100 years of planck's quantum || de broglie waves
TRANSCRIPT
Chapter IV
de BROGLIE: "Le principe de Fermat applique a l'onde de phase est identique au principe de Maupertuis applique au mobile; les trajectoires dynamiquement possibles du mobile sont identique aux rayons possible de l'onde."
"Fermat's Principle applied to the phase of the wave is identical to Maupertuis' Principle applied to the motion; the trajectories dynamically possible for the motion are identical to the possible rays of the wave." Ann. de Phys., 10e serie, t. Ill (Janvier-Fevrier 1925), p.56.
§ rV-1. Introduction. de Broglie's discovery of the dual wave-particle nature of massive material
particles - generalizing and completing Planck's introduction of the light quantum - was unveiled in his 1924 doctoral thesis [2,3] with marvelous depth and sophistication. One must be astounded at the profound consequences which de Broglie deduced from the elemental observation that - contained within the Einstein-Lorentz transformation of space/time and momentum/energy - there is an invariant phase associated with a moving particle, suggesting a wave associated with the motion. But then de Broglie took the courageous creative leap to give primacy to the wave motion and to search in reverse for particle-like attributes in the wave propagation.
It would have been easy for de Broglie to identify - as we now do - a plane wave with a particle in force-free motion [4], but he took a much more powerful and general qualitative approach to the question of particle trajectories from the point of view of the associated wave, de Broglie's far reaching answer, based on Fermat's Principle of ray optics, supposed that the rays of a wave field of given frequency or, equivalently, energy, in a medium of variable refraction must coincide
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Chapter IV. de Broglie Waves 129
with the possible trajectories of the associated particle. Somewhat unfortunately, de Broglie did not deal in specifics. If he had played with simple waves in simple situations, he was poised to anticipate Schrodinger and to invert the actual order of the development of quantum mechanics.
de Broglie's thesis is a treasure trove of results made possible by his Nobel Prize winning introduction of wave-particle duality. He anticipates the Bose derivation of Planck's distribution and the Bose-Einstein quantum gas for material particles in great detail, albeit without mentioning the condensation phenomenon. He was able to derive the Bohr-Sommerfeld quantization condition § pdq = nh. He gave a first explanation of interference and coherence properties of quanta. He gave an elegant covariant derivation of Compton scattering from moving electrons.
de Broglie wandered from his pioneering role and devoted his life to advocacy of his own minority view of wave-particle dualism which finds its modern realization in Bohm's guiding waves accompanying point like material particles [5]. We will return to these minority views in a 1955 review by Heisenberg, who considers them to be a useless excursion to develop an intuitive scenario for quantum mechanics with the express design to agree in every observable result with the more abstract interpretation of the mainstream views (created in large part, of course, by Heisenberg himself).
Footnotes and References: 1) N. Bohr, H.A. Kramers and J.C. Slater, Zeits. f. Phys. 24, 69 (1924) (here as Paper III2) 2) L. de Broglie, Phil. Mag. 24, 446 (1924) (here as Paper IV1). 3) L. de Broglie, Annales de Physique, 10e serie, t. Ill, 22 (1925). 4) In this regard, we refer to footnote [c] following Compton's Biographical Note in Chapter 2; and also Brown and Martins in [5] immediately below. 5) For an accessible critique, see: H.R. Brown and R.de A. Martins, Am. J. Phys. 52(12), 1130 (1984).
Biographical Note on de Broglie: Louis de Broglie (1892-1987) - won the 1929 Nobel Prize for Physics for his
extension of the notion of wave-particle duality to material particles, paving the way especially for Schrodinger's invention of his wave mechanics, an intuitively ac-
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130 100 Years of Planck's Quantum
cessible formulation of quantum mechanics parallel and equivalent to Heisenberg's slightly earlier matrix mechanics. From the London Times (20Mar87) and the NYTimes (21Mar87), we learn that de Broglie was first educated in history, but under the influence of his brother Maurice, seventeen years his senior and already a prominent experimental physicist, he attended the 1911 Brussels Conference on Planck's quantum theory and was inspired to study to physics. He graduated in 1914 in time to serve as a radio-signals officer in the French army of WWI. After graduate studies in the Faculte des Sciences at the Sorbonne, he submitted his 1924 thesis Recherches sur la theorie des quanta which is reproduced at length in the accompanying P a p e r IV-1. He became Professor of Theoretical Physics at the Institut Henri Poincare (1932-62), member (1933) and permanent secretary (1942-75) of the Academy of Sciences, was elected to the Academy Francaise (1945) where he was inducted by his brother, and where his father and grandfather had been members.
