Relativistic Wave Mechanics

By E. Corinaldesi

Geared toward advanced undergraduate and graduate students of physics, this text provides readers with a background in relativistic wave mechanics and prepares them for the study of field theory. The treatment originated as a series of lectures from a course on advanced quantum mechanics that has been further amplified by student contributions.
An introductory section related to particles and wave functions precedes the three-part treatment. An examination of particles of spin zero follows, addressing wave equation, Lagrangian formalism, physical quantities as mean values, translation and rotation operators, spin zero particles in electromagnetic field, pi-mesic atoms, and discontinuous transformations. The second section explores particles of spin one-half in terms of spin operators, the Weyl and Dirac equations, constants of motion, plane wave solutions and invariance properties of the Dirac equation, the Dirac equation for a charged particle in an electromagnetic field, non-relativistic limit of the Dirac equation, and Dirac particle in a central electrostatic field. The final section, on collision and radiation processes, covers time-independent scattering of a spinless particle, non-relativistic steady-state scattering of a particle of spin one-half, time-independent scattering of Dirac particles, non-relativistic time-dependent scattering theory, emission and absorption of electromagnetic radiation, and time-dependent relativistic scattering theory.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 16, 2015
• ISBN: 9780486805733

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INDEX

INTRODUCTION

PARTICLES AND WAVE FUNCTIONS

1. De Broglie’s relation

At the end of the last century it seemed possible to interpret all known physical phenomena in terms of electromagnetic waves and charged particles, but the discoveries of the twentieth century have brought about a radical revision of classical concepts. The theory of blackbody radiation (Planck 1900)* and the explanation of the photoelectric effect (Einstein 1905) were based on the hypothesis that exchanges of energy between matter and radiation took place in quanta

The subsequent discovery of the Compton effect (Compton 1923) showed that a quantum associated with a plane wave has a momentum p given by

where k is the wave propagation vector, (briefly, wavevector), |k| = 2π/λ = ω/c. The relation between the wave properties of radiation and the particle properties of the associated quanta is relativistically invariant. In fact k1, k2, k3 and iω/c are the components of a four-vector, as may be shown from the invariance of the phase of a monochromatic plane wave. On the other hand it is known that the components of the momentum and the energy of a particle can be regarded as the space and time components of the four-momentum pμ = (p, iE/c). Therefore equations (1) and (2) may be condensed in the four-vector equation

equivalent to the four equations pμ = ħkμ (μ = 1, 2, 3, 4).

One notices now that the relativistic relation between energy and momentum of a particle of rest mass m *

can be written in the form **

and, since

we have

for the quanta of the electromagnetic radiation. This shows that the rest mass of a photon is zero.

De Broglie’s postulate of matter waves, which was based on purely theoretical arguments, can be condensed in the statement that the relation p = ħk, where

between the wave four-vector and the four-momentum holds not only for photons, but also for particles of mass not equal to zero. In the case of electromagnetic radiation the wave aspect was discovered much earlier than the corpuscular, for material particles the opposite has been the case.***

NOTE 1. Using the conservation of four-momentum, the expression

for the change in wavelength in the Compton effect, may be cast into a Lorentz invariant form.

This amounts to finding an invariant equation for the four-momenta p and p′ of the electron, and ħk and ħk′ of the photon, before and after collision, which reduces to the given formula in the reference frame in which the electron is initially at rest (p = 0).

From conservation of energy and momentum we have

which, on squaring, becomes

Since

we obtain the invariant equation

In the reference frame in which p = 0, one has

where θ is the angle formed by k and k′.

Thus we have

As a further application of eq. (α), we consider the cases where the electron has:

a) initial non-vanishing momentum in the direction of propagation of the incident photon;

(brick wall system). One can show that |∆| has the meaning of an invariant momentum transfer.

a) From (α) it follows that

from which

b) Since the energy of the electron does not change in the collision,

also the energy of the photon remains unchanged; therefore

The condition |k| = |k′| implies that π · ∆ = 0, so that π and ∆ are orthogonal (Fig. 1).

The momentum transfer is

Fig. 1

On the other hand the invariant

may be calculated in any reference frame, in particular in the brick wall system, in which it has the value |∆|².

2. Phase and group velocity

Waves associated with particles of non-zero mass have properties different from those of electromagnetic waves. For instance the dispersive law λ = 2πc/ω cannot be valid for matter waves. In fact it has been shown above that this is only consistent with de Broglie’s relation if the photon rest mass is zero.

it follows that for a particle of mass m

and also

From this it is seen that the phase velocity vp of a monochromatic matter wave is a function of |k| since

whereas for electromagnetic waves in vacuum the phase velocity is always c (m = 0 in eq. (4)). Therefore, for matter waves associated with a force-free particle, the vacuum is equivalent to a homogeneous isotropic medium of refractive index n given by

In contrast to this, for electromagnetic waves n for most dispersive materials. The corresponding phase velocity is therefore less than c.

