Homework Problems in Quantum Mechanics of a Harmonic Oscillator, Assignments of Quantum Physics

Download Homework Problems in Quantum Mechanics of a Harmonic Oscillator and more Assignments Quantum Physics in PDF only on Docsity! Physics 486 Homework 8 Due March 18, 2005 1) This is Shankar 7.3.4 (page 196) [5 points]. Use the harmonic oscillator wave functions, ψn(x) (see Shankar, p. 195) to show n x̂ m = ! 2m! " #$ % &' 1 2 n +1( ) 1 2 ( n,m+1 + n 1 2( n,m)1 * + , - n p̂ m = i m!! 2 " #$ % &' 1 2 n +1( ) 1 2 ( n,m+1 ) n 1 2( n,m)1 * + , - That is, p̂ and x̂ “connect” adjacent energy states. COMMENT: Why is this result important? Suppose we have solved a problem that involves the motion of a charged particle and know the ψn. Now, add an electric field. The potential gains an extra term, V(x) = -eEx, and the matrix that describes the modified Hamiltonian is no longer diagonal. So, to find the new ψn, we’ll have to rediagonalize the matrix. This will be one of the first problems we do next semester (P487). 2) This is similar to Shankar 7.3.5 (page 196) [5 points]. Use the result of problem 1to show that: a) <x> = 0 and <p> = 0 for every energy eigenstate. b) σxσp = /2 in the ground state (n = 0). This is an exercise in manipulating infinite dimensional matrices. Writing them out as * * ! * * " # ! " # # $ % & & isn’t practical, unless you have a lot of paper. (over)