Particle in magnetic field, classical and quantum view

[Nemanja989]

Hi, I am trying to understand quantum description of electron in B field. What I am looking for is how to relate quantum and classical view?

If in classical treatment electron comes in homogeneous B field, it will go in circular motion around magnetic field lines, with radius R.

From quantum view, we get LHO and quantization of energy. From equation for energy we see that only in z direction (direction along B field) we have dispersion relation for a free electron. That is fine with classical result.

What I cannot see from quantum description, are there orbits around magnetic field lines. Did anyone thought about this, and is there some clear picture about this view?

If electrons are orbiting around B lines, I suppose that radius is quantized? Am I right or wrong?

 

[mfb]

If electrons are orbiting around B lines, I suppose that radius is quantized? Am I right or wrong?

For energy eigenstates, the expectation value of the radius is quantized, indeed. Those eigenstates are "orbits" around field lines.

 

[Nemanja989]

Thank you for your reply. But I would like to ask you from where you see expression for the radius? I was looking for that, but probably I cannot interpret equations in right way :(

 

[mfb]

The expectation value should be the same as the classical radius for the same energy, maybe with a small correction term. You would have to solve the modified Schroedinger equation (with the B-field in the momentum term) and evaluate this to get an exact answer. Alternatively, look for books where this has been done before.

 

[Nemanja989]

Ok, thanks.. On our QM course professor did not give us any physical picture while lecturing quantum LHO (and e in B), and therefore I was a little bit confused. I will to all the calculations.

Thanks once again :)

 

[Bill_K]

The Hamiltonian for a particle in a B field is H = (1/2m) p² where p is the canonical momentum, p = P - eA/c.

Unlike the mechanical momentum, the components of p do not commute: [p_i, p_j] = (ieħ/c) ε_ijkB_k. Specializing to a uniform B field in the z direction, [p_x, p_y] = iħm ω where ω = eB/mc. Also [p_x, H] = iħω p_y, [p_y, H] = - iħω p_x.

Thus defining p_± = p_x ± ip_y,

[p_±, H] = ± ħω p_±
[p₊, p_-] = 2 ħmω

Normalized correctly, p_± satisfy the same commutation relations as the raising and lowering operators for the simple harmonic oscillator.

But note that this oscillation is not between x and p as it is for the SHO, it's between p_x and p_y. Which suggests that the particle is following a circular orbit.