Question

A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic oscillator potential with frequency oo. At time t-0 a weak spatially uniform electric field (E) is turned on, so that the perturbation to the Hamiltonian can be described as R'(t) =-q Exe-t/t for t> 0. Using first order, time-dependent perturbation theory, calculate the following probabilities: (a) the particle is detected in the first excited state after a very long time (t » (b) the particle is detected in the second excited state after a very long time (t » τ) (c) the particle is detected in the first excited state after a very short time (t « τ, dot « 1). Keep only the first non-vanishing term in t. Potentially useful equations: Harmonic oscillator wave functions for the ground and the first two excited states: ψ0(x),' n 片 ψ1(x)-li( )ixe 뽑; ψ2(x) =:/쁩(zm w a!. 1) eー for n even for n odd 0 For any two harmonic oscillator states n'andn: 佒n,le№》=1/2mao (Vn+Ton,n+1+VnOn,n-1)

Answer 1

nsition p itrsh state l 长 Ladder Sperator s for t,,

Here, If lilo 2) aナa 는H 2°-。て ·て hi n. てレ十1 て for wt