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Quantics: Rudiments of Quantum Physics

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JMLL/ ‘The meanings of
h
’, Pavia 2000
1
THE MEANINGS OF PLANCK’S CONSTANT
J
EAN
-M
ARC
L
EVY
-L
EBLOND
Physique théorique, University of Nice
Abstract.
This paper purposes to discuss the many roles played by Planck’s constant in our understanding of quantum theory. It starts by retracing the rise of
h
to the status of a universal constant, which was certainly not obvious to every one at the beginning. The rather advanced views of Planck himself in that respect are stressed, particularly concerning the meaning of
h
as a ‘quantum of action’. It is shown that this point of view, applied to statistical considerations, leads to a very interesting expression of the Pauli principle (Heisenberg-Pauli inequalities). The orthodox Copenhagen interpretation, which sees
h
as a ‘gauge of indeterminism’, is then challenged and reformulated in terms of a standard of quanticity. But the deep meaning of Planck’s constant (in keeping with that of any fundamental constant), is to underly the synthesis leading to the new and specific concepts of quantum theory. It is then pointed out that the change from Planck’s constant
h
to Dirac’s
!
, is much more than a convenient shortcut, as it enables one to obtain correct quantum order-of-magnitude estimates by heuristic considerations. Finally, it is stressed that the role of Planck’s constant is by no means limited to the microscopic domain, and examples are given of its manifestations in the macroscopic (human scale) and megascopic (astronomic scale) domains.
Contents
1
Introduction 2
The rise of
h
to universality 3
The quantum of action 4
Gauge of indeterminism or standard of quanticity? 5
The concept synthetizer 6
From Planck’s to Dirac’s constant 7
The macroscopic level
JMLL/ ‘The meanings of
h
’, Pavia 2000
2
1 Introduction
The
value
of
h
is 6,…
!
10
-34
in the
SI
unit system — or
2
!
in the trade system (
!
=
1
). The
history
of
h
starts with Planck’s (quantum) jump in 1900, comes into full light in 1905 with Einstein, and takes momentum in 1924 with de Broglie. The
role
of
h
is crucial in many areas of physics, from particle physics to solid state. But what is to be said of its
meaning
, or rather, as I wish to stress the plural, of its
meanings
? Questions of meaning are often dismissed in a rather patronizing way by working physicists as “mere philosophy”, as opposed to serious research, that is, theoretical formalism and experimental results. Yet, the purpose of science is (or should be) not only to predict or verify numbers and facts, but to assess and elaborate ideas, without which there is no real “understanding”. It is crucial in that respect to recognize that almost any theoretical notion of physics, and certainly each important one, is prone to various interpretations and is endowed, in the course of time, with widely different meanings, or even with conflicting ones. This can be seen most clearly in the terminological diversity associated with some of these notions, and the ensuing confusion. A worse situation still is that of ancient ways of speaking, kept just out of mental laziness, while their initial strict meaning has long been forgotten. Think for instance of the so-called ‘displacement current’ in Maxwell equations (which was indeed for Maxwell — at the beginning — a true electric current in the ethereal substance, while we now think of it as a genuine field term), or consider the so-called ‘velocity of light’ (the fundamental role of which as a scale standard for the structure of space-time would amply justify a less specific name — for instance, Einstein constant). The importance of these remarks lies in the obvious but overlooked fact that neglecting the diversity of meanings and failing to discuss it explicitly may have strong adverse effects, not only on the spreading of knowledge (in teaching — see the example of the displacement current, or popularizing science — see the example of the velocity of light), but on its development as well; one does not know in advance which of the competing views, if any, will show the greatest fecundity. I now come to the case of Planck’s constant, for which I will discuss, without any claim to completenesss, a handful of the very different meanings it has been given. The argument will be set up in the framework I have proposed for understanding and classifying physical constants in general
1
. One has first to distinguish between fundamental constants, which are to be taken as basic elements of our knowledge, and derived, or phenomenological ones, which we know to be explainable (in priciple at least) from the fundamental ones; the electron charge or Newton’s constant clearly belong to the first category, while Rydberg’s constant or the proton mass (related to the quark masses and coupling constants) belong to the second one. The fundamental constants in turn may be classified under three headings: 1) specific properties of particular objects (say, the electron mass), 2) characteristics of whole classes of phenomena (say, the elementary electric charge which measures the strength of electromagnetic interactions), 3) universal constants which enter universal theories, ruling
all
physical phenomena (say, the limit velocity). I should stress that these distinctions should by no means be taken as giving a closed and atemporal classification; quite the contrary, they permit a detailed discussion of the historical changes in the role and meaning of the constants, as will now be seen.
