(Mo K± line) hν'

hν Foil

S

Fig. 1.5 Schematic of Bothe™s coincidence de-

e

tection of a Compton-scattered X-ray from a

Geiger

thin, metallic foil, and of the recoil electron

counter

from the same scattering event.

½¿

Photons

between the electron and the photon produced by their interaction enforces a tight

correlation”determined by conservation of energy and momentum”upon detection of

each quantum scattering event. It was just such correlations which were ¬rst observed

in the coincidence-counting experiment of Bothe.

1.2 Photons

In one of his three celebrated 1905 papers Einstein (1987a) proposed a new model of

light which explains all of the experimental results discussed in the previous sections.

In this model, light of frequency ω is supposed to consist of a gas of discrete photons

with energy = ω. In common with material particles, photons carry momentum as

well as energy. In the ¬rst paper on relativity, Einstein had already pointed out that

the relativistic transformation laws governing energy and momentum are identical to

those governing the frequency and wavevector of a plane wave (Jackson, 1999, Sec.

11.3D). In other words, the four-component vector (ω, ck) transforms in the same way

as (E, cp) for a material particle. Thus the assumption that the energy of a light

quantum is ω implies that its momentum must be k, where |k| = (ω/c) = (2π/»).

The connection to classical electromagnetic theory is provided by the assumption that

the number density of photons is proportional to the intensity of the light.

This is a far reaching extension of Planck™s idea that energy could only be trans-

ferred between radiation and matter in units of ω. The new proposal ascribes the

quantization entirely to the electromagnetic ¬eld itself, rather than to the mechanism

of energy exchange between light and matter. It is useful to arrange the results of

the model into two groups. The ¬rst group includes the kinematical features of the

model, i.e. those that depend only on the conservation laws for energy and momentum

and other symmetry properties. The second group comprises the dynamical features,

i.e. those that involve explicit assumptions about the fundamental interactions. In

the ¬nal section we will show that even this simple model has interesting practical

applications.

1.2.1 Kinematics

A The photoelectric e¬ect

The ¬rst success of the photon model was its explanation of the puzzling features

of the photoelectric e¬ect. Since absorption of light occurs by transferring discrete

bundles of energy of just the right size, there is no time delay before emission of the

¬rst electron. Absorption of a single photon transfers its entire energy ω to the bound

electron, thereby ejecting it from the metal with energy e given by eqn (1.6), which

now represents the overall conservation of energy. The energy of the ejected electron

therefore depends on the frequency rather than the intensity of the light. Since each

photoelectron emission event is caused by the absorption of a single photon, the number

of electrons emitted per unit time is proportional to the ¬‚ux of photons and thereby

to the intensity of light. The photoelectric equation implied by the photon model is

kinematical in nature, since it only depends on conservation of energy and does not

assume any model for the dynamical interaction between photons and the electrons in

the metal.

½ The quantum nature of light

B Compton scattering

The existence of the second peak in Compton scattering is also predicted by a kine-

matical argument based on conservation of momentum and energy. Consider an X-ray

photon scattering from a weakly bound electron. In this case it is su¬cient to consider

a free electron at rest and impose conservation of energy and momentum to determine

the possible ¬nal states as shown in Fig. 1.6.

For energetic X-rays the electron may recoil at velocities comparable to the velocity

of light, so it is necessary to use relativistic kinematics for this calculation (Jackson,

1999, Sec. 11.5). The relativistic conservation laws for energy and momentum are

mc2 + ω = E + ω , k = k + p, (1.7)

where p and E = m2 c4 + c2 p2 are respectively the ¬nal electron momentum and

energy, |k| = ω/c, and |k | = ω /c. Since the recoil kinetic energy of the scattered

electron (K = E ’ mc2 ) is positive, eqn (1.7) already explains why the scattered

quantum must have a lower frequency (longer wavelength) than the incident quantum.

Combining the two conservation laws yields the Compton shift

∆» ≡ » ’ » = »C (1 ’ cos θ) , (1.8)

in wavelength as a function of the scattering angle θ (the angle between k and k ),

where the electron Compton wavelength is

h

»C = = 0.0048 nm . (1.9)

mc

This simple argument agrees quite accurately with the data in Fig. 1.4, and with

other experiments using a variety of incident wavelengths. The fractional wavelength

shift for Compton scattering is bounded by ∆»/» < 2»C /». This shows that ∆»/» is

negligible for optical wavelengths, » ∼ 103 nm; which explains why X-rays were needed

to observe the Compton shift.

E, p ω', k'

E = mc , p = 0

2

ω, k

Fig. 1.6 Scattering of an incident X-ray quan-

tum from an electron at rest.

½

Photons

The argument leading to eqn (1.8) seems to prove too much, since it leaves no

room for the peak at the incident wavelength, which is also evident in the data. This

is a consequence of the assumption that the electron is weakly bound. In carrying out

the same kinematic analysis for a strongly bound electron, the electron mass m in eqn

(1.9) must be replaced by the mass M of the atom. Since M m, the resulting shift

is negligible even at X-ray wavelengths, and the peak at the incident wavelength is

recovered.

1.2.2 Dynamics

A Emission and absorption of light

The dynamical features of the photon model were added later, in conjunction with the

Bohr model of the atom (Einstein, 1987b, 1987c). The level structure of a real atom

is quite complicated, but for a ¬xed frequency of light only the two levels involved

in a quantum jump describing emission or absorption of light at that frequency are

relevant. This allows us to replace real atoms by idealized two-level atoms which have

a lower state with energy 1 , and a single upper (excited) state with energy 2 . The

combination of conservation of energy with the photoelectric e¬ect makes it reasonable

(following Bohr) to assume that the atoms can absorb and emit radiation of frequency

ω = ( 2 ’ 1 ) / . In this spirit, Einstein assumed the existence of three dynamical

processes, absorption, spontaneous emission, and stimulated emission. The simplest

cases of absorption and emission of a single photon are shown in Fig. 1.7.

