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

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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
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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...
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