2 The rise of
h
to universality
Planck’s constant nowadays is clearly taken as a fundamental constant, and even a universal one, since quantum theory is thought to be universally valid; accordingly, there is no doubt that
h
in principle enters the treatment of any physical phenomenon. In many cases, a classical approximation leading us to forget its underlying presence is possible, but that is
1
Jean-Marc Lévy-Leblond, ‘On the Conceptual Nature of the Physical Constants’,
Riv. Nuovo Cimento
7 (1977), 187.
JMLL/ ‘The meanings of
h
’, Pavia 2000
3
another question (and a difficult one at that, with some surprises in store as will be seen at the end of this paper). Now, this universality was far from characterizing
h
at the beginning of its life. The constant
h
first appeared in Planck’s paper on the blackbody spectrum. It is perhaps not without interest to ask first why Planck chose to denote his new constant by this very letter; such questions, concerning the choices of symbols in the formalism of physics, certainly deserve more attention than they usually receive, as the answers could shed some light on the intellectual (and sometimes psychological) processes at the core of theoretical innovation. In the present case, it must be remarked that in its first printed occurrence (in october 1900), Planck’s formula is written in terms of the wavelength :
e
!
=
C
!
"
5
(
e
c
' /
!
T
"
1)
"
1
, (1) with constants
C
and
c’
(taken from the previously known Wien’s formula) having a purely empirical meaning. It will take a few weeks for Planck to discover a theoretical explanation of this formula
2
. He considers the entropy of a ‘resonator’, starting from the statistical formula
S
=
k
ln
N
(2) where the now so-called ‘Boltzmann constant’ appears in physics
for the first time
, and proceed to compute
S
under the hypothesis that the exchange of energy between radiation and matter is quantized with an ‘Energieelement’ given by:
!
=
h
"
, (3) leading at last to the formula
u
!
=
8
"
hc
#
3
v
3
(
e
h
!
/
kT
#
1)
#
1
. (4) It is to be stressed that Planck introduced the
two
constants
h
and
k
. Later on, he would repeatedly complain that ‘Boltzmann’s constant’ was in fact the
other
Planck’s constant. In any case, the simultaneous appearance of
h
and
k
sheds some light, although a rather trivial one, upon the process of their denomination; it looks as if Planck (who seems never to have given any explanation whatsoever on this point) just chose the two first (and related) letters which did not yet bear too heavy a symbolic role in the accepted conventions of physics. What is more important for the theme of our discussion, is that Planck, commenting upon the expression (3) of the ‘energy element’ and the general formula (2) for the entropy, explicitly stated that “Hierbei sind
h
und
k
universelle Constante.” Contrarily to the usual view of Planck as rather old-minded and reluctant towards the modern aspects of quantum theory (an assessment which could be argued to reason considering his positions in the following decades), this statement about the universality of
h
and
k
certainly proves a more advanced stand than that of most of his contemporaries, for whom the fundamental nature of
h
was far from obvious, not to speak of its universality. Indeed, it must be remembered that for quite a few years,
h
appeared only in considerations related to radiation theory, from the blackbody spectrum (Planck, 1899-1900) to the photoelectric effect (Einstein, 1905). It was only natural, then, to think of
h
as specifically ruling electromagnetic phenomena. In fact, not until Einstein’s 1907 paper on the specific heat of solids, giving a first example of quantum statistical theory, did extend effectively its realm beyond radiation theory. It is all the more interesting to note, first, that Einstein himself, in his 1905 paper
3
, uses neither the expression ‘Planck’s constant’ (which was used for the first time rather late, probably by Millikan around 1915), nor even a specific symbol! Although he of course refers to this srcinal paper by Planck, he does not write the ‘Plancksche formel’ (4), but expresses it in the more empirical form:
!
