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Einstein™s introduction of photons was the ¬rst step toward a true quantum the-
ory of light”just as the Bohr model of the atom was the ¬rst step toward quantum
mechanics”but there is an important di¬erence between these parallel developments.
The transition from classical electromagnetic theory to the photon model is even more
radical than the corresponding transition from classical mechanics to quantum me-
chanics. If one thinks of classical mechanics as a game like chess, the pieces are point
particles and the rules are Newton™s equations of motion. The solution of Newton™s
equations determines a unique trajectory (q (t) , p (t)) for given initial values of the
position q (0) and the momentum p (0) of a point particle. The game of quantum me-
chanics has the same pieces, but di¬erent rules. The initial situation is given by a
wave function ψ (q), and the trajectory is replaced by a time-dependent wave function
ψ (q, t) that satis¬es the Schr¨dinger equation. The situation for classical electrody-
namics is very di¬erent. The pieces for this game are the continuous electric and
magnetic ¬elds E (r, t) and B (r, t), and the rules are provided by Maxwell™s equations.
Einstein™s photons are nowhere to be found; consequently, the quantum version of the
game requires new pieces, as well as new rules. A conceptual change of this magnitude
should be approached with caution.
In order to exercise the caution recommended above, we will discuss the experimen-
tal basis for the quantum theory of light in several stages. Section 1.1 contains brief
descriptions of the experiments usually considered in this connection, together with
a demonstration of the complete failure of classical physics to explain any of them.
In Section 1.2 we will introduce Einstein™s photon model and show that it succeeds
brilliantly in explaining the same experimental results.
In other words, the photon model is su¬cient for the explanation of the experi-
ments in Section 1.1, but the question is whether the introduction of the photon is
necessary for this purpose. The only way to address this question is to construct an
alternative model, and the only candidate presently available is semiclassical elec-
trodynamics. In this approach, the charged particles making up atoms are described
by quantum mechanics, but the electromagnetic ¬eld is still treated classically.
In Section 1.3 we will attempt to explain each experiment in semiclassical terms.
In this connection, it is essential to keep in mind that corrections to the lowest-order
approximation”of the semiclassical theory or the photon model”would not have
been detectable in the early experiments. As we will see, these attempts have varying
degrees of success; so one might ask: Why consider the semiclassical approach at all?
The answer is that the existence of a semiclassical explanation for a given experimental
result implies that the experiment is not sensitive to the indivisibility of photons,
which is a fundamental assumption of Einstein™s model (Einstein, 1987a). In Einstein™s
own words:
According to the assumption to be contemplated here, when a light ray is spreading
from a point, the energy is not distributed continuously over ever-increasing spaces,
but consists of a ¬nite number of energy quanta that are localized in points in space,
move without dividing, and can be absorbed or generated only as a whole.
The early experiments

As an operational test of photon indivisibility, imagine that light containing exactly
one photon falls on a transparent dielectric slab (a beam splitter) at a 45—¦ angle
of incidence. According to classical optics, the light is partly re¬‚ected and partly
transmitted, but in the photon model these two outcomes are mutually exclusive. The
photon must go one way or the other. In Section 1.4 we will describe an experiment that
very convincingly demonstrates this all-or-nothing behavior. This single experiment
excludes all variants of semiclassical electrodynamics. Experiments of this kind had to
wait for technologies, such as atomic beams and coincidence counting, which were not
fully developed until the second half of the twentieth century.

