Free particle in magnetic field / Landau quantization I have a question concerning a possible derivation of the Landau quantization.
In our lecture notes (and some other places as well), the following ansatz is used:
$$
\Psi(x,y,z) = \exp{\left(-j(\beta y + k_z z)\right)}u(x),
$$
which then leads to the known results ($u(x)$ is identified as a harmonical oscillator,...)
But I'm unhappy with the ansatz itself: if I calculate the mean momentum in $y$ direction, I obtain $\hbar \beta$. From classical mechanics, I know the solution can be nothing but a spiral-like movement (should be compatible with quantum mechanics, as I could just use a particle with a huge momentum, which could be directly observed...).
So, what am I missing?? I would also expect some symmetry around some path center?!
Sorry for my maybe rough english, but I'm not too familiar with using it in physical context...
 A: The key insight is that whenever (electro-)magnetism and Hamiltonians are involved, there is always a gauge fixing happening somewhere. The basic issue is that the Hamiltonian is something like $$H = \left|p-eA\right|^2,$$ and for any given magnetic field $B$ there is a variety of potentials $A$ which can fit. For a uniform magnetic field, a frequently used one is the Landau gauge where $$A = \left(0, Bx, 0\right).$$ In this gauge, the Hamiltonian is $$H = p_x^2 + (p_y - eBx)^2.$$ You then stick your ansatz in and get what you already know. The point to remember is that $p_x = i \frac{\partial}{\partial x}$ is not the mechanical momentum. It is the canonical momentum, which is defined to be the one that obey the canonical commutation relation $\left[ x, p_x \right] = i$. Ditto for $y$ and $p_y$. The mechanical momentum is exactly what's in the Hamiltonian: $p - eA$. Fairly sure if you follow that through, you get exactly what you expect, i.e. no average motion. 
For a lot of applications, it's nicer to use the symmetric gauge, where $A = \frac{1}{2}\left(-By, Bx, 0\right)$. In that case, you retain the explicit $x$-$y$ symmetry and things like angular momentum keeps making sense.
Notice that things like $p$ change depending on the gauge you choose. Physical variables such as actual mechanical momentum does not. Indeed, in this case of a magnetic field in the $z$ direction, the mechanical momentum in the $x$ and $y$ directions do not commute! You cannot simultaneously know both.
A: Answer to the spiral-like movement:


*

*High quantum number case
If you want your electron to radiate (via larmor radiation), you need to implement the electron's contribution above the background A field to allow it to radiate (or else the total A field will stay static all the time -> no radiation). Note that the A field itself doesn't have to be quantum mechanical if you don't care about spontaneous emission.
-> This also applies to the classical case whenever you want to implement radiation.

*Low quantum number case
The radiated light has to be quantized if you want to stay consistent with the equation.
At ground state the larmor radiation term and the vacuum fluctuation term exactly cancels each other.