"
=
#"
3
(
e
$"
/
T
%
1)
%
1
(5) where neither
h
nor
k
appear. It is even more striking to look at Einstein’s formula for the photoelectric effect, since he writes:
2
Max Planck, ‘Ueber das Gesetz der Energieverteilung in Normaspektrum’,
Ann. d. Phys.
4 (1901), 553.
3
Albert Einstein, ‘Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristichen Gesichtspunkt’,
Ann. d. Phys.
17 (1905), 132.
JMLL/ ‘The meanings of
h
’, Pavia 2000
4
“Die kinetische Energie solcher Elektronen ist
(
R
/
N
)
!"
#
P
.” (where
P
is the extraction potential). This hiding away of
h
can certainly be taken as displaying at least some skepticism concerning the relevance of the theoretical derivation by Planck, as well as about the fundamental role of the constants
h
and
k
. The idea was rather common in these days that
h
was not a fundamental constant, expressing the inception of a radically new theory, and that it could be explained away by some mechanical model leading to a more or less classical explanation of the quantization of energye exchanges. Born later on recalled the ‘apple-tree model of Planck’s quantization formula’, an admittedly farcical model, which was discussed at the time
4
. Imagine an apple-tree with the property that the stems of the apples decrease with their height above ground; more specifically, let us suppose that the length
l
of the stems is inversely proprtional to the square of their height
H
, that is,
l
!
H
"
2
. Then, the frequency of free oscillations of the apples considered as pendula is
!
=
(2
"
)
#
1
(
g
/
l
)
1 / 2
$
H
. Suppose now that the tree is shaking in the wind. Pressure waves with frequency
!
will excite specifically the oscillations of those apples which lie at the corresponding height
H
!
"
. These apples only will fall to the ground, transferring to it an energy
E
=
mgH
!
"
; in other words, the exchange of energy between the tree and the ground is quantized in apple-units with an energy
E
=
h
!
, where the constant
h
may be expressed in terms of more fundamental quantities (the apple mass, the acceleration of gravity, and the constant of the length-height relationship). In a more serious vein, there were several quite explicit attempts to relate
h
with suposedly more fundamental magnitudes of the atomic realm, like the Haas model based on the Thomson atom (which was favorably quoted by Sommerfeld as late as 1911). Einstein himself made a start in the same direction in 1909. He remarked that
h
had the same physical dimensionality as the combination
e
2
/
c
and looked upon this coincidence as indicating the possibility of a specifically electromagnetic mechanism
5
(note that this happened the very same year when he himself showed, through his theory of the specific heat of solids,
h
to have a general relevance, beyond purely electromagnetic phenomena!). Lorentz, in a letter dated 6 May 1909, expressed serious doubts, based upon the numerical gap (three orders of magnitudes) between the two expressions, writing “I could imagine to have a factor of
4
!
or so intervening, but a factor 900, that is really too much.” To what Einstein carefreely answered that “a factor like
6(4
!
)
2
is not so extraordinary.” But unless someone produces a theory of the fine structure constant
!
yielding its numerical value, thus giving
h
as equal to
2
!"
(
e
2
/
c
)
, but above all explaining its seemingly ubiquitous role (beyond electromagnetism), we have better think of Planck’s constant, in the very terms of Planck himself, as a universal constant. However, within this general conception, there remains a variety of possible views, as will now be seen.
3 The quantum of action
A first concrete understanding of
h
as a universal constant was put forward by Planck himself. He had for long noticed that
h
had the dimensionality of classical ‘action’. In his contribution to the 1911 Solvay conference, he introduced the very fecund idea that
h
in fact defined the magnitude of irreducible quantities of action, “elementaren Wirkungsquanten” in Planck’s srcinal terms. There was born the wording ‘quantum of action’ for
h
. This new vision of quantization was in turn expressed as the existence of elementary areas in phase space, that is, cells
A
with finite extension:
dpdq
!
h
A
"
. (6) Planck was clearly aware that this point of view called for a renunciation to all attempts at classical interpretations of
h
(of the type mentioned above). He wrote, in his Solvay paper: “The framework of classical dynamics, even if combined with the Lorentz-Einstein principle
4
See Max Jammer,
The Conceptual Development of Quantum Mechanics
(McGraw Hill, 1966).