The fact, that the phase velocity of matter waves is greater than the speed of light, is at first sight somewhat strange. Although the formalism used so far is invariant under Lorentz transformations, it might seem that matter waves infringe one of the basic postulates of relativity i.e. that a signal cannot propagate with a speed greater than c. On closer inspection, though, it is seen that the phase velocity characterizes a plane wave of infinite extension, which therefore cannot be regarded as a signal. In order to speak of a signal one must have a wave function ψ(x, t) which is zero outside a certain region of space, the location (and generally shape and size) of which changes with time.

A wave packet may be constructed as a superposition of plane waves and expressed in the form of a triple Fourier integral *

It must be stressed that (5) is not a four-dimensional Fourier integral as might be used to give an integral representation of any function of the variables xμ, (μ = 1, 2, 3, 4), (see Note 4 below).

In writing (5) it has tacitly been assumed that the principle of superposition holds also for matter waves and that the general ideas on interference and diffraction of electromagnetic waves may be extended to matter waves.

In order to invert e–ik·x and integrate with respect to x

Remembering the properties of the three-dimensional α function (see Note 2), one has

The problem of finding the Fourier transform of a given wave function ψ(x, t) is thus reduced to a simple integration.

Defining

which may be interpreted as the wave function in k space, one can write more symmetrically

NOTE 2. We summarize some of the basic properties of the Dirac δ function. For a more complete treatment see P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford, 1958), p. 58, and L. Schwartz, Théorie des Distributions (Hermann, Paris, 1950).

The Dirac function -may be defined by the properties

or by the equivalent property

(f(x) is a continuous function). No ordinary function with these properties exists. δ(x – x0) may be conceived as the limit of an ordinary function, e.g.

The Fourier representation of the δ function is

(Here the equality is to be interpreted in the sense that the integral is equivalent to δ(x), when it occurs as the factor of an integrand.)

The following is a non-rigorous proof that

We write

Now

for any K, and therefore also in the limit K→∞. Moreover for x = 0 the expression on the right of equation (β) becomes infinitely large. For x ≠ 0 its oscillations become infinitely fast, so that its contribution to the integral over x reduces to that of the points in neighbourhood of x = 0, and f(x) may be replaced by f(0) and taken out of the integral sign.

Likewise one can define a three-dimensional δ function

with the Fourier representation

NOTE 3. Much use will be made of the formula

where f(xn) = 0, f′(xn) ≠ 0.

In order to prove this, the x axis is divided into N intervals by the points a1= –∞, b1= a2, b2, . . . bN–1 bN= ∞ Each interval contains only one zero of f(x).

For an arbitrary continuous function F(x) we have

For a generic interval

where y =f(x), max = max [f(bn),f(an)], min = min [f(bn),f(an)]. Hence it follows that

(where xn are the zero’s of y = f(x)), showing that δ(f(x)) has the same properties as the linear combination of Dirac functions

For f(x) = x² – a² and a > 0

NOTE 4. Using the formula given in the preceding note, it can be shown that the triple integral

can be written in the form of a fourfold integral

where

In fact

Thus

The function η (k) eliminates the negative frequencies which would result from δ(k0 + ω/c). Because of the relativistic dispersive law (3) one has

Remembering that δ(– x) = δ(xit follows that

The exponential, and the argument of the δ function, are invariant, as k1, k2, k3, k4 transform like the components of a four-vector. The 6 function limits the domain of integration to time-like four-vectors whose square is – k². Since the sign of the fourth component of a time-like vector is invariant, the function η(k) of such a four-vector is invariant. The volume element d⁴k is also invariant under Lorentz transformations.

Now suppose that, for a certain wave packet, the modulus of A(k)= |A(k)| eiϕ(k), (ϕ real), has a maximum for a certain value of k, and is appreciably different from zero only in the finite domain (ki–η, ki + η) (i = 1, 2, 3) with η |k|. In the neighbourhood of the maximum the wave function has the form

where the symbol ∇k denotes a vector with components

Therefore the amplitude of the wave function is appreciable only at the value k for which |A(k′)| has a maximum and at the value of x for which

This last equation defines, as a function of time, the position of the centre of the wave packet, which moves with the velocity

This is called the group velocity. This expression is acceptable since |νg| < c in accordance with the postulate of relativity theory.

In the classical limit when diffraction effects may be neglected, a particle, even if subject to forces, can be represented by a wave packet of finite extension which moves in space with a group velocity determined at any time by the wave vector for which |A(k, t)| is a maximum. The momentum p = ħk and the energy E = ħω associated with this wavevector may be identified with the momentum and the energy of the particle. From equations (7) one sees that vg is the velocity of a particle of momentum p and energy E

Notice that this argument may be reversed. If one requires that the group velocity vg = ∇ħkħω must coincide with the velocity of a particle v = ∇PE, one has

where c is independent of k. Now a relation of the type pi = ħki + ci (i being a linear homogeneous combination of the ci ′s.

