8.7 Problems

Problem 8.1

In first-order time-dependent perturbation theory a particle initially in eigenstate n of the unperturbed Hamiltonian scatters into state m with probability a _m( t) ² after the perturbation is applied.

1. Show that if the perturbation is applied at time t = 0 then the time dependent coefficient a _m( t) is

   
   

   where the matrix element W _mn = mW( x, t) n and ? ? _mn = E _m ? E _n is the difference in eigenenergies of the states m and n .

2. A particle of mass m ₀ is initially in the ground state of a one dimensional harmonic oscillator. At time t = 0 a perturbation ( x, t) = V ₀ x ³ _e ^{?t / ?} is applied where V ₀ and ? are constants. Using the result in part (a), calculate the probability of transition to each excited state of the system in the long time limit, t ? ?.

Problem 8.2

An electron is in the ground state of a one-dimensional rectangular potential well for which V( x) = 0 in the range 0 < x < L and V( x) = ? elsewhere. It is decided to control the state of the electron by applying a pulse of...