THE MECHANICS OF THE ATOM

MAX BORN

Translated from the German by J. W. FISHER, Ph.D.
and revised by D. R. HARTREE, Ph.D.


FREDERICK UNGAR PUBLISHING CO. NEW YORK

c. 1960


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According to the classical theory, when a resonator oscillates, it
emits an electromagnetic wave, which carries away energy; in
consequence the energy of the oscillation steadily decreases. But
according to the quantum theory, the energy of the resonator remains
constant during the oscillation and equal to n.k\nu; a change in the
energy of the resonator can occur only as the result of a process in
which n changes by a whole number, a "quantum jump."

A radically new connection between radiation and the oscillation of
the resonator must therefore be devised. This may be accomplished in
two ways. We may either assume that the resonator does not radiate at
all during the oscillation, and that it gives out radiation of
frequency \nu only when a quantum jump takes place, there being some
yet unexplained process by which energy lost or gained by the
resonator is given to or taken away from the ether. The energy
principle is then satisfied in each elementary process. Or we may
assume that the resonator radiates during the oscillation, but retains
its energy in spite of this. The energy principle is then no longer
obeyed by the individual processes; it can only be maintained on an
average provided that a suitable relation exists between the radiation
and the probabilities of transitions between the states of constant
energy.

The first conception was long the prevailing one; the second
hypothesis was put forward by Bohr, Kramers, and Slater,^l but new
experiments by Bothe and Geiger,^2 and by Compton and Simon,^3 have
provided strong evidence against it. The investigations of this book
will, in general, be independent of a decision in favour of either of
these two assumptions. The existence of states of motion with constant
energy (Bohr's "stationary states") is the root of the problems with
which we are concerned in the following pages.

§ 2.-General Conception of the Quantum Theory

By consideration of Planck's formula W_0 = h\nu, Einstein was led to
interpret phenomena of another type in terms of the quantum theory,
thus giving rise to a new conception of this equation which has proved
very fruitful. The phenomenon in question is the photoelectric effect.
If light of frequency \tilde\nu falls on a metallic surface,^4
electrons are set free and it is found that the intensity of the light
influences

1 ZeilBchr.f. Physik, vol. xxiv, p. 69, 1924; Phil. Mag., vol. xlvii,
p. 785, 1924.
2 W. Bothe and H. Geiger, Zeietchr. f. Physik, vol. xxxii, p. 639,
1925.
3 A. H. Compton and W. Simon, Phys. Rev., vol. xxv, p. 306,1925.
4 When the symbols \nu and \tilde\nu are employed concurrently,
\tilde\nu always refers to the frequency of the radiation, the symbol
\nu to a frequency within the atom. (Translator's note.)

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the number of electrons emitted but not their velocity. The latter
depends entirely on the frequency of the incident light. Einstein
suggested that the velocity v of the emitted electrons should be given
by the formula

1/2 mv^2 = h\tilde\nu,

which has been verified for high frequencies (X-rays), while for low
frequencies the work done in escaping from the surface must be taken
into consideration.

We have then an electron, loosely bound in the metal, ejected by the
incident light of frequency \tilde\nu and receiving the kinetic
energy h\tilde\nu; the atomic process is thus entirely different from
that in the case of the resonator, and does not contain a frequency at
all. The essential point appears to be, that the alteration in the
energy of an atomic system is connected with the frequency of a
light-wave by the equation

(1) h\tilde\nu = W_l - W_2

no matter whether the atomic system possesses the same frequency
\tilde\nu or some other frequency, or indeed has any frequency at all.
Planck's equation

W = n.W_0; W_0 = h\tilde\nu

gives a relation between the frequency of oscillation \nu of a
resonator and its energy in the stationary states, the Einstein
equation (1) gives a relation between the change in the energy of an
atomic system for a transition from one state to another and the
frequency \tilde\nu of the monochromatic light with the emission or
absorption of which the transition is connected.

Whereas Einstein applied this relation solely to the case of the
liberation of electrons by incident light and to the converse process,
viz. the production of light (or rather X-rays) by electronic
bombardment, Bohr recognised the general significance of this quantum
principle for all processes in which systems with stationary states
interact with radiation. In fact the meaning of the equation is
independent of any special assumptions regarding the atomic system.
Since Bohr demonstrated its great fertility in connection with the
hydrogen atom, equation (1) has been called Bohr's Frequency
condition.

