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## Relativistic Quantum Mechanics.

An Introduction

Walter Pfeifer

Switzerland

2004, revised 2008

### Contents

Preface 2

1 Elements of the theory of special relativity 3

1.1 Lorentz transformation for inertial systems in uniform relative motion 3

1.2 Partial derivatives and quantum mechanical operators 5

1.3 Electromagnetic quantities 7

2 The Dirac equation 11

2.1 The equation 11

2.2 Lorentz-covariance of the Dirac equation (form invariance) 12

2.3 Probability density, current density 12

2.4 Nonrelativistic limit of the Dirac equation with electromagnetic fields. The
Pauli term and the spin-orbit energy 12

2.5 The Dirac equation for particles with an anomalous magnetic moment 12

3 Wave functions of Dirac particles 12

3.1 The wave function of a free particle, helicity, the velocity operator 12

3.2 A Dirac particle in a homogeneous magnetic field 12

3.3 Dirac equation with central potential. Parity and total angular momentum 12

3.4 Separation of the variables for the Dirac equation with central potential 12

3.5 Solution of the radial equations for a Dirac particle in a Coulomb field 12

3.6 Massless Dirac particles 12

3.7 Particles with arbitrary spin 12

3.8 Charge conjugation. The positron 12

4 Other relativistic quantum mechanical equations 12

4.1 The Klein-Gordon equation 12

4.2 The Klein-Gordon Schrödinger equation 12

References 12

Index 12

### Preface

Relativistic quantum mechanics are used to describe high-energy particles and highly ionised atoms. They give a consistent formalism for spin-½ particles and provide finer details of atomic and molecular spectra. In short, they are an important tool of modern physics.

This book deals mainly with the Dirac equation, its properties, its applications and its limiting cases. A formalism for particles with arbitrary spin and remarks on other relativistic quantum mechanical equations are given.

This publication is an introduction and is directed towards students of physics and interested physicists. The detailed developments and the numerous references to preceding places make it easier to follow. However, knowledge of the elements of quantum mechanics, relativistic mechanics and electrodynamics is a prerequisite. In order to relieve the reader, we don't deal with rotations of the coordinate system and not with Lorentz groups either. We have no renaming of matrices, no Feynman daggers, no Einstein convention of summation over repeated indices, no quantum field theory, no second quantisation and no natural units with . The system SI (MKSA system) is used without exception.

We use the following symbols: operators are written in bold letters, a three- dimensional vector is marked with an arrow, two- or four-spinors or -vectors are underlined and the symbols for matrices are doubly underlined.

### 1 Elements of the theory of special relativity

### 1.1 Lorentz transformation for inertial systems in uniform relative motion

The coordinate system
is given parallel to a system
and has a constant relative velocity
in the direction
. At the time
both origins coincide.

For (=velocity of light) the classical, nonrelativistic relations

(1.1.1)

hold between the space-time coordinates of
both systems for a given event. If the velocity
is not negligible with respect to
, i.e. for the relativistic case, a Lorentz transformation
has to be performed as follows

(1.1.2)

Since in relativistic transformations
space- and time coordinates are interrelated linearly, the following variables
are used

(1.1.3)

with corresponding expressions for
Equations (1.1.2) read now

(1.1.4)

Introducing the matrix

(1.1.5)

with elements
, equations (1.1.4) can be written as

(1.1.6)

and by means of the column vectors
and
we have

(1.1.7)

Eq. (1.1.6) implies

(1.1.8)

Of course, motions in the x- or y-direction
result in corresponding transformation formulas. Four-vectors which transform
like
are named contravariant
. With the help of (1.1.4) we calculate the gauge

(1.1.9)

The first and the last line show that the
form of this expression isn't changed by the Lorentz transformation (1.1.4). The
expression (1.1.9) is said to be form invariant.

There are other linear transformations
which preserve the Lorentz metric (1.1.9): a rotation of the coordinate
system, the reflection of the space coordinates
and the time reversal
. These transformations and the translational transformation
of the type (1.1.2) can be coupled individually resulting in a transformation
with the same Lorentz gauge. That is, all these transformations form a group:
the inhomogeneous Lorentz group
.

Transformations to a frame with parallel
axes but moving in an arbitrary direction are called boost
s.

