momentum’ : Generalized statistical interpretation Probability of measuring momentum p: Phys 451 Example Harmonic ocillator Pb 3.11

The Dirac notation Different notations to express the wave function: Projection on the energy eigenstates Projection on the position eigenstates Projection on the momentum eigenstates Phys 451