1.1 The early experiments
1.1.1 The Planck spectrum
In the last half of the nineteenth century, a considerable experimental e¬ort was made
to obtain precise measurements of the spectrum of radiation emitted by a so-called
blackbody, an idealized object which absorbs all radiation falling on it. In practice,
this idealized body is replaced by a blackbody cavity, i.e. a void surrounded by a
wall, pierced by a small aperture that allows radiation to enter and exit. The interior
area of the cavity is much larger than the area of the hole, so a ray of light entering
the cavity would bounce from the interior walls many times before it could escape
through the entry point. Thus the radiation would almost certainly be absorbed before
it could exit. In this way the cavity closely approximates the perfect absorptivity of
an ideal blackbody. Even when no light is incident from the outside, light is seen
to escape through the small aperture. This shows that the interior of a cavity with
heated walls is ¬lled with radiation. The blackbody cavity, which is a simpli¬cation of
the furnaces used in the ancient art of ceramics, is not only an accurate representation
of the experimental setup used to observe the spectrum of blackbody radiation; it
also captures the essential features of the blackbody problem in a way that allows for
simple theoretical analysis.
Determining the spectral composition (that is, the distribution of radiant energy
into di¬erent wavelengths) of the light emitted by a cavity with walls at temperature
T is an important experimental goal. The wavelength, », is related to the circular
frequency ω by » = c/ν = 2πc/ω, so this information is contained in the spectral
function ρ (ω, T ), where ρ (ω, T ) ∆ω is the radiant energy per unit volume in the
frequency interval ω to ω + ∆ω. The power per unit frequency interval emitted from
the aperture area σ is cρ (ω, T ) σ/4 (see Exercise 1.1). In order to measure this quantity,
the various frequency components must be spectrally separated before detection, for
example, by refracting the light through a prism. If the prism is strongly dispersive
(that is, the index of refraction of the prism material is a strong function of the
wavelength) distinct wavelength components will be refracted at di¬erent angles.
For moderate temperatures, a signi¬cant part of the blackbody radiation lies in
the infrared, so it was necessary to develop new techniques of infrared spectroscopy
in order to achieve the required spectral separation. This e¬ort was aided by the
discovery that prisms cut from single crystals of salt are strongly dispersive in the
infrared part of the spectrum. The concurrent development of infrared detectors in
The quantum nature of light

Fig. 1.1 Distribution of energy in the spec-
trum of a blackbody at various temperatures.
(Reproduced from Richtmyer et al. (1955,
Chap. 4, Sec. 64).)

the form of sensitive bolometers2 allowed an accurate measurement of the blackbody
spectrum. The experimental e¬ort to measure this spectrum was initiated in Berlin
around 1875 by Kirchho¬, and culminated in the painstaking work of Lummer and
Pringsheim in 1899, in which the blackbody spectrum was carefully measured in the
temperature range 998 K to 1646 K. Typical results are shown in Fig. 1.1.
The theoretical interpretation of the experimental measurements also required a
considerable e¬ort. The ¬rst step is a thermodynamic argument which shows that the
blackbody spectrum must be a universal function of temperature; in other words, the
spectrum is entirely independent of the size and shape of the cavity, and of the material
composition of its walls. Consider two separate cavities having small apertures of
identical size and shape, which are butted against each other so that the two apertures
coincide exactly, as indicated in Fig. 1.2. In this way, all the radiation escaping from
each cavity enters the other. The two cavities can have interiors of di¬erent volumes
and arbitrarily irregular shapes (provided that their interior areas are su¬ciently large
compared to the aperture area), and their walls can be composed of entirely di¬erent
materials. We will assume that the two cavities are in thermodynamic equilibrium at
the common temperature T .
Now suppose that the blackbody spectrum were not universal, but depended, for
example, on the material of the walls. If the left cavity were to emit a greater amount
of radiation than the right cavity, then there would be a net ¬‚ow of energy from left
to right. The right cavity would then heat up, while the left cavity would cool down.
The ¬‚ow of heat between the cavities could be used to extract useful work from two
bodies at the same temperature. This would constitute a perpetual motion machine of
the second kind, which is forbidden by the second law of thermodynamics (Zemansky,
1951, Chap. 7.5). The total ¬‚ow of energy out of each cavity is given by the integral of

2 These devices exploit the temperature dependence of the resistivity of certain metals to measure
the deposited energy by the change in an electrical signal.
The early experiments

Fig. 1.2 Cavities ± and β coupled through a
Temperature = 6 common aperture.

its spectral function over all frequencies, so this argument shows that the integrated
spectral functions of the two cavities must be exactly the same.
This still leaves open the possibility that the spectral functions could di¬er in
certain frequency intervals, provided that their integrals are the same. Thus we must
also prove that net ¬‚ows of energy cannot occur in any frequency interval of the
blackbody spectrum. This can be seen from the following argument based on the
principle of detailed balance. Suppose that the spectral functions of the two cavities,
ρ± and ρβ , are di¬erent in the small interval ω to ω + ∆ω; for example, suppose that
ρ± (ω, T ) > ρβ (ω, T ). Then the net power ¬‚owing from ± to β, in this frequency
interval, is
c [ρ± (ω, T ) ’ ρβ (ω, T )] σ∆ω > 0 , (1.1)
where σ is the common area of the apertures. If we position absorbers in both ± and β
that only absorb at frequency ω, then the absorber in β will heat up compared to that
in ±. The two absorbers then provide the high- and low-temperature reservoirs of a
heat engine (Halliday et al., 1993, Chap. 22“6) that could deliver continuous external
work, with no other change in the system. Again, this would constitute a perpetual
motion machine of the second kind. Therefore the equality

