Solve Quantum Mechanics 3-Level System with Time-Dependent Perturbation

[zheng89120]

Homework Statement

Consider a three level system, the time-dependent perturbation is:

W(t) = W cos(wt) e^(-t^2/T^2)

at time t=-infinity, it's in the ground state (w/ E₁)

the pertubation frequency is:

w = (w₂₁+w₃₁)/2

non-zero matrix elements are: <1|W|2>=<1|W|3>=<2|W|1>=<3|W|1>=$$\gamma$$

Show that the eigenstate is:

|$$\psi$$(t)> = e^-iw₁t [ ||$$\psi$$>
-iAe^-iw₂₁t |$$\psi$$>
-iAe^-iw₃₁t |$$\psi$$₃> ]

Homework Equations

fourier transform of a gaussian

The Attempt at a Solution

tried to take Fourier transform of cos wt * e^(-t^2/T^2), but couldn't get the form Ae^iwt |$$\psi$$_2/3>

 

[mark726]

Dear fellow scientist,

After carefully analyzing the problem at hand, I have come up with the following solution:

First, we need to use the time-dependent Schrodinger equation to find the time evolution of the three level system. This can be written as:

iħ d/dt |Ψ(t)> = H(t)|Ψ(t)>

where H(t) is the time-dependent Hamiltonian. Since we are dealing with a time-dependent perturbation, the Hamiltonian can be written as:

H(t) = H0 + W(t)

where H0 is the unperturbed Hamiltonian and W(t) is the perturbation.

Next, we can write the eigenstate |Ψ(t)> in terms of the unperturbed eigenstates |ψi> as:

|Ψ(t)> = Σi ci(t)|ψi>

where ci(t) is the time-dependent coefficient of the ith eigenstate.

Substituting this in the Schrodinger equation and using the fact that the unperturbed eigenstates satisfy the time-independent Schrodinger equation (H0|ψi> = Ei|ψi>), we get the following differential equation for ci(t):

iħ d/dt ci(t) = Σj Wji(t) ci(t)

where Wji(t) = <ψj|W(t)|ψi> is the matrix element of the perturbation between the ith and jth eigenstates.

Since the perturbation is time-dependent, we can write it in terms of its Fourier transform as:

W(t) = ∫ dω W(ω) e-iωt

where W(ω) is the Fourier transform of W(t). In our case, W(ω) is a Gaussian function centered at the perturbation frequency w:

W(ω) = W cos(ωt) e^(-t^2/T^2) = W cos(ωt) e^(-(ω-ω0)^2/T^2)

where ω0 = (w21+w31)/2 is the perturbation frequency.

Using this in the expression for Wji(t), we get:

Wji(t) = ∫ dω W(ω) <ψj|eiωt|ψi>

Since the unperturbed eigenstates are stationary states, we can write:

<ψj|eiωt|ψi> = ei