1. The problem statement, all variables and given/known data A single spin-one-half system has Hamiltonian [tex]H=\alpha*s_x+\beta*s_y[/tex], where [tex]\alpha[/tex] and [tex]\beta[/tex] are real numbers, and [tex]s_x[/tex] and [tex]s_y[/tex] are the x and y components of spin . a) Using the representation of the spin components as Pauli spin matrices, find an expression for [tex]H^2[/tex] in termms of the above parameters. b) used the result from part(a) to find the energy eigenvalues. c) Find the eigenvectors of H in equation [tex]H=\alpha*s_x+\beta*s_y[/tex] in the Pauli spin matrix representation. d) Supposed that a t time t=0 the system is an eigenstate of [tex]s_z[/tex], with eigen value [tex]+\h-bar/2[/tex]. Find the state vector as a function of time in the Pauli spin matrix representation. e) Suppose the z-component of the spin in the state found in part d) is measured at time t>0 . Find probability that the result is [tex]+\hbar/2[/tex] 2. Relevant equations [tex] s=(\hbar)*(\sigma)/2[/tex] [tex](\sigma_x)[/tex],[tex](\sigma_y)[/tex], and [tex](\sigma_z)[/tex] 3. The attempt at a solution a) Just multiply H twice right? but just need to insert matrix of x-component and y component for spin x and spin y b) No idea what the energy eigenvalue is; Wouldn't it be H ? could they mean : U=exp(-i*H*t/(h-bar))? c)Do they want me to just write the equation H out explicitly, i.e. with the matrix components of x and y ? d) No idea what the state vector is; is it [tex] \phi=\varphi_x+[/tex]? is [tex]\varphi_x+= \hbar/2[/tex]? e) I probably need to square the state vector which would be [tex](\hbar^2)/4[/tex] if my state vector in d is correct. What do you think of my approach?