hw04 - 2 2 2 2 mL n E n =(a Find C for all n(5 pts(b If at...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Quantum and Statistical Mechanics Homework 4 Prob. 4.1 For a 2 × 2 Hermitian matrix d c b a , (a) Prove that a and d are real, and b = c* (5 pts) (b) Find the eigenvalues and prove that they are real. (5 pts) (c) Find the corresponding eigenvectors and prove that they are orthogonal. (10 pts) Prob. 4.2 A free particle with the dispersion relation as ( 29 m k k 2 2 F = ϖ is described by the wave function ( 29 ( 29 ( 29 ( 29 ( 29 t x k i B t x k i A t x 0 0 0 0 exp exp , ψ - - + - = (a) Show that (x,t) is an eigenfunction of H ˆ , but not p ˆ . (5 pts) (b) Is (x,t) a stationary state? (5 pts) (c) Find p ˆ . Does p ˆ have a time dependence (is the expectation value of the momentum conserved)? Why or why not? (10 pts) Prob. 4.3 Consider a particle of mass m confined by an infinite potential square box, i.e., ( 29 < < = elsewhere L x x V 0 0 . The eigenfunctions are known as = L x n C n π φ sin , n = 1, 2, … with the corresponding eigenenergy as 2
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 2 2 2 mL n E n = (a) Find C for all n . (5 pts) (b) If at t = 0, ( 29 L x elsewhere L x i L x A x < < + = 3 sin 2 sin , , find A . (5 pts) (c) Find x ˆ at t = 0. (5 pts) (d) Find p ˆ at t = 0. (5 pts) (e) Derive the expression for (x,t) . (5 pts) (f) Find ) ( ˆ t E for t > 0. (5 pts) Prob. 4.4. Following the description of Prob. 4.3 of the infinite-potential well, if at t = 0, the particle is only in the ground and first excited states with equal probability, (a) Find (x,0) with proper normalization. (5 pts) (b) Find P(x,t) and sketch P(x,t) against x for a few t ’s to show the motion. (5 pts) (c) Find x ˆ and p ˆ at t = 0. (10 pts) (d) Find x dt d (5 pts) (e) Find ) ( ˆ t E for t > 0. (5 pts) 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online