High-Mag Theory Journal: Education:
Lectures

Electrons in Magnetic Fields

I.D.Vagner

Universite Josef Fourier - Max Planck Institute

Lecture 1: Landau Levels

Chapter 4: QUANTUM MECHANICS OF THE ORBITAL MOTION

4.1 Gauge Invariant Formulation

The Hamiltonian of a charge in a magnetic field is:

(79)

A
very important property of the Hamiltonian Eq.(79) is its gauge
invariance, namely it is invariant under a simultaneous transformation of the
potentials:

(80)

and
of the wave function:

(81)

where: (See Flügge, 125).
In what
follows two different gauges will be used:
a) the Landau gauge:

(82)

and:
b) the Symmetric
gauge:

(83)

These
are particular cases of the Coulomb gauge: A=0.

Indeed:

(84)

and:

(85)

4.1.1 Transformation from the Symmetric to
the Landau gauge

(86)

where:

(87)

If
, then: and

if A is or .

(88)

4.1.2 Classical rotation and QM
correspondence

Figure 10: Classical orbit of an electron in a magnetic field

4.1.3 Classical rotation

Let us start
with the classical description of the Larmor orbit and let us define the
classical values, which than we will generalize to quantum operators.
Classically:
and are the Larmor radius and the cyclotron frequency. In the Fig.1 also
defined are the vectors of the center of rotation: and the radius-vecto of the rotating point
All these
vectors are connected by the equation of motion:

(89)

where
.

4.4 Quantum mechanical correspondence

Using the
quantum mechanical correspondence to the classical variables we define the
following operators:
Operator of
the center of orbit:

(90)

where:

(91)

are
the operators of coordinates of the center of motion, defined by Eq.(89), and the Larmor
radius operator:

(92)

4.5 Commutation relations

4.6 Velocity operator

Using the
operator identity: the velocity operator can be obtained to be:

(93)

4.7 Conserved quantities

The following
operators:

(94)

are
integrals of motion, since they commute with the Hamiltonian, Eq.(79)
Let us show
now that the commutation relations Eq.(121) and (127) are analogous
to those of the harmonic oscillator and explore this similarity in order to find
the energy spectrum of the electron in a magnetic field.

4.8 Correspondence between the Larmor rotation and
harmonic oscillator

4.9 P,Q
coordinates
The
Hamiltonian of a harmonic oscillator can be written in P,Q coordinates:

(95)

where
operators and satisfy the following commutation relation :

(96)

The
energy spectrum in this case is:

(97)

here
the oscillator frequency is:

(98)

4.10 Correspondence with an
oscillator

Using the commutation relations Eq.(121) and Eq.(127) we define the
following correspondence between the operators, corresponding to the Larmor
rotation and to harmonic oscillator:

(99)

The
corresponding parameters of the Larmor rotation and the oscillatory motion are
connected as follows:

(100)

Using this correspondence, we arrive at:

4.11 Energy spectrum

Since the
Hamiltonian is:

(101)

his
spectrum will be:

Figure 11: Landau Levels (3D)

4.12 Spectrum of the Larmor Radius operator

(102)

4.2 Landau Gauge

4.2.1 Hamiltonian

The
Schrödinger Equation for an electron in a magnetic field, in the Landau gauge,
is:

(103)

is separable in z,x and
y components:

(104)

i.e.:

(105)

4.2.2 Eigenfunctions and eigenvalues

Eq. (105) is an equation
for a harmonic oscillator with the eigenvalues:

(106)

and
the eigenfunctions are:

(107)

where is the magnetic length, and is the Landau level index.

(108)

are
the Hermitte polinomials.

4.2.3 Degeneracy of a Landau level

Since the
electron is assumed to be in the sample, should stay inside the sample of dimensions and therefore: . If we assume periodic boundary conditions , the possible num k values, i.e. the degeneracy of n, is: There remains the motion in the z-direction and the
energy is and therefore the density-of-states in 3D for the Landau level n
is

(109)

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