ρ± (ω, T ) = ρβ (ω, T ) (1.2)

must be exact, for all values of the frequency ω and for all values of the temperature T .
We conclude that the blackbody spectral function is universal; it does not depend on
the material composition, size, shape, etc., of the two cavities. This strongly suggests
that the universal spectral function should be regarded as a property of the radiation
¬eld itself, rather than a joint property of the radiation ¬eld and of the matter with
which it is in equilibrium.
The thermodynamic argument given above shows that the spectral function is uni-
versal, but it gives no clues about its form. In classical physics this can be determined
by using the principle of equipartition of energy. For an ideal gas, this states that
the average energy associated with each degree of freedom is kB T /2, where T is the
temperature and kB is Boltzmann™s constant. For a collection of harmonic oscillators,
the kinetic and potential energy each contribute kB T /2, so the thermal energy for
each degree of freedom is kB T .
In order to apply these rules to blackbody radiation, we ¬rst need to identify and
count the number of degrees of freedom in the electromagnetic ¬eld. The thermal
radiation in the cavity can be analyzed in terms of plane waves eks exp (ik · r), where
The quantum nature of light

eks is the unit polarization vector and the propagation vector k satis¬es |k| = ω/c and
k·eks = 0. There are two linearly independent polarization states for each k, so s takes
on two values. The boundary conditions at the walls only allow certain discrete values
for k. In particular, for a cubical cavity with sides L subject to periodic boundary
conditions the spacing of allowed k values in the x-direction is ∆kx = 2π/L, etc.
Another way of saying this is that each mode occupies a volume (2π/L) in k-space,
so that the number of modes in the volume element d3 k is 2 (2π/L) d3 k, where the
factor 2 accounts for the two polarizations. The ¬eld is completely determined by the
amplitudes of the independent modes, so it is natural to identity the modes as the
degrees of freedom of the ¬eld. Furthermore, we will see in Section 2.1.1-D that the
contribution of each mode to the total energy is mathematically identical to the energy
of a harmonic oscillator. The identi¬cation of modes with degrees of freedom shows
that the number of degrees of freedom dnω in the frequency interval ω to ω + dω is

k 2 dk L3 k 2
dnω = 2 dθ sin θ dφ 3 = π 2 c dω , (1.3)

where θ and φ specify the direction of k. The equipartition theorem for harmonic
oscillators shows that the thermal energy per mode is kB T . The spectral function is
the product of dnω and the thermal energy density kB T /L3 , so we ¬nd the classical
Rayleigh“Jeans law:
ρ (ω, T ) dω = kB T 2 3 dω . (1.4)
This ¬ts the low-frequency data quite well, but it is disastrously wrong at high
frequencies. The ω-integral of this spectral function diverges; consequently, the total
energy density is in¬nite for any temperature T . Since the divergence of the integral
occurs at high frequencies, this is called the ultraviolet catastrophe.
In an e¬ort to ¬nd a replacement for the Rayleigh“Jeans law, Planck (1959) con-
centrated on the atoms in the walls, which he modeled as a family of harmonic oscil-
lators in equilibrium with the radiation ¬eld. In classical mechanics, each oscillator is
described by a pair of numbers (Q, P ), where Q is the coordinate and P is the momen-
tum. These pairs de¬ne the points of the classical oscillator phase space (Chandler,
1987, Chap. 3.1). The average energy per oscillator is given by an integral over the
oscillator phase space, which Planck approximated by a sum over phase space elements
of area h. Usually, the value of the integral would be found by taking the limit h ’ 0,
but Planck discovered that he could ¬t the data over the whole frequency range by
instead assigning the particular nonzero value h ≈ 6.6 — 10’34 J s. He attempted to
explain this amazing fact by assuming that the atoms could only transfer energy to
the ¬eld in units of hν = ω, where ≡ h/2π. This is completely contrary to a clas-
sical description of the atoms, which would allow continuous energy transfers of any
This achievement marks the birth of quantum theory, and Planck™s constant h
became a new universal constant. In Planck™s model, the quantization of energy is a
property of the atoms”or, more precisely, of the interaction between the atoms and
the ¬eld”and the electromagnetic ¬eld is still treated classically. The derivation of the
The early experiments

spectral function from this model is quite involved, and the fact that the result is in-
dependent of the material properties only appears late in the calculation. Fortunately,
Einstein later showed that the functional form of ρ (ω, T ) can be derived very simply
from his quantum model of radiation, in which the electromagnetic ¬eld itself consists
of discrete quanta. Therefore we will ¬rst consider the other early experiments before
calculating ρ (ω, T ).

