wave-particle duality || wave mechanics and relativity

CHAPTER 9

WAVE MECHANICS AND RELATIVITY

GEORGES LoCHAK AND REGIS DUTHEIL

Many papers have already been published concerning de Broglie's ideas on matter waves, including papers by one of us (G.L.) and D. Fargue in an earlier collective book devoted to the foundations of wave mechanics. (1) One can find, in the latter, some important points such as the law of phase accordance, considered by de Broglie as his basic idea, the hypothesis of a permanent localization of the particle in the wave, upon which de Broglie based his theory of measurement, and the attempts to find a nonlinear wave equation which would be able to describe the wave-particle dualism in a synthetic way.

In the same papers, the importance of relativity for the development of wave mechanics was already strongly emphasized. Nevertheless, we shall come back again, with new arguments, to this unique problem in the present chapter, because we think, as Louis de Broglie, that relativity is the cornerstone of wave mechanics and that every attempt to better understand this theory must be based on it.

It must not be forgotten that, when de Broglie found his famous connection between the principles of Maupertuis and Fermat, his approach was purely relativistic and historically, it was even the first application of the principle of relativistic covariance in quantum mechanics. (2) But two years later, in one of his famous papers, (3) Schrodinger obtained some of the basic formulas of wave mechanics (especially wave length and group velocity) making use only of classical mechanics. He was especially very satisfied that group velocity could be obtained without relativity, contrary to what de Broglie did in his thesis. Later on, following SchrOdinger, de Broglie himself often introduced, in some of his books, these formulas from classical mechanics. (4-6) But it was for him only a heuristic process which made it possible to shorten an introduction: afterwards, all of his fundamental works were based on relativity.

GEORGES LocHAK and REGIS DuTHEn.. • Foundation Louis de Broglie, F-75006 Paris, France.

Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.

157

F. Selleri (ed.), Wave-Particle Duality© Plenum Press, New York 1992

158 GEORGES LocHAK AND REGIS DuTHEIL

However, many theoreticians seem to consider that despite a historical role played in the birth of wave mechanics, relativity has not a crucial importance for its foundations. In particular, almost all works about causal interpretation do not make use of relativity: among many examples, see for instance Refs. 7-9. de Broglie was the only one who always started from Klein-Gordon and Dirac relativist equations in this kind of work. He considered the SchrOdinger equation only as an approximation, excepting for the problem of systems of particles.

For these reasons, we would like to show, in this chapter, that relativity is really unavoidable when we want to obtain some fundamental formulas of wave mechanics.

1. CAN WE REACH WAVE MECHANICS STARTING FROM CLASSICAL MECHANICS?

In this section, we shall examine (in a slightly different form) some calculations of de Broglie and SchrOdinger, based on classical mechanics. But first of all, we are going to recall briefly one of de Broglie's initial relativistic reasonings.

Introducing relativity directly he set up a phase four-vector of the wave, which he likened to the energy-momentum four-vector of the particle. He used Planck's law:

E = hv (1)

in order to identify time components vic and Elc of the two four-vectors. From this followed, by relativistic covariance, the identity of the space components 1IA and p; hence the famous formula of wavelength:

h A=

mv (2)

Owing to the identification of the two four-vectors, de Broglie deduced the identity of Maupertuis's and Fermat's principles and he got the famous groupvelocity theorem. Note that in the preceding formula, m must be considered as the relativistic mass (depending on v).

Now suppose that we' ignore relativity. Consequently, we cannot use the above reasoning because we cannot introduce the four-vectors and thus we have to proceed differently. In this paragraph, we are going to do almost the same thing as SchrOdinger and de Broglie in Refs. 3-6: we shall start form the identification of Maupertuis's and Fermat's principles (as Hamilton did, though in a different form, which we shall examine at the end of this chapter).

Let us then postulate the identity of the two extremal formulas:

f B fBds 8 mvds = 8 ~ = 0

A A I\. (3)

WAVE MECHANICS AND RELATIVITY 159

However, contrary to what Hamilton did, we shall not consider the particle and wave descriptions as alternative. In accordance with de Broglie, we assume an association between wave and particle within the same physical object. It will be understood that we consider only the case of a particular and its wave in the vacuum, in the absence of an external field.

The left-hand integral in (3) is taken along the particle's trajectory and the right-hand one along a radius of the associated wave: we postulate, with de Broglie, that these two curves coincide. In order that this identity should hold for any motion, the two integrants must be identical up to a universal constant, which implies de Broglie's formula (2) (now m is a constant because we are in classical mechanics).

In saying that, we assume implicitly that the constant of proportionality between the two sides of (3) is the Planck constant, but doing so we are only anticipating experimental results which prove it, without violating the logic of the reasoning.

