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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!.
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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
For a given transformation T, there are “special” vectors for which: is transformed into a scalar multiple of itself is an eigenvector of T l is an eigenvalue of T Quantum mechanics Eigenvectors & eigenvalues
Find the N roots Spectrum Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a Nth polynomial in l: characteristic equation 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 • No proof of theorem 1 and 2… but intuition for: • Eigenvalues being real • Orthogonality between eigenstates • Compliteness of the eigenstates Orthogonalization Pb 3.7
For real eigenvalue p: • Dirac orthonormality • Eigenfunctions are complete Wave length – momentum: de Broglie formulae Quantum mechanics Continuous spectra of eigenvalues Momentum operator:
Quantum mechanics Continuous spectra of eigenvalues Position operator: • - Eigenvalue must be real • Dirac orthonormality • Eigenfunctions are complete
but • If eigenvalues are 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
We measure an observable (Hermitian operator) • Operator’s eigenstates: Discrete spectrum Continuous spectrum eigenvector eigenvalue Phys 451 Generalized statistical interpretation • Particle in a given state Eigenvectors are complete:
Operator’s eigenstates: orthonormal Phys 451 Generalized statistical interpretation Particle in a given state • Normalization: • Expectation value
Phys 451 Quiz 18 If you measure an observable Q on a particle in a certain state , what result will you get? • the expectation value • one of the eigenvalues of Q • the average of all eigenvalues • A combination of eigenvalues • with their respective probabilities
Phys 451 Generalized statistical interpretation Operator ‘position’: Probability of finding the particle at x=y:
Phys 451 Generalized statistical interpretation Operator ‘momentum’: Probability of measuring momentum p: Example Harmonic ocillator Pb 3.11
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