Prince Louis Victor Pierre Raymond de Broglie [a], born at Dieppe, of a conservative military noble family originally from the Piedmont in Italy in the 17 century. The family has produced marshals, governors, ministers of s tate and his grandfather a Prime Minister of France in the 1870's. With a courtesy title Prince most of his life, Louis de Broglie succeeded his brother as 1th due de Broglie in 1960.
The family was established by Francois-Marie Broglia (1611-1656) [b], count of Revel in the Piedmont, who rose in French service to be governor of the Bastille and a marshal, and was able to purchase the marquisat de Senonches (for a million livres, on an annual pension of 12000 livres) before being killed in the siege of Valencia on 2 July 1656. His older son Victor-Maurice became the count de Broglie, and Victor's third son Francois-Maurice, also a marshal, became count and then the l s ( due de Broglie.
Among Louis de Broglie's namesakes one finds [c] his great-great-grandfather Victor-Francois, 2nd due de Broglie (1718-1804) [d], marshal, son and grandson of marshals, in action at 15, cavalry commander at 16, general at 24, marshal at 40. Minister of War for Louis XVI in 1789, he was forced to admit that his troops were unable to guarantee the king's safety and were unreliable to oppose the revolution.
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Chapter IV. de Broglie Waves 131
At the head of an army of Prussians and emigres, he invaded Champagne in 1792. Finally, on the verge of a reconciliation with Napoleon, he died in exile at Munster. His son Charles-Louis-Victor (1756-94) served in the American war of independence, and later as a colonel on the staff of Metz in the Bourbon army. As deputy to the constituent assembly representing the nobility, he defended the cause of the people and frequently voted with the left. He was president of the assembly in 1891, and then returned to active service in the army of the Rhine under the command of Luckner. He refused to support the death sentence of the king in 1792, retiring instead. He was soon arrested, imprisoned, released for a short time, rearrested, tried by the revolutionary tribunal, condemned, and immediately executed. His brother, the physicists' great-grandfather, Victor-Amedee-Marie (1785-1870), 3 r d due de Broglie, fought against the revolution with the 'Army of the Princes' eventually as colonel of the White Rose regiment, in campaigns against France in years IV and V of the revolution, became Chavalier de Saint-Louis in year VII, and then Gentleman of Honor to the Duke of Angueleme. Returning to France, he refused to serve Napoleon, suffered electoral defeat after the restoration and retired to his chateau at Ranes. Victor's son Albert (1821-1901), 4th due de Broglie [e] and grandfather of the physicists Maurice and Louis, demonstrated the rightist politics of his father and the literary brilliance of his grandmother, Madame de Stael [f]! He was a prolific and intellectual writer on a wide range of subjects - from Leibnitz to the later Roman Empire. Elected to the National Assembly in 1871, he helped overthrow the conservative republican government, served as prime minister (1873-74), supported the monarchist cause of Philippe d'Orleans, but was finally defeated in a second term in 1877, and with him the monarchist delusions ended. His political defeat stifled the political career of his son Louis Alphonse Victor (1846-1906), father of the physicists and 5th due de Broglie, who had served in his father's cabinet but retired and only reappeared as a deputy in 1893.
These were the aristocratic origins of the man who gave us de Broglie waves. The name de Broglie cannot be correctly pronounced by native English speak
ers, but an acceptable approximation evidently is "duh-BROY-ee". The debate seems to revolve around a subtle gargling sound centered on the u-gl-", which is
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132 100 Years of Planck's Quantum
possible only for native French speakers; and even amongst them there is no precise agreement. The name evolved from the Italian Broglio and has been spelled Broglia, Broglie, and even Broille which reflects the way it is often pronounced.
Footnotes and References: a) Who's Who in France 1983-1984 (Jacque Lafitte S.A. 75008 Paris, France, 1982), p.220. b) Dictionnaire du Grand Siecle (Fayard Press, Paris, 1990) Francois Bluche, ed., p.241. c) Dictionnaire Historique et Bibliographique de la Revolution et de Empire 1789-1815 (Paris, Kraus Reprint, Nedeln/Lichtenstein), p.284. d) Historical Dictionary of the French Revolution 1789-1799 (Greenwood Press, Westport CT, 1985) S.F. Scott and B. Rothaus, eds., p.127. e) Historical Dictionary of the 3rd French Republic 1870-1940 (Greenwood Press, Westport CT, 1986) P.H. Hutton, ed., p.138. f) Les Broglie (Paris, Fasquelle, 1950), p.160.