Let us introduce phase velocity V through the formula

Identifying (2) and (4) we get

A=~ v

hv V'=

mv

(4)

(5)

Since we are in the case of inertial motion, the particle energy is entirely kinetic and according to Planck's law and classical mechanics, we obtain

E = hv = !mv2 (6)

Introducing (6) in (5), we find a relation between the phase velocity V of the wave and particle velocity v:

v V=-

2 (7)

Oddly enough, this trivial relation is never written. And yet it is interesting because it is very different from the de Broglie formula:

c2 V=

V (8)

This is a crucial point to which we shall return. Nevertheless, (7) shows a dispersion of phase waves, because according to (6), v becomes a function of

160 GEORGES LocHAK AND REGIS DuTHEn.

frequency and therefore V too. Now, let us consider the group velocity U of this wave through Rayleigh's classical formula:

Introducing (5) and (6) in (9), we find

Consequently,

lId d 1 - = --(mv) = -(v) = -U h dv vdv v

U=v

(9)

(10)

(11)

We see that, starting from a purely classical calculation, equivalent to SchrOdinger's, we reach a result that de Broglie obtained using relativity: the group velocity of waves is equal to the particle velocity. SchrOdinger underlined the fact that this relation "does not necessarily follow from relativity, but remains valid for any conservative system of classical mechanics. "(3) We can see that, remaining in the framework of classical mechanics and wave theory, postulating Planck's law and identifying Maupertuis's and Fermat's principles, we obtain:

1. de Broglie's wavelength (2) 2. A relation between phase velocity of the wave and particle velocity 3. Group velocity (11)

Among these relations, which ones can be experimentally checked? As long as we observe only material particles, like electrons, only the wavelength will be accessible and for low energies, formula (2), with constant m, holds. But we shall not be able to check the formula (11) of group velocity, because experiment only gives particle velocity, which does not prove that particle velocity is equal to group velocity. Moreover, energy E, phase velocity V, and frequency v still remain inaccessible.

We cannot even claim verify Planck's formula, because E and v might be wrong; likewise we cannot check equality (6) since v is measured, but E and v are not; and above all we cannot say anything about the relation (7) between phase velocity V and particle velocity v. SchrOdinger was satisfied to obtain the same group velocity as de Broglie, but this result proves nothing about phase velocity. We observe also that by measuring A, (4) gives only the ratio V/v: we must therefore measure one of these two quantities experimentally in order to deduce the other.


Since neither V nor v is measurable directly for a material particle, let us introduce one of de Broglie's essential ideas which was at the basis of his work: wave mechanics must be a unifying theory of matter and light. In fact, his starting point was a mechanical theory of the photon and not a wave mechanics of matter. * Moreover, since Fermat's principle derives from optics and Maupertuis's principle from mechanics, unifying them involves a synthesis between these two fields of science, which we will assume from now on.

Can we apply to light the formulas deduced above? This can be answered experimentally because with light, and contrary to material particles, phase and group velocity (and even frequency) can be measured. Now these velocities are equal in vacuum, which invalidates (7). Besides, if (7) had been correct, it would have been known as far back as the 18th century, because the light velocity measured by Romer would have been twice the one measured by Bradley: indeed, Romer based himself on the occultation of Jupiter's satellites and measured group velocity, whereas Bradley measured phase velocity through astronomic aberration. (11,12) Of course, the distinction between these two velocities was unknown at the time because it was hidden by the equality of results in vacuum: it was observed later in refracting media. t To conclude, since (7) is incorrect, we cannot start from classical mechanics to find wave mechanics.

2. MINIMAL CONDITIONS OF WAVE-PARTICLE DUALISM: THEY ARE INCOMPATIBLE WITH CLASSICAL MECHANICS AND REQUIRE RELATIVITY

Let us now leave classical mechanics, but without introducing relativity yet. Still remaining within inertial motion, let us define wave-particle dualism with the following postulates:

1. Identity (3) of Maupertuis's and Fermat's principles 2. Planck's law (1) 3. Equality (11) between group velocity of waves and corpuscle velocity 4. Leaving classical mechanics and consequently (6), we replace it by the

following general formula (in which we recall Planck's law already postulated):

*This means that, for de Broglie, the photon is an ordinary particle. Its mass is nonzero but only very small. Remember that the masslessness of the photon is not a physical fact but only a theoretical consequence of the principle of gauge invariance. Experiment cannot prove that a mass is zero but can only give an upper bound for it, which is the case for the photon.

tRecall that all of the methods that rely on the measurement of a light signal give the group velocity (Fizeau's cogwheel, Kerr's cell, Foucault's rotating mirror, Michelson's interferometer). Besides astronomical aberration, phase velocity appears in the refraction index and in the ratio of the electric and magnetic units. (13) It can also be measured in electromagnetic resonant cavities. (II)

162 GEORGES LocHAK AND REGIS DvTHEIL

E = hv = m(v)F(v) (12)

Here v is the particle velocity. Thus, we only suppose that energy is proportional to mass, but we suppose that the latter-and the unknown factor Fmay be dependent on velocity. Observe that in a homogeneous space and in the absence of field, if mass is not a constant it can only depend on v.

