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Concept
Landau quantization
Summary
In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.
Landau quantization is directly responsible for the electronic susceptibility of metals, known as Landau diamagnetism. Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials as a function of the applied magnetic field known as the De Haas–Van Alphen and Shubnikov–de Haas effects.
Landau quantization is a key ingredient to explain the integer quantum Hall effect.
Consider a system of non-interacting particles with charge q and spin S confined to an area A = LxLy in the x-y plane. Apply a uniform magnetic field along the z-axis. In SI units, the Hamiltonian of this system (here, the effects of spin are neglected) is
Here, is the canonical momentum operator and is the operator for the electromagnetic vector potential (in position space ).
The vector potential is related to the magnetic field by
There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to A changes the overall phase of the wave function by an amount corresponding to the scalar field. But physical properties are not influenced by the specific choice of gauge.
From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.
When then is a possible solution in the Landau gauge.
In this gauge, the Hamiltonian is
The operator commutes with this Hamiltonian, since the operator ŷ is absent by the choice of gauge. Thus the operator can be replaced by its eigenvalue ħky.
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