1. The problem statement, all variables and given/known data A box containing a particle is divided into a left and right compartment by a thin foil. The two orthonormal base kets |L> and |R> stand for the particle being in either the left or the right compartment, respectively. Hence, any state ket in our system can be decomposed as |psi> = |R> <R|psi> + |L> <L|psi> Tunneling between them, through the foil is characterised by a quantity , with units of energy, such that the corresponding Hamiltonian reads H = V [|L> <R| + |R> <L|] : (a) Write the Hamiltonian in matrix form. (b) Find the normalised energy eigenkets and the corresponding energy eigenvalues. (c) Find the state ket |psi(t)> = psiL(t)|L> + psiR(t)|R> in the Schrodinger picture, if Psi L, R(t = 0) = psi L, R are known. 2. Relevant equations Eigenvalue equations Av = yv Time evolution operator U(t,t0) = exp(iH(t-t0)E/hbar) 3. The attempt at a solution I don't have a problem with part a) or b). I simply constructed my 2x2 matrix for the hamiltonian by using scalar products. This gave a matrix that looked like: 0 V V 0 I then calculated the eigenvalues of the matrix and associated eigenvectors, which I then normalised for part b). I ended up with eigenvalues of +V and -V, and eigen vectors of 1/sqrt(2) (1 1) and 1/sqrt(2) (1 -1). Now for part C I know I need to use the time evolution operator, but it won't work very well for a matrix that isn't diagonal and H isn't in the original basis. So consequently I understand that I need to perform the time independent time evolution operator (I'm assuming the operator itself is time independent since we are in the Schrodinger picture) on the basis of eigen vectors just calculated which will yield a diagonal matrix of eigenvalues: V 0 0 -V this is about as far as I have got. I really don't quite understand how to represent the wavefunction in terms of this new basis of eigenvectors. I think the notation is a bit of a problem for me, I don't really know how to deal with the |psi> vs psi notation.