Let us underline that postulates 1 and 3 which describe the essential part of wave-particle dualism were, in de Broglie's work, theorems that were deduced from relativity. But it cannot be the case here, because relativity is not yet included in our hypotheses. However, we will see that relativity is the only possible mechanics, provided an essential condition, without which we could not calculate the functions m(v) and F(v), is fulfilled: the principle of invariance of light velocity in vacuum.

One could assert that this principle is not a relativist postulate but an independent physical fact deduced, for instance, from the observation of double stars, as was shown by De Sitter. All the same, this is the principle which paved the way for relativity.

According to the previous paragraph we know that the first two postulates we introduced lead to de Broglie's wavelength (2) and to the relation (5) we shall need.

Taking into account postulates 1 and 3 and using (12), we introduce (1) and (11) in Rayleigh's formula (9). Hence:

d(mv) d(hv)

d(mv) d(mF)

(13)

from which we find a first equation between the unknown function m( v) and F( v)

d(mF) d(mv) --=v--

dv dv (14)

Notice that if we supposed mass was not dependent on v, we would find:

dF m = em ::;. - = v ::;. F = tv2

dv (15)

and we would come back to the case of classical mechanics, which we excluded earlier. Therefore, we will admit that m(v) is really a function of v.

Let Ro be a reference frame where the particle is at rest and R a reference frame with velocity v with respect to Ro. Space and time are denoted (xo' to) in Ro and (X,f) in R. Consider a stationary wave in Ro:

(16)


With respect to R, it will be a propagating wave with a phase velocity V and a frequency v:

'" = exp 2i'lTV (t - ~)

V is given by (5) which, taking account of (12), yields:

F V=

v

(17)

(18)

Insert this formula in (17) and identify (16) and (17). This gives us the following transformation law for the time variable:

_ m(v)F(v) ( _~) to - hvo t F(v) (19)

In order to find the space transformation, imagine a light signal which travels along the interval (xo' to) in Ro and the corresponding interval (x, t) in R. The velocity of light being the same in both reference frames (this is where the postulate interferes), the time necessary to cover the distance will be equal to

x" x. t = --'L in R and t =- 10 R o c 0 c (20)

Inserting (20) in (19), we get the law of space transformation:

x = m(v)F(v) (x _ vc2t) o hvo F(v)

(21)

It is easily seen that the transformations (19) and (21) form a group and that their composition law implies the following composition law of velocities which already looks like the relativistic one:

v" vIF(v) + Vi IF(v')

1 + [vv l c2IF(v)F(v ' )] (22)

F(v")

As v is the translation velocity of reference frame R with respect to the reference frame Ro' if we compose two transformations of opposite velocities v and Vi = -v, we should obtain the unit transformation (with zero velocity); this implies that the function F(v) is even, because:

V Vi V v F(v) + F(v') = F(v) - F(-v) = 0 from which F(v) = F(-v) (23)

164 GEORGES LocHAK AND REGIS DlrrHEIL

Then, inverting transformations (19) and (21), we find

hvo 1 ( VXn) t= t +--"- . m(v)F(v) 1 - (v2e2fF2) 0 F(v) ,

hvo 1 ( Ve2~) x = x + m(v)F(v) 1 - (v2e2/F2) 0 F(v)

(24)

These inverse formulas should coincide with those which obtained by changing v in -v in (19) and (21); as F(v) is even, this gives a second equation between m(v) and F(v):

m(v)F(v) hvo 1 hvo m(v)F(v) 1 - (v2e2fF2)

(25)

Thus:

(26)

From which follows, taking the derivative:

mF d(mF) = e2mv d(mv) dv dv

(27)

And taking account of (14):

F(v) = e2 (28)

This entails, in virtue of (18), de Broglie's relation between phase velocity and particle velocity:

Vv = e2 (29)

According to postulate 3 this is also, by definition, the relation which links phase and group velocity.

