I am trying to understand the derivation for the wavefunctions and levels of free electrons in a 2D surface perpendicular to a magnetic field. The usual prescription is to take a vector potential $\vec{A}=Bx\hat{y}$ and write the Schroedinger equation, using the momentum operator $\vec{p}=-i\hbar\nabla +e\vec{A} $

Then you take the ansatz $\Psi(x,y)=\exp(ik_y y)\phi(x)$, which turns the equation for $\phi(x)$ into an analogue of the harmonic oscillator.

But I don't understand this. Why choose the $y$-direction to have the exponential part of the equation? And once you've done this, it looks to me like the solution is free electrons in $y$-direction, but bound in $x$ in a harmonic potential. I know that in reality the electrons are "bound" in both directions and are orbiting a given $x,y$ value, given by $\frac{\hbar \vec{k}}{eB}$. So how does using that ansatz and getting a harmonic oscillator in 1 dimension with free electrons in the other, translate to this physical picture?

Obviously I am misunderstanding something basic about the derivation. Any suggestions?


If you write down the Hamiltonian with the vector potential you provided, you should see that the Hamiltonian commutes with $p_{y} = -i\hbar\partial_{y}$. Given that, you know that the eigenfunctions of $H$ must also be eigenfunctions of $p_{y}$ - namely, the $y$-dependence of $\Psi$ must be of the form of a plane wave, as you have correctly written it.

The next part of your question is addressed by gauge invariance. As I'm sure you know, you can change $\vec{A}$ by the gradient of any scalar function and recover the same electromagnetic field configurations. That is, the gauge-transformed field $\vec{A}' = \vec{A}-\nabla f(\vec{x},t)$ is physically equivalent to $\vec{A}$. However, since the potentials (NOT the fields) are the objects that appear in the quantization procedure, it can be very non-trivial to verify that physics is the same in different gauges in quantum mechanics. For practice, you should try to find a gauge transformation which gives plane waves in the $x$-direction and a harmonic oscillator in the $y$ direction.

Another thing to try if you're feeling ambitious is to find a gauge that includes both $x$ and $y$ - this is typically called symmetric gauge. It is the closest in spirit to capturing the classical physics underlying charged particle motion in a magnetic field (namely, cyclotron motion), but it is dramatically different from the two gauges mentioned thus far (which are both Landau gauges).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.