Eigenvalues Of An Hermitian Operator-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

Download Eigenvalues Of An Hermitian Operator-Quantum Physics and Mechanics-Lecture Slides and more Slides Quantum Mechanics in PDF only on Docsity! Announcements Phys 451 Review 2 on Monday Oct 24 Practice test 2: Mo Oct 24 – Wed Oct 26 No Class on Friday Oct 21 docsity.com Quantum mechanics Eigenvalues of an Hermitian operator Finite space Generalization of Determinate state: Q q  operator eigenstate eigenvalue Hermitian operator: †Q Q 1. The eigenvalues are real 2. The eigenvectors corresponding to distinct eigenvalues are orthogonal 3. The eigenvectors span the space docsity.com 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 docsity.com Quantum mechanics Gram-Schmidt Orthogonalization procedure Discrete spectra Degenerate states More than one eigenstate for the same eigenvalue See problem A4 docsity.com 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 docsity.com 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 docsity.com Quantum mechanics Generalized statistical interpretation • Operator’s eigenstates: n n nQ q  eigenvector eigenvalue • Particle in a given state  • We measure an observable Q (Hermitian operator) Eigenvectors are complete: Discrete spectrum 1 n n n c     Continuous spectrum ( ) ( )qc q x dq      docsity.com Quantum mechanics Quiz 17 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 Y, what will you get? Q 2 1 n n n c q    1 n n q    docsity.com Quantum mechanics    p p d f x pf x i dx  Operator ‘momentum’: Generalized statistical interpretation  / 1 ( ) ( , ) , 2 ipxc p e x t dx p t      Y  Probability of measuring momentum p: 2 2 ( ) ( , )c p p t       px c p f x dp     docsity.com Quantum mechanics Different notations to express the wave function: • Projection onto the energy eigenstates • Projection onto the position eigenstates • Projection onto the momentum eigenstates Generalized statistical interpretation docsity.com Quantum mechanics The uncertainty principle   2 2 2 , 2 A B A B i            Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables docsity.com