Transition probability from two states

[CAF123]

Homework Statement

A system has two independent states $|1\rangle$ and $|2\rangle$ represented by column matrices $|1\rangle \rightarrow (1,0)$ and $|2\rangle \rightarrow (0.1)$. With respect to these two states, the Hamiltonian has a time independent matrix representation $$\begin{pmatrix} E&U\\U&E \end{pmatrix},$$ E and U both real. Show that the probability of a transition from state $|1\rangle$ to state $|2\rangle$ in a time interval $t$ is given by (without any approximation) $p(t) = \sin^2(Ut/\hbar)$

Homework Equations

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Time dependent Schrodinger equation

The Attempt at a Solution

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Reexpress the states in terms of energy eigenstates, so can write the general evolution of an arbritary state. The eigenvectors of the Hamiltonian are $\frac{1}{\sqrt{2}}(1,1) = |u_1\rangle$ and $\frac{1}{\sqrt{2}}(1,-1) = |u_2\rangle$. Then $|1\rangle = \frac{1}{\sqrt{2}}(|u_1\rangle + |u_2 \rangle )$ while $|2\rangle = \frac{1}{\sqrt{2}}(|u_1\rangle - |u_2 \rangle ).$ So generic state is $$|\Psi,t_o \rangle = C_1 (1,0) + C_2 (0,1) \Rightarrow |\Psi, t\rangle = \frac{C_1}{\sqrt{2}} (|u_1\rangle e^{-iE_1 t/\hbar} + |u_2 \rangle e^{-iE_2 t/\hbar} ) + \frac{C_2}{\sqrt{2}} (|u_1\rangle e^{-iE_1 t/\hbar} - |u_2 \rangle e^{-iE_2 t/\hbar}).$$ But I am not sure how to progress. I am looking to compute $\langle \Psi, t | 1 \rangle$ and from that extract the probability of finding state |2>.
Thanks!

 

[assed]

You don't need to write a generic state. The problem asks about a transition from 1 to 2.
All you need to do is write (as you have already done) 1 and 2 in the eigenvector basis. Then evolve 1 in time t and project it into 2, to find the probability of finding 2.

 

[CAF123]

Hi assed,

You don't need to write a generic state. The problem asks about a transition from 1 to 2.
All you need to do is write (as you have already done) 1 and 2 in the eigenvector basis. Then evolve 1 in time t and project it into 2, to find the probability of finding 2.

Thanks, I see. I noticed that the result obtained from the first order time dependent theory gives exactly the same result. So the first order correction to the transition probability induced by the given time independent hamiltonian is exact. Is there any reason why we would expect this or does this imply that the approximation is true regardless of size of U? Thanks!

 

[CAF123]

Woops, ignore my last reply, the first order term coming from perturbation theory gives the first term in the expansion of sin^2 Ut/h, which is more sensible. So the approximation is valid for a time interval $t$ such that $P^{(1)}(t) = P_{1 \rightarrow 2} + P_{1 \rightarrow 1} = 1$ holds. Is that correct?