1.1.2 The photoelectric e¬ect
The infrared part of atomic spectra, contributing to the blackbody radiation discussed
in the last section, does not typically display sharp spectral lines. In this and the
following two sections we will consider e¬ects caused by radiation with a sharply
de¬ned frequency. One of the most celebrated of these is the photoelectric e¬ect:
ultraviolet light falling on a properly cleaned metallic surface causes the emission of
electrons. In the early days of spectroscopy, the source of this ultraviolet light was
typically a sharp mercury line”at 253.6 nm”excited in a mercury arc.
In order to simplify the classical analysis of this e¬ect, we will replace the complex-
ities of actual metals by a model in which the electron is trapped in a potential well.
According to Maxwell™s theory, the incident light is an electromagnetic plane wave
with |E| = c |B|, and the electron is exposed to the Lorentz force F = ’e (E + v — B).
Work is done only by the electric ¬eld on the electron. Hence it will take time for
the electron to absorb su¬cient energy from the ¬eld to overcome the binding energy
to the metal, and thus escape from the surface. The time required would necessarily
increase as the ¬eld strength decreases. Since the kinetic energy of the emitted electron
is the di¬erence between the work done and the binding energy, it would also depend
on the intensity of the light. This leads to the following two predictions. (P1) There
will be an intensity-dependent time interval between the onset of the radiation and
the ¬rst emission of an electron. (P2) The energy of the emitted electrons will depend
on the intensity.
Let us now consider an experimental arrangement that can measure the kinetic
energy of the ejected photoelectrons and the time delay between the arrival of the
light and the ¬rst emission of electrons. Both objectives can be realized by positioning
a collector plate at a short distance from the surface. The plate is maintained at a
negative potential ’Vstop , with respect to the surface, and the potential is adjusted to
a value just su¬cient to stop the emitted electrons. The photoelectron™s kinetic energy
can then be determined through the energy-conservation equation

mv 2 = (’e) (’Vstop ) . (1.5)

The onset of the current induced by the capture of the photoelectrons determines the
time delay between the arrival of the radiation pulse and the start of photoelectron
emission. The amplitude of the current is proportional to the rate at which electrons
are ejected. The experimental results are as follows. (E1) There is no measurable time
delay before the emission of the ¬rst electron. (E2) The ejected photoelectron™s kinetic
energy is independent of the intensity of the light. Instead, the observed values of
½¼ The quantum nature of light

the energy depend on the frequency. They are very accurately ¬tted by the empirical
e = eVstop = mv = ω ’ W ,
where ω is the frequency of the light. The constant W is called the work function; it is
the energy required to free an electron from the metal. The value of W depends on the
metal, but the constant is universal. (E3) The rate at which electrons are emitted”
but not their energies”is proportional to the ¬eld intensity. The stark contrast between
the theoretical predictions (P1) and (P2) and the experimental results (E1)“(E3) posed
another serious challenge to classical physics. The relation (1.6) is called Einstein™s
photoelectric equation, for reasons which will become clear in Section 1.2.
In the early experiments on the photoelectric e¬ect it was di¬cult to determine
whether the photoelectron energy was better ¬t by a linear or a quadratic dependence
on the frequency of the light. This di¬culty was resolved by Millikan™s beautiful ex-
periment (Millikan, 1916), in which he veri¬ed eqn (1.6) by using alkali metals, which
were prepared with clean surfaces inside a vacuum system by means of an in vacuo
metal-shaving technique. These clean alkali metal surfaces had a su¬ciently small work
function W , so that even light towards the red part of the visible spectrum was able
to eject photoelectrons. In this way, he was able to measure the photoelectric e¬ect
from the red to the ultraviolet part of the spectrum”nearly a threefold increase over
the previously observed frequency range. This made it possible to verify the linear
dependence of the increment in the photoelectron™s ejection energy as a function of
the frequency of the incident light. Furthermore, Millikan had already measured very
accurately the value of the electron charge e in his oil drop experiment. Combining
this with the slope h/e of Vstop versus ν from eqn (1.6) he was able to deduce a value
of Planck™s constant h which is within 1% of the best modern measurements.

