Quantum Mechanics, time independant solution in Dirac notati

[Ben Whelan]

Homework Statement

Consider the Hamiltonian:
$$\hat{H}=C*(\vec{B} \cdot \vec{S})$$
where $C$ is a constant and the magnetic field is given by

$$\vec{B} = (0,B,0) $$

and the spin is

$$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$

with$$\hat{S}_{x} =\frac{\hbar}{2}∣↑⟩⟨↓∣+ \frac{\hbar}{2}∣↓⟩⟨↑∣ $$$$\hat{S}_{y} =−\frac{i\hbar}{2}∣↑⟩⟨↓∣+ \frac{i\hbar}{2}∣↓⟩⟨↑∣ $$$$\hat{S}_{z} =\frac{\hbar}{2}∣↑⟩⟨↑∣− \frac{\hbar}{2}∣↓⟩⟨↓∣,$$

where the basis vectors are assumed orthogonal and normalised. Knowing that at t=0

$$ ∣ψ(0)⟩= \frac{1}{2}∣ψ_1⟩+ \frac{\sqrt{3}}{2}∣ψ_2⟩$$

where the ${∣ψ_i⟩}$ are the eigenvectors of the Hamiltonian, solve the time-dependent Schrödinger equation. Write the solution in terms of the basis vectors $∣↑⟩$ and $∣↓⟩$.

Homework Equations

The Attempt at a Solution

From the question I understand that the Hamiltonian will cancel out the x and z spin operators through the dot product. I have tried following this forward and using a presumed general solution to the time dependent Schrödinger equation but I got no where. Could someone just explain to me how to look at attempting this question?

 

[DrClaude]

I would start by calculating the eigenstates of $\hat{H}$.