P a p e r IV-1: Excerpt from Phil. Mag. 47 , 446 (1924).
XXXV. A Tentative Theory of Light Quanta
by LOUIS DE BROGLIE*
I . The Light Quantum.
The experimental evidence accumulated in recent years seems to be quite conclusive in favor of the actual reality of light quanta. The photoelectric effect, • • • the recent results of A.H. Compton • • • Bohr's theory • • • tha t atoms can only emit radiant energy of frequency / by finite amounts of energy hf ■ ■ •
I shall assume the real existence of light quanta, and try to see how to reconcile it with the strong experimental evidence on which the wave theory was based.
II. The Black Radiation as a Gas of Light Quanta.
This is obviously Wien's limiting form of the radiation law. Two years ago [1] I was able to show that it was possible, by using the hypothesis made by Planck that the unit phase-space volume was dxdydzdpxdpydpz/h3, to find for the
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Chapter IV. de Broglie Waves 133
radiant energy density the value
ufdf=^f3e-h""rdf.
This was an encouraging result, but not quite complete. The assumption of the unit of phase-space volume seemed to have a somewhat arbitrary and mysterious character. Moreover, Wien's law is only a limiting form of the actual radiation law, and I was obliged to suppose some kind of quanta aggregation for explaining the other terms of the series.
It seems that these difficulties are now removed, but we shall first of all explain many other ideas; we shall later on return to the "black radiation" gas.
I I I . An Important Theorem on the Motion of Particles.
Let us consider a moving particle of rest mass mo, moving with respect to a given observer with velocity v = j3c (/3 < 1), and containing internal energy TUQC2 . The quantum relation suggests that we ascribe to this internal energy a wave of frequency /o = TUQC2 /h. For the fixed observer, the whole energy is moC2/\/l — /32
and the corresponding frequency is
1 TOQC2 / _ / o J ~ h yr^F v v 7 ! ^ ,
But if the fixed observer is looking at the internal period, he will see its frequency lowered and equal to f\ = /o \ / l - P2, so the wave seems to him to vary as sin 2-KJ\t. The frequency f\ is widely different from the frequency / ; but they are related by an important theorem which gives us the physical interpretation of / .
Suppose that at time t = 0 the moving particle coincides in space with a wave of the frequency / given above and which spreads with a phase velocity
c c2
V*=li= — p v
According to Einstein's ideas, however, this wave cannot carry energy.
Our theorem is the following: - "//, at the beginning, the internal wave of the moving particle is in phase with the spreading wave, this phase agreement will
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134 100 Years of Planck's Quantum
always persist." In fact, at time t, the moving particle is a distance x = vt from the origin and its internal wave is sin 2ivfix/v\ at the same place the expanding wave is
sir in 2*/(*-£)= sin 2,/* ( i - £ ) . The two sines will be equal and the agreement of phase will always occur for fi and / as defined above:
/» = / (i - P2) (= M / I - / ? 2 )
This important result is implicitly contained in the Lorentz time transformation. If r is the local time of the moving particle, the internal wave is sin 2TV/OT. By the Lorentz transformation, the fixed observer describes the same wave by
s in2?r /vrM'-?)' which can be interpreted as a wave of frequency / = / o / \ / l — /32 spreading along the x axis with phase velocity Vj, = c/0.
We then recognize that any moving particle can be associated uniquely with a propagating wave.
* * * * * * * * * * * * * * * * * * * * In more familiar and contemporary terms (with h and c set to one), we associate with a particle of mass m moving with velocity v (energy E = m/y/1 — v2, and momentum p = \jE2 — m2 = Ev) the Lorentz invariant de Broglie phase $ = (Et — px). From this we recognize the phase velocity V̂ = E/p and the group velocity Vg = dE/dp = p/E = v. de Broglie's three equivalent expressions for the phase are
and in the particle rest frame, with (Et — px) = mr,
2nf\t = mr = 2wfoT.
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Chapter IV. de Broglie Waves 135
But moving clocks run slow, so r = t v l — u2, and we get de Broglie's result
* * * * * * * * * * * * * * * * * * * *
This idea can also be expressed in another way. A group of waves whose frequencies are very nearly equal has a group velocity Vg, which is the velocity of energy propagation. This group velocity is related to the phase velocity V^ by
vg df [v*
if 1 m0c2 c
f=h7T^W and V* = H' we find Vg = f3c - that is, "The velocity of the moving particle v = /3c is the energy velocity Vg of a group of waves having frequencies
f = -—. and phase velocities VA — —,
corresponding to very slightly different values of /?."
