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When looking at the Schrödinger equation one can quickly observe that there is no reason for this differential equation to result in discrete eigenenergies unless boundary conditions, which confine the motion of a particle, are taken into account. Numerous discussions on stackexchange have covered that topic, see e.g. this one, this one or that one.

I am however wondering where the quantization of energies in the derivation of Landau levels in a 2-dimensional electron gas (2DEG) in a magnetic field comes from? To my knowledge, in this case the free electron gas is not confined in any way, but nevertheless the spectrum of the respective Hamiltonian is discrete and looks like

$$E = \hbar \omega_c \left(n+\frac{1}{2}\right)$$

where the index $n$ takes integer values, but I can't quite figure out why that is since the system was never assumed to be bound (Yes, it is true that I am limiting the system to two dimensions, but that should not change the result much as a similarly quantized spectrum arises in a three dimensional electron gas exposed to a magnetic field).

What is the fundamental reason Landau levels are discrete if it is not the boundary?

Qmechanic
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1 Answers1

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  1. This is perhaps easiest to see in the Landau gauge $\vec{A}=(0,Bx,0)$. Then the 2D Hamiltonian operator becomes $$ \hat{H}~=~\frac{\hat{p}_x^2}{2m} + \frac{(\hat{p}_y-qB\hat{x})^2}{2m}.\tag{1}$$

  2. The operator $\hat{y}$ is a cyclic variable, so the corresponding momentum $\hat{p}_y$ is a COM, i.e. $(\hat{y},\hat{p}_y)$ decouples. Hence the 2D system is effectively a shifted 1D SHO.

  3. Alternatively, perform a symplectomorphism conserving the CCRs: $$ \begin{align} \hat{X}~=~&\hat{x}-\frac{\hat{p}_y}{qB}, \cr \hat{Y}~=~&\hat{y}-\frac{\hat{p}_x}{qB}, \cr \hat{P}_x~=~&\hat{p}_x, \cr \hat{P}_y~=~&\hat{p}_y. \cr \end{align}\tag{2}$$ (Classically, this symplectomorphism has a type-2 generating function $$F_2(x,y,P_x,P_y)~=~xP_x+yP_y-\frac{P_xP_y}{qB},\tag{3}$$ in case the reader is wondering.) Then the 2D Hamiltonian operator becomes $$ \hat{H}~\stackrel{(1)+(2)}{=}~\frac{\hat{P}_x^2}{2m} + \frac{(qB\hat{X})^2}{2m}.\tag{4}$$ Hence the 2D system decomposes into 2 independent subsystems:

    • a 1D SHO [with a compact phase space $(\hat{X},\hat{P}_x)$ for a given energy-level, and hence a discrete spectrum, cf. e.g. this Phys.SE post], and

    • a 1D system with a zero Hamiltonian [with a non-compact phase space $(\hat{Y},\hat{P}_y)$, but which nevertheless clearly only has a zero energy eigenvalue].

Qmechanic
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