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Physics 451 Quantum mechanics I Fall 2012 Oct 17, 2012 Karine Chesnel Phys 451 Announcements 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 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: T a a a is transformed into a scalar multiple of itself a is an eigenvector of T is an eigenvalue of T Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: T I a 0 det T I 0 We get a Nth polynomial in : characteristic equation Find the N roots 1, 2 ,...N Spectrum Pb A18, A25, A 26 Quantum mechanics Discrete spectra Degenerate states More than one eigenstate for the same eigenvalue Gram-Schmidt Orthogonalization procedure 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 Q̂f x f x 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: 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 Quantum mechanics Continuous spectra of eigenvalues Eigenfunctions are not normalizable Do NOT belong to Hilbert space Do not represent physical states but If eigenvalues are real: - Dirac orthonormality - Eigenfunctions are complete Phys 451 Generalized statistical interpretation • Particle in a given state • We measure an observable • Operator’s eigenstates: Q (Hermitian operator) Q n qn n eigenvalue eigenvector Eigenvectors are complete: Discrete spectrum cn n n 1 Continuous spectrum c(q) q ( x)dq Phys 451 Generalized statistical interpretation Particle in a given state cn n n 1 Operator’s eigenstates: Q n qn n • Normalization: cn 2 n 1 • Expectation value 2 Q Q cn qn n 1 orthonormal Phys 451 Quiz 18 If you measure an observable Q on a particle in a certain state cn n , n what result will you get? A. the expectation value Q B. one of the eigenvalues of Q C. the average of all eigenvalues D. A combination of eigenvalues with their respective probabilities q n 1 n c n 1 n 2 qn Phys 451 Generalized statistical interpretation Operator ‘position’: ˆ y x yf y x xf c( y ) ( x y) ( x, t )dx ( y, t ) c( y ) ( y, t ) 2 Probability of finding the particle at x=y: 2 Phys 451 Generalized statistical interpretation Operator ‘momentum’: d f p x pf p x i dx x c p f x dp p 1 c( p) 2 ipx / e ( x, t )dx p, t c ( p ) ( p, t ) 2 Probability of measuring momentum p: Example Harmonic ocillator Pb 3.11 2 Phys 451 The Dirac notation Different notations to express the wave function: • Projection on the position eigenstates • Projection on the momentum eigenstates • Projection on the energy eigenstates x, t y, t x y dy eipx / p, t dp 2 cn n ( x)eiEnt / n