* * * ♦ * ♦ * ♦ * * ♦ * ♦ ♦ * ♦ * * ♦ *
Here de Broglie introduces the concept of wave-particle duality for material particles.
* * * * * * * * * * * * * * * * * * * *
IV. Dynamics and Geometric Optics.
To extend these ideas to the case of variable velocity is a difficult but suggestive problem. If a particle moves on a curved path, we say that there is a field of force; at each point the potential energy can be calculated, and the particle when crossing this point has a velocity determined by the constant value of its total energy. It seems natural to suppose that the phase wave at any point must have a velocity and a frequency fixed by the value which 0 would have if the particle were there. During its propagation the phase wave has a constant frequency / and
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136 100 Years of Planck's Quantum
a continuously variable phase velocity V .̂ it seems that we already know the final result:" The rays of the phase wave are identical with the paths which are dynamically possible." In fact, the paths of the rays can be computed as in a medium of variable dispersion by Fermat's Principle:
J J X J V* J h^/T^J*
in agreement with the Maupertuis form of the Principle of Least Action which gives the dynamical path of the particle by the equation
;Jfit_hiT , sJ^f^-JT^) dt
confirming the above theorem on the phase coincidence.
This theory suggests an explanation of Bohr's quantization condition. At time t = 0 the electron is at point A of its trajectory. The phase wave starting at this instant will propagate around the orbit and meet the electron again at A, where it must again be in phase with the electron. That is to say: "The motion can only be stable if the phase wave is compatible with the length of the path." The requirement is:
ds [T m0P2c2 . =dt = n, J X Jo
with n a positive integer and T the period of the motion.
V. The Propagation of Light Quanta and the Coherence Problem.
• • • The light quantum is in some manner a part of the wave, but for explaining interferences and other phenomena of wave optics it is necessary to see how several light quanta can be parts of the same wave. This is the coherence problem.
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Chapter IV. de Broglie Waves 137
* * * * * * * * * * * * * * * * * * * * de Broglie here seems not quite ready to admit the dual character of quanta - particle and wave - and interference of a photon only with itself; but then immediately (in the next section, in italics) makes the leap. * * * * * * * * * * * * * * * * * * * *
V I . Diffraction and the Inertia Principle.
Here the theory of light quanta meets a great difficulty, known since the time of Newton. The light rays passing near an edge are no longer straight but penet ra te into the geometric shadow. Newton ascribed this deflection to the action of some force exerted by the edge on the light corpuscle. It seems to me that this phenomenon deserves a more general explanation. Since an intimate connection exists between the motion of particles and the propagation of waves, and since the rays of the phase wave are the possible paths of the energy quanta, we are inclined to give up the inertia principle and to say: "A moving particle must always follow the same ray of its phase wave." In the continuous spreading of the wave, the surfaces of equal phase will change continuously and the particle will always follow the common perpendicular to two infinitely near surfaces.
When Fermat 's Principle is valid for computing the ray path, then the Principle of Least Action is also valid for computing the particle path. These ideas are a synthesis of wave optics and particle dynamics.
the ray - which assumes an important physical significance - is defined by the continuous spreading of a small part of the phase wave: it cannot be defined • • • by the • • • "energy or Poynting's vector." • • •
V I I . A New Explanation of Interference Fringes.
Consider how we detect light at a point in space • • • It seems tha t all these means can, in fact, be reduced to photoelectric actions • • •
Next consider Young's Interference Experiment. Light quanta pass through the holes and diffract along the rays of the neighboring parts of their phase waves. In the space behind the wall, their capacity for photoelectric action will vary from
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138 100 Years of Planck's Quantum
point to point depending on the interference state of the two phase waves from the two holes. We will see interference fringes however few the number of diffracted quanta and however small the incident light intensity. (Note: Italics added.) The light quanta do cross all the dark and bright fringes; only their ability to act on matter is constantly changing. This explanation, which removes the objections against light quanta and against energy propagation through dark fringes, may be generalized for all interference and diffraction phenomena.
♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ■ f t
de Broglie presents the quantum interpretation of interference phenomena.
* * * * * * * * * * * * * * * * * + + *
VII I . The Quanta and the Kinetic Theory of Gases.
To calculate the absolute entropy, Planck and Nernst were forced to introduce the quantum idea into the kinetic theory of gases. Planck assumed a phase-space volume element equal to
■j-^dxdydzdpzdpydpz or T T ^ O V^wdwdxdydz.
We shall now justify this assumption.