1.1.3 Compton scattering
As the study of the interaction of light and matter was extended to shorter wavelengths,
another puzzling result occurred in an experiment on the scattering of monochromatic
X-rays (the K± line from a molybdenum X-ray tube) by a graphite target (Compton,
1923). A schematic of the experimental setup is shown in Fig. 1.3 for the special

angle Lead box
θ = 135ο
target Crystal
X-ray tube
(source of
Fig. 1.3 Schematic of the setup used to ob-
Mo K± line)
serve Compton scattering.
The early experiments

case when the scattering angle θ is 135—¦ . The wavelength of the scattered radiation
is measured by means of a Bragg crystal spectrometer using the relation 2d sin φ =
m», where φ is the Bragg scattering angle, d is the lattice spacing of the crystal,
and m is an integer corresponding to the di¬raction order (Tipler, 1978, Chap. 3“
6). Compton™s experiment was arranged so that m = 1. The Bragg spectrometer
which Compton constructed for his experiment consisted of a tiltable calcite crystal
(oriented at a Bragg angle φ) placed inside a lead box, which was used as a shield
against unwanted background X-rays. The detector, also placed inside this box, was
an ionization chamber placed behind a series of collimating slits to de¬ne the angles
θ and φ.
A simple classical model of the experiment consists of an electromagnetic ¬eld of
frequency ω falling on an atomic electron. According to classical theory, the incident
¬eld will cause the electron to oscillate with frequency ω, and this will in turn generate
radiation at the same frequency. This process is called Thompson scattering (Jack-
son, 1999, Sec. 14.8). In reality the incident radiation is not perfectly monochromatic,
but the spectrum does have a single well-de¬ned peak. The classical prediction is that
the spectrum of the scattered radiation should also have a single peak at the same
The experimental results”shown in Fig. 1.4 for the scattering angles of θ = 45—¦ ,
90—¦ , and 135—¦ ”do exhibit a peak at the incident wavelength, but at each scattering

(a) (c)
Molybdenum Scattered at
K± line 90o

Scattered by
graphite at

6o 30' 7o 7o 30' 6o 30' 7o 7o 30'
Angle from calcite Angle from calcite
Fig. 1.4 Data from the Compton scattering experiment sketched in Fig. 1.3. A calcite crystal
was used as the dispersive element in the Bragg spectrometer. (Adapted from Compton
½¾ The quantum nature of light

angle there is an additional peak at longer wavelengths which cannot be explained by
the classical theory.

1.1.4 Bothe™s coincidence-counting experiment
During the early development of the quantum theory, Bohr, Kramers, and Slater raised
the possibility that energy and momentum are not conserved in each elementary quan-
tum event”such as Compton scattering”but only on the average over many such
events (Bohr et al., 1924). However, by introducing the extremely important method
of coincidence detection”in this case of the scattered X-ray photon and of the
recoiling electron in each scattering event”Bothe performed a decisive experiment
showing that the Bohr“Kramers“Slater hypothesis is incorrect in the case of Compton
scattering; in fact, energy and momentum are both conserved in every single quantum
event (Bothe, 1926). In the experiment sketched in Fig. 1.5, X-rays are Compton-
scattered from a thin, metallic foil, and registered in the upper Geiger counter. The
thin foil allows the recoiling electron to escape, so that it registers in the lower Geiger
When viewed in the wave picture, the scattered X-rays are emitted in a spherically
expanding wavefront, but a single detection at the upper Geiger counter registers the
absorption of the full energy ω of the X-ray photon, and the displacement vector
linking the scattering point to the Geiger counter de¬nes a unique direction for the
momentum k of the scattered X-ray. This is an example of the famous collapse of
the wave packet.
When viewed in the particle picture, both the photon and the electron are treated
like colliding billiard balls, and the principles of the conservation of energy and mo-
mentum ¬x the momentum p of the recoiling electron. The detection of the scattered
X-ray is therefore always accompanied by the detection of the recoiling electron at the
lower Geiger counter, provided that the second counter is carefully aligned along the
uniquely de¬ned direction of the electron momentum p. Coincidence detection became
possible with the advent, in the 1920s, of fast electronics using vacuum tubes (triodes),
which open a narrow time window de¬ning the approximately simultaneous detection
of a pair of pulses from the upper and lower Geiger counters.
Later we will see the central importance in quantum theory of the concept of an
entangled state, for example, a superposition of products of the plane-wave states
of two free particles. In the case of Compton scattering, the scattered X-ray pho-
ton and the recoiling electron are produced in just such a state. The entanglement

Low-pressure box
Source of Geiger

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