What Determines Spin State Energies in a Magnetic Field?

[ehrenfest]

[SOLVED] energies of spin states

Homework Statement

An electron is in a constant magnetic field with magnitude B_0, aligned in the -z direction. My book says without explanation:

"the spin-up state has energy $-\mu_B B_0$"

where $\mu_B$ is the Bohr magneton

I looked back in the Spin Angular Momentum chapter and I cannot find where this was derived.

I am thinking that they used the equation [tex] H = B_0 \mu_B \sigma_z [/itex] and just calculated the energy of the spin up state using the TISE and the vector representation of spin-up in z. Is there an a priori way of knowing the energy of a spin state of it is spin up or down in x, y,or z in a magnetic field?

EDIT: OK. So I think that if you actually calculate the energies for the spins using H |\psi> = E | \psi>, you get that the energy is always $-\mu_B B_0$ when the spin is antiparallel to the magnetic field vector and $\mu_B B_0$ when the spin is parallel to the magnetic field vector. I explicitly checked that this holds for x, y, and z. Could I have obtained this result without calculating with Pauli matrices, though? Does this result hold when the spin and magnetic field are parallel but not along x,y,z,-x,-y,-z?

Homework Equations

The Attempt at a Solution

 

[malawi_glenn]

the Hamiltonian for a (spin)particle in magnetic field:

$$H = - (q/mc) \vec{S} \cdot \vec{B}$$ (sakurai eq 2.1.49)
(q = -e for the electron)

the magnetic moment for an electron is of course: $$\mu_B = e\hbar /2mc$$

Now you can simply relate the spin matrices to the hamiltonian and see what energy eigenvalues different states have.

for your B-field: $$\vec{B} = -B_0 \hat{z}$$
You will get this hamiltonian:
$$H = - (e/2mc) \sigma _z B_0$$ and the pauli matrix property:
$$\sigma _z \chi _+ = \hbar \chi _+$$
So the energy for this particle (spin in +z and magnetic field in -z) is $-\mu_B B_0$

 

[ehrenfest]

So, I guess the answers to my questions in the EDIT are no and yes.

 

[ehrenfest]

Right?

 

[malawi_glenn]

You must use the spin- (pauli) matrices for this, the form of the hamiltonian follows from basic electromagnetism.

For your second "question" , i don't know what you ask for, but it is very easy to evaluate the energy eigen values for a specific state in a certain magnetic field.