Each atom of velocity v = (5c is equivalent to waves of phase velocity V-^ = c/(3, frequency / = m0c2/h^/l - /J2, and group velocity Vg = 0c. The state of the gas can only be stable for standing waves. Following Jeans, we find the number of waves per unit volume with frequencies in the interval / to / ■+- df to be:
i!!_/V=—"'2-v0vJfdJ c3' n<df=v^fdf=^Pfdf-If w is the kinetic energy of an atom and / the corresponding frequency, then:
moc2 2 2
hJ ~ /■, ^ =w + moC = n*oc (1 + a), vi - P
with a = w/m0c2. Then njdf is given by:
n/df = —mlc(l + a)yja(2 + a)dw
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Chapter IV. de Broglie Waves 139
Each phase wave can carry with it one, two, or more atoms, so that, according to the canonical law, the number of atoms whose energy is hf, will be proportional to:
njdfdxdydz ( £ e-nhj/kT\
Consider first a classical gas whose atoms have large mass and small velocities, so we can ignore all terms in the sum except the first, and can set 1 + a — 1. The number of atoms whose kinetic energy is w will be proportional to
^m30/2V2^dwdxdydze-w/kT,
which justifies Planck's method and leads to Maxwell's distribution.
In the case of light quanta, a is large and we must keep the whole series, and also double the result to include both polarization states. We find the radiant energy density proportional to:
C2 ehf/kT_ ia->-
A method developed in the Journal de Physique, of November 1922, shows that the proportionality factor is unity, so that we obtain the actual radiation law.
♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ i t * * * *
de Broglie anticipated Bose's derivation of the Planck distribution (Zeits. f. Phys. 26, 178 (1924)) by some two years, as partially acknowledged by Einstein (S.B.d. Preuss. Akad. Wiss. Ber. 22, 261 (1924)). Einstein, however, referred only to de Broglie's 1924 Thesis. * * * * * * + * * * * * * * * * * * * *
IX. Open Questions.
The ideas stated in this paper, if correct, will require a wide modification of electromagnetic theory. The so called "electric and magnetic energies" must be only an average value, all the real energy of the fields being of a fine-grained quantum structure. The construction of a new theory seems to be a very difficult task,
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140 100 Years of Planck's Quantum
but we have one guide: according to the correspondence principle, the defining vectors of the old theory should give the probability of reactions in the fine-grained theory.
There seems to be a great analogy between the scattering of radiation and the scattering of particles explaining optical dispersion will be more difficult • • • What occurs when an atom passes from a stable s tate to another, and how does it eject a single quantum? How can we introduce the • • • quantum • • • into • • • elastic waves and into Debye's theory of specific heats?
Finally, we must remark tha t the quantum remains a postulate defining the constant h whose actual significance is not at all cleared up; but it seems that the quantum enigma is now reduced to this unique point.
♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ H i
Where it remains still. ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦
Summary.
It is assumed that light is made up of quanta. It is shown that the Lorentz-Einstein transformation together with the Planck quantum relation leads us necessarily to associate particle motion and wave propagation, and tha t this idea gives a physical interpretation of Bohr's quantization condition. Diffraction is shown to be consistent with an extension of Newtonian dynamics. It is then possible to have both the particle and the wave character of light, and, by means of hypotheses suggested by electromagnetic theory and the correspondence principle, to give a plausible explanation of coherence and interference fringes. Finally, it is shown why quanta must take part in the kinetic theory of gases and how Planck's distribution is the limiting form of Maxwell's distribution for a gas of light quanta.
Many of these ideas may be criticized and perhaps modified, but little doubt can now remain of the real existence of light quanta. Moreover, if our ideas are correct, based as they are on the relativity of time, all the enormous experimental evidence of the "quantum" will support Einstein's conceptions.
1 October, 1923.
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Chapter IV. de Broglie Waves 141
Note.- Since I wrote this paper, I have found a much more general result. The Principle of Least Action for a point particle can be written in the space-time notation as : -
r 4
6/22 Jidx' = 0, J l
where the J{ are the four-dimensional energy-momentum vector of the particle.
Similarly, for the propagation of waves, we write:-
t 4
6 / £©,•<**'= 0, ^ l
where the 0{ are the covariant components of the four-dimensional vector whose time component is the frequency f/c, and whose space components are a vector along the ray, of magnitude f/V^ = 1/A (Vj, is the phase velocity). The quantum condition says that
J4 = h04.
More generally, I suggest putting
/ = he.
From this statement, the identity of the Fermat and Maupertuis s tatements of the Principle of Least Action follows immediately, and it is possible to deduce rigorously the velocity of the phase wave in any electromagnetic field.
* - Communicated by R.H. Fowler, M.A.
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