The general translation where the systems
are positioned arbitrarily relative to the
velocity
, can be constructed with a preceding instant rotation, a
translation along the
- axis (boost) and following rotations. In most cases it is
not necessary to deal with the mentioned (instant) rotations, i.e. the axes
can be chosen in the direction of
. In this book we will restrict ourselves to this case, and
the transformation formulas (1.1.2) up to (1.1.7) will be used. Consequently the
Lorentz group will not be investigated.

### 1.2 Partial derivatives and quantum mechanical operators

The inversion of the transformation relations (1.1.4) reads

(1.2.1)

(1.2.2)

which can easily be checked. We set up the
partial derivatives

(1.2.3)

In short,

(1.2.4)

holds, where (1.2.2) has been used. The
inverse relation reads

.

In electrodynamics the expression

appears. Applying eq. (1.2.3) repeatedly
one obtains the Lorentz form invariance

(1.2.5)

in analogy with (1.1.9).

In quantum mechanics the energy operator
reads

. (1.2.6)

Therefore we write

where we have defined
. The *x*-component of the momentum operator
reads

(1.2.7)

and we have

Inserting in (1.2.3) and defining
we obtain the following relation between
operators

(1.2.8)

That is, the four-vector
transforms like
(cf.(1.1.4)). Consequently it is contravariant.
Due to the strong analogy with (1.1.9) the operator expression
must be form invariant in a Lorentz
transformation, i.e.

(1.2.9)

A similar expression comes from the
relativistic relation between energy and momentum (as quantities) of a free
particle

(1.2.10)

Following the rules of quantum mechanics we
substitute
and
by the corresponding operators
and
, (1.2.6) and (1.2.7), and obtain

(1.2.11)

Because the left hand side of (1.2.11) is
form invariant, the rest mass
is the same in every inertial system as we
expect.

1.3 Electromagnetic quantities

In the
theory of electromagnetism the electric field
strength
and the magnetic field strength
can be calculated starting from the scalar
potential
and a vector potential
like this

(1.3.1)

If a charged particle of charge *e*
moves in an electromagnetic field, the Lorentz force

(1.3.2)

acts on the particle. Our formulas and
quantities are written in the system SI. The potentials
and
obey differential equations which contain the
electric charge density
and the electric current density
. The equations read

(1.3.3)

. (1.3.4)

The quantities
and
are the permeability
and the dielectricity
respectively of the vacuum. An additional
constraint can be chosen. We take the “Lorenz gage”:

. (1.3.5)

In the framework of special relativity it
is natural to introduce the contravariant four-vectors

(1.3.6)

(1.3.7)

whose components transform as

(1.3.8)

(1.3.9)

when changing over from the inertial system
to
(c.f. (1.1.4).

We now show that with these transformation
rules the differential equations (1.3.3) up to (1.3.5) are form invariant, i.e.
in both systems they have the same form, that is, the same electromagnetic laws
hold.

Making use of the definitions (1.1.3),
(1.3.6) and (1.3.7) we rewrite (1.3.3) in the system

(1.3.10)

where we have applied the basic relation

(1.3.11)

With the help of (1.2.5), (1.3.8) and
(1.3.9) we obtain from (1.3.10)

(1.3.12)

In the same way we deal with the *z*-component
of eq. (1.3.4)

(1.3.13)

(1.3.14)

We multiply (1.3.14) by* v*/*c*
and add it to (1.3.12), which yields

(1.3.15)

which is form invariant with respect to
(1.3.10). Multiplying (1.3.12) by *v*/*c* and adding it to (1.3.14)
results in

(1.3.16)

in form invariant accordance to (1.3.13).
The *x*- or *y*-component of eq. (1.3.4) in the system
reads

(1.3.17)

When handled in the same way as the
preceding equations, it results immediately in the form invariant form
affiliated to the coordinate system
.

Finally we write eq. (1.3.5) with the help
of (1.1.3) and (1.3.6) in the system

(1.3.18)

With the help of (1.2.3) and (1.3.8) we
obtain

which results finally in

, (1.3.19)

which has the same form as (1.3.18). Thus,
we have proven that the field equations (1.3.3), (1.3.4) relating the
electromagnetic potential
with the sources
and the Lorenz gauge condition (1.3.5) are form
invariant in our Lorentz transform.

This is only an excerpt. The whole publication can be ordered from the category «order for free» in this site.