[Einstein's equation has been usurped by Bohr via the then generation
of both physicists, neither Einstein nor Bohr seeming to have anything
to say about it.]

Taking into account the new experiments by Bothe and Geiger, and by
Compton and Simon, which have been mentioned above, we have to assume
that the frequency \tilde\nu is radiated during the transition and the
waves carry with them precisely the energy h\tilde\nu

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(light quantum); there is at present no theoretical indication of the
detailed nature of the transition process.

If Bohr's frequency relation (1) be applied to the resonator we are
faced by alternatives which will now be considered. The change of
energy which takes place when the resonator passes from the state with
the energy n_lh\nu to that with the energy n_2h\nu, viz.:

(n_l-n_2)h\nu,

is, in general, a multiple of the energy quantum, h\nu, of the
resonator. According now to Bohr and Einstein, this change in energy
must be connected with the frequency of the emitted monochromatic
radiation by the equation

h\tilde\nu = (n_l-n_2)h\nu.

This admits of two possibilities only: either we may require that, as
in the classical theory, the radiated frequency shall correspond with
that of the radiator, in which case only transitions between
neighbouring states, for which

n_l - n_2 = 1

are possible, or we may assume that the frequency of the radiation
differs from that of the resonator, being a multiple of it. In the
latter case the emitted light will not be monochromatic, on account of
the possibility of different transitions. The decision between these
two possibilities has been attained in the course of the further
development of Bohr's atomic theory, the conclusion being that the
emitted radiation is strictly monochromatic, with the frequency given
by the condition (1), but that the agreement between the frequency of
the radiation and the frequency of oscillation of the resonator (i.e.
n_l - n_2 = 1) is brought about by an additional principle, which
provides a criterion for the occurrence of transitions between the
different states, and is called the Correspondence Principle.

A fundamental difference between the quantum theory and the classical
theory is that, in the present stage of our knowledge of the
elementary processes, we cannot assign a "cause" for the individual
quantum jumps. In the classical theory, the transition from one state
to another occurs causally, in accordance with the differential
equations of mechanics or electrodynamics. The only connection in
which probability considerations find a place on the older theory is
in the determination of the probable properties of systems of many
degrees of freedom (e.g. distribution laws in the kinetic theory of
gases). In the quantum theory, the differential equations for the
transitions between stationary states are given up, so that in this
case special rules must be sought. These transitions are analogous in

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atoms. In as far as they can be comprehended on the basis of our
theoretical conceptions we can draw conclusions regarding the
processes taking place in the exterior portions of atoms only; they
afford us little or no information about those occurring in the inner
regions. The most important means of investigating the internal
structure of the atom is the study of the X-ray spectra. Our theory of
the motion of an electron in a central field of force is applicable
also to these, since it may be inferred from the observations that we
are here concerned with quantum transitions of the atom in which one
electron (corresponding to the series electron in the optical spectra)
changes its position in the interior of the atom while the rest of the
atom remains approximately a structure possessing central symmetry.

Before we follow out these ideas in detail, we shall give a brief
summary of some of the results of observations on X-ray spectra. Since
the discovery of v. Laue, the natural gratings of crystals have been
available for the analysis of these spectra. Each X-ray spectrum
consists of a continuous band and a series of lines.

The continuous spectrum has a short-wave limit, whose frequency
\nu_max is related to the kinetic energy of the generating cathode
rays by the equation

h\tilde\nu = m/2 v^2.

This result can be looked upon as a kind of converse to the
photoelectric effect, on the assumption that the incident cathode rays
are retarded in the anti-cathode and that their energy is transformed
into radiation according to the Einstein law (§ 2); the highest
frequency emitted corresponds then to the total loss of kinetic energy
of the incident electrons.

The line spectrum is characteristic of the radiating matter, and is
called, therefore, "characteristic radiation." The most important fact
relating to it is that every element exhibits the same arrangement of
lines, and that with increasing atomic number the lines shift towards
the shorter wave-lengths. This line spectrum contains various groups
of lines: a short-wave group (called K-radiation) has already been
found in the case of the light elements (from elements in the
neighbourhood Na and onwards). These become continually shorter for
the heavier elements, and are followed by a group of longer waves
(L-radiation); behind this group follows, in the case of still heavier
elements, a group of still longer wave-lengths (M-radiation).