410 THE QUANTUM THEORYTechnically, the following issue arises. If a molecule emits or absorbs an
amount e of radiative energy all of which moves in the same direction, then it
experiences a recoil of magnitude (./ c. There is no recoil if the radiation is not
directed at all, as for a spherical wave. Question: What can one say about the
degree of directedness of the emitted or absorbed radiation for the system under
consideration? Einstein began the discussion of this question in the same way he
had treated the mirror problem in 1909. Instead of the mirror, he now considered
molecules that all move in the same direction. Then there is again a drag force,
PUT, and a fluctuation term, A. Equipartition gives again m(v^2 ) = kT, and one
arrives once more at Eq. 21.16.
Next comes the issue of compatibility. With the help of Eqs. 21.7 and 21.8,
Einstein could compute separately expressions for (A^2 ) as well as for P in terms
of the A and B terms and p, where p is now given by Planck's law.* I shall not
reproduce the details of these calculations, but do note the crux of the matter. In
order to obtain the same answer for the quantities on both sides of Eq. 21.16, he
had to invoke a condition of directedness: 'if a bundle of radiation causes a mole-
cule to emit or absorb an energy amount hv, then a momentum hv/c is transferred
to the molecule, directed along the bundle for absorption and opposite the bundle
for [induced] emission' [El 1]. (The question of spontaneous emission is discussed
below.) Thus Einstein found that consistency with the Planck distribution (and
Eqs. 21.7 and 21.8) requires that the radiation be fully directed (this is often called
Nadelstrahlung). And so with the help of his trusted and beloved fluctuation
methods, Einstein once again produced a major insight, the association of momen-
tum quanta with energy quanta. Indeed, if we leave aside the question of spin,
we may say that Einstein abstracted not only the light-quantum but also the more
general photon concept entirely from statistical mechanical considerations.
21d. Earliest Unbehagen about Chance
Einstein prefaced his statement about photon momentum just quoted with the
remark that this conclusion can be considered 'als ziemlich sicher erwiesen,' as
fairly certainly proven. If he had some lingering reservations, they were mainly
due to his having derived some of his equations on the basis of 'the quantum
theory, [which is] incompatible with the Maxwell theory of the electromagnetic
field' [Ell]. Moreover, his momentum condition was a sufficient, not a necessary,
condition, as was emphasized by Pauli in a review article completed in 1924:
'From Einstein's considerations, it could. .. not be seen with complete certainty
that his assumptions were the only ones that guarantee thermodynamic-statistical
equilibrium' [PI]. Nevertheless, his 1917 results led Einstein to drop his caution
and reticence about light-quanta. They had become real to him. In a letter to*In 1910, Einstein had made a related calculation, together with Hopf [E12]. At that time, he used
the classical electromagnetic theory to compute (A^2 ) and P. This cast Eq. 21.16 into a differential
equation for p. Its solution is Eq. 19.17.