High-Mag Theory Journal: Education:
Lectures

Electrons in Magnetic Fields

I.D.Vagner

Universite Josef Fourier - Max Planck Institute

Lecture 1: Landau Levels

Chapter 1: CHARGES AND FIELDS

1.1 Lagrangian Formalism

1.1.1. Action

The motion of a body (its trajectory and velocity at each instant) is
defined by a principle of the Least Action (Maupertui principle) [1,2],
and can be found by minimising the Action
functional S:

(1)

where:

(2)

1.1.2 Lagrangian

The function of coordinates and velocities:

(3)

minimising the Action is called the
Lagrangian. Here:

(4)

are the generalised
coordinates
Since
Lagrangian functional is defined as an integrand of the Action functional, Eq. (
2
), it is defined up to a total time derivative of an arbitrary function of the
coordinates and time. This freedom is not influencing the classical equations of
motion. In quantum mechanics, as we will see later, the spectrum of a system may
be influenced by such transformation (Berry phase, Bohm-Aharonov effect etc.).
This is the case in toplogically nontrivial configuration space.

1.1.3 Euler-Lagrange equations

The Equations of motion, following from the principle of least
action Eq. ( 1
) are the Euler-Lagrange’s equations:

(5)

In a
D-dimensional space , is a D-dimensional vector and the Eq. ( 5 ) is
a set of D differential equations of the second order.

1.1.4 Free Particle

The
Lagrangian of a nonrelativistic free particle is:

(6)

This
can be checked by substitution into the Euler-Lagrange equations, to give
correct Newton equation for a free nonrelativistic particle:

(7)

Using
Eq. ( 6)
we can find the relativistic Lagrangian for a free particle in a following way.

The
action function for a free particle is:

(8)

here
we used:

(9)

Therefore the relativistic Lagrangian for a free particle is:

(10)

Expanding this Lagrangian in we obtain:

(11)

Comparing this expression with Eq. ( 6)
we obtain the following value for :

(12)

Therefore the relativistic Action is:

(13)

and
the relativistic Lagrangian for a free particle is:

(14)

1.1.5 Charged particle in a field

The Lagrangian function for a relativistic charged particle in an
electromagnetic field should satisfy the following conditions :
1.
Linear function of charges involved;
2.
Linear in the electromagnetic potentials;
3.
Translationally invariant;
4.
Should not contain higher than the first time derivatives of the particle
coordinates.
It is
easy to verify that all these conditions are satisfied by the following
Lagrangian:

(15)

The
terms: describe the interaction of the charge with the field.

Here
is the light velocity, e -s the electron charge and x are the four-dimensional coordinates (x,y,z,t)
and are the electromagnetic potentials defined as:

(16)

and

(17)

Here
is a polar vector (electric field), and is an axial vector (magnetic field).

1.1.6 Action function

For a
charge in an electromagnetic field the Action function has the following form:

(18)

In
three-dimensional notations:

(19)

1.2 Equations of motion for a particle in a field

Using
the Lagrangian function Eq. 30 in the Eq. 5 we arrive at the
following Equation of motion for an electron in electric and magnetic fields:

(20)

where
is the Lorentz force.
In the nonrelativistic limit :

(21)

1.2.1. The equations of motion are invariant

with respect to simultaneous change in sign of time and of the magnetic
field:

(22)

Thus,
if a certain motion is possible in an electromagnetic field, then the reversed
motion is possible in a field in which the direction of is reversed.

1.3 Hamiltonian formalism

1.3.1. Definitions

The
Hamiltonian, , is a function of independent variables and and is defined by the relation:

(23)

Here
the Generalised momentum:

(24)

was
introduced.

1.3.2. Hamilton’s Equations

The
Lagrangian’s equations, Eq.(5) could be
substituted by a system of 2D differential equations of the first order, the
Hamilton equations (the canonical equations):

(25)

and

(26)

1.3.3 Charge in a field

1.3.4 Free particle

Using
the Lagrangian for a free particle, Eq. (14), we obtain for
the generalised momentum:

(27)

In
the nonrelativistic limit:

(28)

1.3.5 Inclusion of fields

:

(29)

At the Eq. ( 15) goes to:

(30)

and

(31)

1.3.6. The Hamiltonian function

is, now:

(32)

Comparing it to Eq.() we arrive at:

(33)

1.3.7 Nonrelativistic Hamiltonian

Finally for vc:

(34)

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