Neutron in a magnetic field (Schrödinger Equation, Eigenstates, Eigenvalues).

Question

Consider the spin of a neutron in a magnetic field $\vec{B}$. A neutron is a neutral particle with the mass of a proton and the spin $\frac{1}{2}$. The Hamiltonian is

$H=\mu_n\vec{S}\cdot\vec{B}$ ,

whereas $\mu_n$ is the magnetic moment of the neutron. Consider a constant magnetic field along the z-axis. Thus the Hamiltonian is

$H=\omega S_z$ with $\omega=\mu_n B$.

a) What are the eigenvalues and eigenstates of the system?

b) At t=0 the system is in the state

$\left|\alpha(t=0)\right>=\frac{1}{\sqrt{2}}\left|+\right>+\frac{1}{\sqrt{2}}\left|-\right >$

$\left|+\right> =\left|j=\frac{1}{2},m=+\frac{1}{2}\right>$ and $\left|-\right>=\left|j=\frac{1}{2},m=-\frac{1}{2}\right>$.

Use the time-dependent Schrödinger Equation to get $\left|\alpha(t)\right>$.

I'm studying for an upcoming exam on QM and this is one of the exercises I'm doing dealing with right now.

a): I don't know how to start this. My general approach when I get the Hamiltonian in a problem is to write down the Schrödinger Equation.

$H\psi=E\psi$

In this case $\omega S_z\psi=E\psi$. Since $\omega S_z$ is not really an operator it means that $\omega S_z$ is already the eigenvalue? Not really sure about the eigenstates.

b) I don't know how one would get $\left|\alpha(t)\right>$ from using the Schrödinger Equation. Usually my approach to find a unitary operator and get it from $U\left|\alpha(t=0)\right>=\left|\alpha(t)\right>$ since $\left|\alpha(t=0)\right>$ is already given. Do I get the unitary operator by solving the corresponding Schrödinger Equation?

Sorry for my dumb rumblings, I'm not comfortable using the dirac notation with the Schrödinger Equation. I don't know what the problem is. I can solve the Schrödinger Equation when there's a typical problem involved and the wave function is a function and not something like $\left|\alpha(t=0)\right>=\frac{1}{\sqrt{2}}\left|+\right>+\frac{1}{\sqrt{2}}\left|-\right >$ . When I first solved the Schrödinger Equation for the potential well problem I thought I had the hang of it, but when I see this type of problem I just can't come up with any approach.

Answer 1

a) $S_z$ in this case is the spin operator which is equal to $S_z=\frac{\hbar}{2}\sigma_{z}$, where $\sigma_{z}$ is the z-Pauli matrix. The procedure you suggested should work now.

b) Once you have the eigenstates and eigenvalues of the Hamiltonian, you can write down an equation of motion for the eigenstates by solving the time dependent Schrodinger equation for them. Then, you can find a representation of the initial state given in the question in terms of these eigenstates and simply write down an equation of motion by substituting in your equations of motion for the eigenstates.