5
Albert Einstein, ‘Zum gegenwärtigen Stand der Strahlungsproblem’,
Phys. Zeits.
10 (1909), 185.
JMLL/ ‘The meanings of
h
’, Pavia 2000
5
of relativity, is too narrow to account for all those physical phenomena which are not directly accessible to our coarse senses.”
6
Planck’s idea was seized upon by Sommerfeld, who, inspired by what he called the “most fortunate” naming of
h
as a quantum of action, related it more precisely to Hamilton’s action function which enabled him to develop the Old Quantum Theory. Later on, the formal development of quantum theory was to be built upon another, more general, meaning of
h
(see below). Nevertheless, despite the limitations of the ‘quantum of action’ point of view, it has not lost its fecundity, and can be put to good use, at least as a heuristic tool for understanding some aspects of quantum behviour. As an example, it leads to a very picturesque and useful way of expressing the Pauli principle, as I now proceed to show.
The Heisenberg-Pauli inequalities
Consider a system of
N
one-dimensional particles, classical ones to start with. The state of the system can be represented at a given time, by a collection of
N
points in the two-dimensional one-particle phase-space
(p,q)
. Contemplate now a system of
N
quantons (quantum ‘particles’). In accordance with Planck’s idea about the quantum of action, an individual state can no longer be represented by a point, but is to be associated to an extended cell with an area of order
h
, or rather
!
; indeed, as will stressed below, one should definitely
use the Dirac constant
!
:
=
h
/ 2
!
as soon as numerical evaluations are contemplated. Denoting by
!
p
and
!
q
respectively the extension of an individual cell in momentum and position respectively (in order of magnitude), the expression (6) for the minimal area of the cell constitures but a rewriting (and an interpretation) of the Heisenberg inequality:
!
p
!
q
! "
(7) (where the symbol
!
denotes inequality up to a constant numerical factor of order unity). The collective state then consists of an ensemble of
N
such cells. Let us suppose now that the quantons are identical fermions. The effect of Fermi statistics may be expressed by the Pauli exclusion principle which requires that no two individual states be identical. It is traightforward to translate this constraint as requiring the
N
cells now to be disjoint (Figure 2c). One sees vividly how the Pauli principle requires the collective state of the system to have a much greater minimal extension in phase-space. Indeed, for a system with
N
quantons
not
obeying the Pauli principle, the inequality (7) for the individual regions in phase-space applies as well to the total region, since nothing prevents the individual states to be one and the same. For identical fermions however, the total region, consisting of non-overlapping cells, must have a minimal area at least equal to
N
times that of an individual cell, leading to an inequality characterizing the collective extensions in position and momentum of a system with
N
identical fermions in one dimension:
!
p
!
q
!
N
"
. (1-D) (8) I call it the ‘Heisenberg-Pauli inequality’. It can easily be generalized to three (or more…) dimensions. The individual cells now have a minimal volume
!
3
and the collective state with
N
identical fermions must occupy a region of volume at least
N
!
3
. Hence the Heisenberg-Pauli inequality in three dimensions:
!
p
!
q
!
N
1/ 3
"
. (3-D) (9) It should be emphasized that this heuristic derivation may be given a fully rigorous form, and that the inequality (9) may be proved formally
7
. The best way to understand the Heisenberg-Pauli inequalities (8) and (9), is to compare them to the standard Heisenberg inequality (7), and to consider that the effect of the Pauli principle is to replace the fundamental quantum constant
!
by an effective fermionic one,
!
f
(
d
)
(
N
)
=
N
1/
d
!
, where
d
is the dimensionality of space (which shows, by the way, that the effect of the exclusion principle is all the more powerful, the lower is the dimensionality of space).
6
Max Jammer,
op. cit.
7
Jean-Marc Lévy-Leblond, ‘Generalized Uncertainty Relations for Many-Fermion Systems’,
Physics Letters
26A, 540 (1968).

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