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-

o

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

1

c [ρ± (ω, T ) ’ ρβ (ω, T )] σ∆ω > 0 , (1.1)

4

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.

3

Another way of saying this is that each mode occupies a volume (2π/L) in k-space,

’3

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)

(2π/L)

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:

ω2

ρ (ω, T ) dω = kB T 2 3 dω . (1.4)

πc

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

amount.

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

1

mv 2 = (’e) (’Vstop ) . (1.5)

2

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

relation

1

e = eVstop = mv = ω ’ W ,

2

(1.6)

2

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

Scattering

angle Lead box

θ = 135ο

Detector

Graphite

target Crystal

spectrometer

X-ray tube

(source of

Fig. 1.3 Schematic of the setup used to ob-

Mo K± line)

serve Compton scattering.

½½

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

frequency.

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

primary

(d)

(b)

135o

Scattered by

graphite at

45o

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

(1923).)

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

counter.

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