Landau quantization

The Landau levels (after Lev Davidovich Landau ) represent a quantization of energy of charged particles moving in homogeneous magnetic fields. It can be shown that the energy of the mass of a charged particle (for example, an electron) and charge e, which moves parallel to a magnetic field B in the z- direction is m as follows:

Thereby pz is the ( unquantized ) momentum of the particle in the z direction, the cyclotron and the reduced Planck's constant. , The charged particles and a spin on, this results in an additional splitting of the levels according to the quantum number of the z-component σz ( = magnetic field direction ) of the spin:

This means that (such as to the right in the indicated figure) are allowed only certain particle trajectories, which are characterized by the two quantum numbers PZ and n ( and, if the spinning σz ). The movement One can also imagine such a way that the particle propagates freely longitudinally and transversely ( radially ) to a harmonic motion with (see harmonic oscillator ( quantum mechanics) ). This corresponds to a total of a helical orbit around the magnetic field lines. In the transverse momentum space (only px, py component) the motion on a circle for each quantum number n remains limited, in 3-dimensional momentum space, the states thus lie on cylinders ( Landau - cylinder).

The split into Landau levels can be, for example, in solid state physics measure (de Haas -van Alphen - effect). There, the transverse pulses are quantized due to the crystal lattice. It can be shown then that are located on each Landau - cylinder exactly the same number of states.

Theoretical derivation using the Schrödinger equation

The derivation presented here is based on the references and the original work.

Conditions and tasks

Consider a simple situation: a particle of mass m and charge q is located in a homogeneous magnetic field having only one component in the z- direction. This field can also be represented by the following vector potential:

One can easily show that this results in over again the above magnetic field.

One then obtains the ( initially classical ) Hamiltonian of this system to:

By the position and momentum variables are replaced by the corresponding quantum mechanical operators ( → correspondence principle ), one obtains the Hamiltonian of the system. In the last part of the above equation, a velocity was defined (a " velocity operator" in the Hamiltonian ), which has the following form:

From the classical treatment, we know that the solution of the problem is a helical motion ( helical motion, see figure above) in the z- direction. That's why it makes sense (which will be shown in the later invoices ), the following division of the Hamiltonian into a longitudinal ( along the magnetic field direction) and to the transverse part ( in the classical view that finds in this plane rotational motion instead of, which results in a helical motion ) to be made:

Obtained for the " velocity operator" following commutation relation:

The cyclotron frequency is used. Further can be seen in the definition of light that

So also with each other and swap and there is a basis of eigenvectors common to and.

Eigenvalues ​​of H | |

There is the following commutation relation:

In order for a theorem on operators which interchange according to the above relation is (ie as the canonical position and momentum operators swap ), applicable and we can conclude that a continuous spectrum of eigenvalues ​​has vz. Furthermore, all eigenvectors are also eigenvectors of. The eigenvalues ​​of can thus be written in the following form:

In analogy to classical mechanics so thus describes the free propagation of a particle in the z direction.

Eigenvalues ​​of H ⊥

To obtain the energy eigenvalues ​​of (and thus the so-called Landau levels ), one introduces the following operators with their commutation relation a:

Thus then in the form of a quantum harmonic oscillator which oscillates at the cyclotron frequency? C.

The energy eigenvalues ​​of are therefore

Eigenvalues ​​of H

The total energy is obtained from the sum of self- energies and of:

These levels are called Landau levels. You are unendlichfach degenerate by the continuous speed spectrum.

Depending on the applied magnetic field is thus obtained for a fixed speed different level spacings:

More

It can be shown that the degeneracy of the Landau levels is proportional to the magnetic flux density. With the above finding that the level distance to also proportional to, the oscillations occur in De Haas van Alphen effect in physical quantities, which depend on the density of states to explain: If the magnetic field is increased, the energy of the rising Landau levels, while simultaneously increasing their degeneration. Electrons are therefore migrate to a deeper level located. Therefore, if the topmost first occupied Landau level (ie, the former Fermi level ) was completely emptied, the next lowest Landau level suddenly to the Fermi level.