Since the relation between p and k must and this is only possible if ci = 0.

This last approach was followed by de Broglie to establish the relationship between momentum and wavevector. The approach is independent of relativity, and is in the spirit of the correspondence principle. Moreover it may be adapted to cases in which the particle is subject to forces.

3. Charged particles in an electromagnetic field

We review some of the basic equations of relativistic electrodynamics, which will frequently be used in the following. It is well known that Maxwell’s equations may be written in tensor form

where

is the four-dimensional Ricci-Levi Civita tensor, antisymmetric in all its indices

and with ε1234 = 1 (see, however, Chapter VII, § 1).

It may be shown that a necessary and sufficient condition for the second of equations (8) to be satisfied is that Fμv may be written as a four-dimensional curl*

From the tensor character of Fμv it follows that the four-potential Aμ is a four-vector. The first of equations (8) becomes

is the d’ Alembert operator.

If the Lorenz condition

is imposed, eq. (10) becomes

For a given Fμv the four-potential Aμ is not uniquely determined by eq. (9). In fact the electromagnetic tensor is invariant under the gauge transformation of the second kind

Only the field strengths have a direct physical meaning, and they do not change under a gauge transformation. Any relation involving electromagnetic quantities must be gauge invariant, if it is to have a physical meaning. From an arbitrary four-potential, one satisfying the Lorentz condition can be obtained by making a gauge transformation with Λ satisfying

In the case of eqs. (10′), the only permissible gauge transformations are those for which

(special gauge transformations).

The Lagrangian of a particle of charge – e in a static electromagnetic field is

Thus the canonical momentum is

and the Hamiltonian

Hence the equations of motion for a particle of energy E

The last equation is not invariant under the static gauge transformation A → A + ∇x Λ(x), but it becomes so if e i/c is added to both sides. In fact one gets

where we took into account that A is an implicit function of t through x = x(t). In order to give a more significant form to the right-hand side, we consider the three-dimensional Ricci-Levi Civita tensor εikl, by means of which the vector product may be written in the form

Therefore

Using the identity

where δab is the Kronecker symbol, one finally obtains

and therefore

which is the Lorentz force.

In order that the equation

may be invariant –as is physically necessary –under the gauge transformation A → A + ∇xΛ, the latter must be accompanied by the momentum transformation

For a particle in an electrostatic field it is easy to establish de Broglie’s relation. Assuming that E = ħω, then on requiring that v = ∇kω, one gets

as for a free particle.

The wavelength λ = 2π/|k| is given by

and is a function of position.

For small velocities (|p| mc), energy and momentum of a free particle are given by the Taylor expansions of mγc² and mγv with respect to v/c terminated at the second term, p = mv + terms of third order in v/c, Correspondingly we have

and

The last of these equations is precisely the relation between wavelength and velocity as was found by de Broglie.

Notice that the term mc²/ħ in the expression for ω gives the same periodic factor exp [– imc²/ħ] for all plane waves, and, therefore, also for an arbitrary wave function ψ(x, t). Thus one may redefine the zero point of both energy and angular frequency

For an electron in an electrostatic field

These non-relativistic expressions may be used in discussing experiments on diffraction of electrons (Davisson and Germer (1927), reflection on mono-crystals; G. P. Thomson (1928) and Rupp (1928), Debye-Scherrer rings from diffraction by crystal powders).

It is convenient to measure the energy of electrons in electron volts and the wavelength in Ångstroms. Writing

and remembering that the electron Compton wavelength, i.e. the change of wavelength of a photon scattered 90° by an electron at rest, is *

and the rest energy of an electron is

we have

where Eev is the kinetic energy of the electron measured in eV.

4. Principle of least action and Fermat’s principle

In giving the first formulation of wave mechanics, de Broglie made use of the analogy between the principle of least action of mechanics and Fermat’s principle of geometrical optics. The first principle states that, of all motions x = x(t) leading from the position P1 = P(t1) to P2 = P(t2), the one actually realized makes the action, i.e. the integral

stationary (Hamilton’s principle). That is, for variations x(t) → x(t) + δx(t) subject to the condition δP(t1) = δP(t2) = 0, we must have

and thus Lagrange’s equations hold:

The principle of least action can also be formulated by restricting the variations x(t) → x(t) + δx to those which do not change the energy (Maupertuis’ in this case we have

(where p must be regarded as a function of x), with the subsidiary condition

In this form the principle of least action is analogous to Fermat’s principle of geometrical optics

where k = 2πvnv(x)k/c |k|,nv being the refractive index for light of frequency v (isotropic medium). This principle determines a ray from P1 to P2 as a line for which the optical path is stationary.

Now classical mechanics is the limit of wave mechanics, just as geometrical