The problem consists of 2 parts,the first one(I have done it) is on the following website:
Q1: I calculated the desired result p(t) = sin^2(Ut/h). However,I don't understand why <1,t | 2 > will give the coefficient corresponds to the transition probability from state 1 to state 2 in Time t. My initial guess is similar to that of OP in the above post, but take the scalar product of the general state with |2> .
In the second part,I was asked to compute the transition probability using 1st-order time-dependent perturbation theory.Using the result from my notes:
where P_mk means the transition probability from energy eigenstate k to m at time t.
H'_mk = < m(0) | H' | k(0) > , where H' denotes the perturbation.
w_mk = w_m - w_k = ( E_m - E_k )/ h
Q2: I was a bit confused when I calculate H'_mk. Using the definition of scalar product in matrix
representation, I have H'_21 = < u_2 | H' | u_1> , where |u_i> is defined as in the above post.
However,(1 -1)(E U U E) (1 1)(forgive me for the matrix notation...) is 0 since |u_1> and |u_2>
are orthogonal. So the P_21 (t) vanishes which is not the thing I want.
Interestingly, if instead, I take the state m and k to be |2> and |1> respectively, I would have
(0 1)(E U U E) (1 0) = U ,and following the formula I found that P_21 (t) = sin^2(Ut/h)
However,(0 1) and (1 0) are NOT energy eigenstates,but H'_mk works with energy eigenstate.Why would they produce the desired result?
2. Homework Equations
Note that I have calculated that E_1 = E+U and E_2 = E-U where E_1 is the (energy) eigenvalue of |u_1> and etc.
For a clearer version of the original question:
The Attempt at a Solution
Incorporated in question