Then according to (28), the composition law (22) for velocities gives the relativistic formula:

v" v + v' e2 1 + (vv' fe2)

(30)

On the other hand, the general law (12) becomes Einstein's law:

E = me2 (31)


If we introduce (28) into (26) we get:

hv me2 = 0

VI - (v2/e2) (32)

From this, using (31) and defining the rest mass by applying Planck's law in the system at rest, there follows the usual relativist dependence of mass with respect to velocity:

m m = 0

VI - (v2/e2) (33)

Finally, introducing (28) and (32) in (19) and (21), we recover the Lorentz transformation:

t - (vx/e2) x - vt to = VI _ (v2/e2); Xo = -yrI=-=(v2:::;:/=e2==) (34)

Let us summarize the argument:

1. We first defined wave-particle dualism through three postulates which were nothing else than Planck's and de Broglie's laws.

2. We then postulated the proportionality between energy and mass in every inertial system.

3. This form admits classical mechanics and relativity as particular cases but it could a priori correspond with many conservative dynamics, which would be in accordance with de Broglie's law for the group velocity. But we have shown that there are actually only two dynamics compatible with all of these four postulates: classical mechanics and relativity. Now the theory of light compels us to reject classical mechanics: thus, there remains only relativity as de Broglie always asserted.

3. A NOTE ABOUT A REASONING OF HAMll..TON

It is well known that Hamilton having defined the surfaces of constant action,

s = enl (35)

noticed that the length of the vector:

p = mv = VS (36)

166 GEORGES LoCHAK AND REGIS DurHEIL

varies as the inverse of the velocity V (Le., the phase velocity) of the wave front defined by (35). Therefore, he called it the vector of normal slowness of the front. (10,14)

This remark led Hamilton to set

1 V=

v (37)

Relation (37) lies at the basis of his analogy between optics and mechanics: specifically, the relationship between corpuscular and wave optics.

Of course, in Hamilton's work group velocity is not mentioned, as the concept was unknown at that time. But we shall see that relation (37) which at first looks like de Broglie's relation (8) is in fact inconsistent with Planck's law and in any case could not lead to wave mechanics. Indeed, Hamilton seemed to consider (37) as a direct consequence of the identification of principles of Maupertuis and Fermat. But this cannot be the case because this would lead to set in (3) and (4):

m = ent and v = ent (38)

Actually, introducing (4) into (3) and eliminating the now superfluous factors m and v, the identity of extremal principles could be written under the form:

18 181 8 vds = 8 -Vds = 0 A A

(39)

This equality (which by the way is inhomogeneous) leads indeed to relation (37) (which is not homogeneous either). But conditions (38) cannot be accepted because:

1. Already in classical mechanics (the Hamilton one) the second equality (38) contradicts Planck's law since it contradicts the equality (6).

2. In relativist mechanics, equalities (38) go against both Planck's law and equivalence of mass and energy.

Therefore, it would be wrong to assert that to derive formula (8) it is enough to identify Maupertuis's and Fermat's principles. However, we have shown at the beginning of this chapter that identification, when properly formulated, leads to the wavelength formula. But as we have seen, the true problem is the one of phase velocity, which forces us to choose relativity.

REFERENCES

1. The Wave-Particle Dualism (S. DiNER, D. FARGUE, G. LoCHAK, and R SELLER!, eds.), Reidel, Dordrecht (1984).

WAVE MECHANICS AND RELATNITY 167

2. L. DE BROGLIE, Thesis (1924). 3. E. SCHRODINGER, Ann. Phys. (4) 79, 489 (1926). 4. L. DE BROGLIE, Theorie de la quantification dans la nouvelle mecanique, Hermann, Paris (1932). 5. L. DE BROGLIE, Nonlinear Wave Mechanics, Elsevier, Amsterdam (1960). 6. L. DE BROGLIE, Les incertitudes d' Heisenberg et l'interpretation probabiliste de la mecanique

ondulatoire, Gauthier-Villars, Paris (1982). 7. D. BOHM, Phys. Rev. 85, 166, 180 (1952). 8. A. BELINFANTE, A Survey of Hidden Variables Theories, Pergamon, Oxford (1973). 9. A. HARUT and M. BOZIC, Ann. Fond. L. de Broglie, 15, 67 (1990).

10. W. R. HAMn..ToN, Mathematical Papers, Cambridge University Press, London (1940). 11. P. DRUDE, Precis d'Optique, Gauthier-Villars, Paris (1911). 12. G. BRUHAT, Optique, Masson, Paris (1965). 13. M. BORN and E. WOLF, Principles of Optics, Pergamon, Oxford (1964). 14. V. ARNOLD, Les methodes mathematiques de la mecanique classique, "Mir", Moscow (1976).

wave-particle duality || wave mechanics and relativity

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