How Does Time-Dependent Perturbation Theory Explain Quantum State Evolution?

[kakarukeys]

Given a system,
H = H_0 + V
V is a small perturbation that does not depend on time.

the system is in |E_0> at time t_0
H_0 |E_n> = E_n |E_n>
H_0 |E_0> = E_0 |E_0>

Let |\Psi(t)> be the solution of the system.
Let |\Phi(t)> be the solution of the system without perturbation.
Let |u(t)> = |\Psi(t)> - |\Phi(t)>.

Show that |<E_n|u(t)>|^2 = 4 |V_{n0}|^2 [{{\sin(t - t_0)(E_n - E_0)/\hbar}\over{E_n - E_0}}]^2

at lowest order
No matter how many times I try, the answer I get is

|<E_n|u(t)>|^2 = 2 |V_{n0}|^2 [{{1 - \cos(t - t_0)(E_n - E_0)/\hbar}\over{E_n - E_0}}]^2

Please help!

 

[quantumworld]

I think you got the right answer, but u don't have the correct question, it should be show that ...sin(.../2), there is a divide by 2 missing in the sine function. I looked it up in my quantum book.

 

[kakarukeys]

Sorry,
I typed wrongly,

My answer was
|<E_n|u(t)>|^2 = 2 |V_{n0}|^2 [{{1 - \cos(t - t_0)(E_n - E_0)/\hbar}\over{(E_n - E_0)^2}}]

Ya, you are brilliantly right,

Since,
2\sin^2\theta/2 = 1 - \cos\theta

There should be a "divided by 2" inside the Sine

 

[koshyninan]

How can i get certain solved papers in quantum mechanics?
thanks