- #1
CAF123
Gold Member
- 2,948
- 88
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
[/B]
Time dependent Schrodinger equation
The Attempt at a Solution
[/B]
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!