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

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.

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.