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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.


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