# Probability transition without approximation

I am given that a system has two independent states (not eigen states of Hamiltonian) |+> and |->, represented as \begin{bmatrix}1 \\ 0 \\ \end{bmatrix} and \begin{bmatrix}0 \\ 1\end{bmatrix} respectively. In this basis, the Hamiltonian of this system (time-independent) is represented as \begin{pmatrix}E & U\\ U & E\end{pmatrix} both E and U are real. The question asks me to show that the transition from state |+> to state |-> in a time interval t is given by: p(t) = $\sin^2{(Ut/ ħ)}$.

My attempt: I first found the eigen values and eigen states of the Hamiltonian and then expressed the eigen states in the given bases. Then, I wrote the wavefunction at time t in terms of the time evolution operator and eigen states (later changed to the given basis states) and then substituted the wavefunction into time dependent schrodinger equation. I am just getting nowhere.

What is wrong in my attempt?

• I think you're close. Change basis so that the Hamiltonian is diagonal, then use the time evolution operator to evolve the + state expressed in the new basis. Project this vector along - and take the modulus squared to get from the amplitude to the probability. – Phoenix87 Mar 20 '18 at 22:53
• Since my eigen values were E-U and E+U, the diagonalized Hamiltonian would be \begin{pmatrix}E-U & 0 \\ 0 & E+U\end{pmatrix}. So are you suggesting me that I now use this Hamiltonian and act on the two states expressed in terms of my eigen states? – Ufomammut Mar 20 '18 at 22:59