How Does an External Electric Force Affect Quantum Harmonic Oscillator States?

[mark18]

Homework Statement

H = p^2/2m + (kx^2)/2 - qAx (THis is a harmonic potentional with external electric force in 1D)

Braket:
Definitions:
|0, A=0 > = |0>_0 for t=0 (ground state)
|0, A not 0 > = |0> for t=0 (ground state)

2. Question
1. Find the probability of being in the state |0, A not 0> for t>=0 when you are in the state |0>_0
2. Find the coefficients for the station |0>_0 = Sigma_k c_k |k>

The Attempt at a Solution

1.
P = |<0, A not 0|e^(-iE_0t)|0, A=0 >|^2 = <0, A not 0||0, A=0 >^2 = ? (how to find this, do i go over to position representation?)
i've tried that but got an answer which didn't depend of t? since t factor is being canceled out by the complex conjugation.

2. c_k = < k | | 0 >_0 = ?

 

[Jhero]

(how to find this, do i go over to position representation?)

1. To find the probability, we need to use the time evolution operator, which is given by e^(-iHt), where H is the Hamiltonian. In this case, the Hamiltonian is the harmonic potential with an external electric force. So, we have:

P = |<0, A not 0|e^(-iHt)|0, A=0>|^2

To find the matrix element <0, A not 0|e^(-iHt)|0, A=0>, we need to express the states in the position representation. So, we have:

|0, A not 0> = <x|0, A not 0> and |0, A=0> = <x|0, A=0>

Substituting this in the expression for P, we get:

P = |<x|0, A not 0>e^(-iHt)|x><x|0, A=0>|^2

Since we know that <x|0, A=0> is the ground state wavefunction, we can write it as <x|0, A=0> = ψ_0(x). Similarly, we can write <x|0, A not 0> = ψ_A(x), where ψ_A(x) is the wavefunction for the state |0, A not 0>.

Substituting these in the expression for P, we get:

P = |ψ_A(x)e^(-iHt)ψ_0(x)|^2

Now, we can use the time evolution operator to find the new wavefunction ψ_A(x) at time t. This can be done by solving the time-dependent Schrodinger equation:

iħ∂ψ_A(x)/∂t = Hψ_A(x)

Solving this equation, we get:

ψ_A(x, t) = ψ_A(x)e^(-iHt/ħ)

Substituting this in the expression for P, we get:

P = |ψ_A(x)e^(-iHt)ψ_0(x)|^2 = |ψ_A(x, t)ψ_0(x)|^2

Therefore, the probability of being in the state |0, A not 0> at