Einstein originally introduced the notion of spontaneous emission by analogy with

radioactive decay, but the existence of spontaneous emission is implied by the princi-

ple of time-reversal invariance: i.e. the time-reversed ¬nal state evolves into the time-

reversed initial state. We will encounter this principle later on in connection with

Maxwell™s equations and quantum theory. In fact, time-reversal invariance holds for

all microscopic physical phenomena, with the exception of the weak interactions. These

photon atom atom

BEFORE AFTER

(a) Absorption of a single photon

photon

atom atom

AFTER

BEFORE

(b) Spontaneous emission

Fig. 1.7 (a) An atom in the ground state jumps to the excited state after absorbing a single

photon. (b) An atom in the excited state jumps to the ground state and emits a single photon.

½ The quantum nature of light

very small e¬ects will be ignored for the purposes of this book. For the present, we

will simply illustrate the idea of time reversal by considering the motion of classi-

cal particles (such as perfectly elastic billiard balls). Since Newton™s equations are

second order in time, the evolution of the mechanical system is determined by the

initial positions and velocities of the particles, (r (0) , v (0)). Suppose that at time

t = „ , each velocity is somehow reversed3 while the positions are unchanged so that

(r („ ) , v („ )) ’ (r („ ) , ’v („ )). More details on this operation”which is called time

reversal”are found in Appendix B.3.3. With this new initial state, the particles will

exactly reverse their motions during the interval („, 2„ ) to arrive at (r (2„ ) , v (2„ )) =

(r (0) , ’v (0)), which is the time-reversed form of the initial state. A mathematical

proof of this statement, which also depends on the fact that the Newtonian equations

are second order in time, can be found in standard texts; see, for example, Bransden

and Joachain (1989, Sec. 5.9).

In the photon model, the reversal of velocities is replaced by the reversal of the

propagation directions of the photons. With this in mind, it is clear that Fig. 1.7(b) is

the time-reversed form of Fig. 1.7(a). Absorption of light is a well understood process

in classical electromagnetic theory, and in principle the intensity of the ¬eld can be

made arbitrarily small. This is not the case in Einstein™s model, since the discreteness

of photons means that the weakest nonzero ¬eld is one describing exactly one photon,

as in Fig. 1.7(a). If we extrapolate the classical result to the absorption of a single

incident photon, then time-reversal invariance requires the existence of the process of

spontaneous emission, pictured in Fig. 1.7(b).

This argument can also be applied to the situation illustrated in Fig. 1.8, in which

many photons in the same mode are incident on an atom in the ground state. The

absorption event shown in Fig. 1.8(a) is evidently the time-reversed version of the

process shown in Fig. 1.8(b). Consequently, the principle of time-reversal invariance

implies the necessity of the second process, which is called stimulated emission.

Since the N photons in Fig. 1.8(a) are all in the same mode, this argument also shows

that the stimulated photon must be emitted into the same mode as the N ’ 1 incident

photons. Thus the stimulated photon must have the same wavevector k, frequency ω,

and polarization s as the incident photons. The identical values of these parameters”

which completely specify the state of the photon”for the stimulated and stimulating

photons implies a perfect ampli¬cation of the incident light beam by the process of

stimulated emission (ignoring, for the moment, the process of spontaneous emission).

This is the microscopic origin of the nearly perfect directionality, monochromaticity,

and polarization of a laser beam.

B The Planck distribution

We now consider the rates of these processes. Absorption and stimulated emission

both vanish in the absence of atoms and of light, so for low densities of atoms and

low intensities of radiation it is natural to assume that the absorption rate W1’2 from

the lower level 1 to the upper level 2, and the stimulated emission rate W2’1 ”from

the upper level 2 to the lower level 1”are both jointly proportional to the density of

3 This is hard to do in reality, but easy to simulate. A movie of the particle motions in the interval

(0, „ ) will display the time-reversed behavior in the interval („, 2„ ) when run backwards.

½

Photons

photon

photons atom

atom

BEFORE AFTER

(a) Absorption from a multi-photon state

photon

atom photons atom

BEFORE AFTER

(b) Stimulated emission

Fig. 1.8 (a) An atom in the ground state jumps to the excited state after absorbing one

of the N incident photons. (b) An atom in the excited state illuminated by N ’ 1 incident

photons jumps to the ground state and leaves N photons in the ¬nal state.

atoms and the intensity of the light. We further assume that the two-level atoms are

placed inside a cavity at temperature T , so that the light intensity is proportional to

the spectral function ρ (ω, T ). Therefore we expect that

W1’2 = B1’2 N1 ρ (ω, T ) , (1.10)

W2’1 = B2’1 N2 ρ (ω, T ) , (1.11)

where N1 and N2 are respectively the number of atoms in the lower level 1 and the

upper level 2. The rate S2’1 of spontaneous emission can only depend on N2 :

S2’1 = A2’1 N2 , (1.12)

since spontaneous emission occurs in the absence of any incident photons. The phe-

nomenological Einstein A and B coe¬cients, A2’1 , B2’1 , and B1’2 , are assumed to

be properties of the individual atoms which are independent of N1 , N2 , and ρ (ω, T ).

By studying the situation in which the atoms and the radiation ¬eld are in thermal

equilibrium, it is possible to derive other useful relations between the rate coe¬cients,

and thus to determine the form of ρ (ω, T ). The total rate T2’1 for transitions from

the upper state to the lower state is the sum of the spontaneous and stimulated rates,

T2’1 = A2’1 N2 + B2’1 N2 ρ (ω, T ) , (1.13)

and the condition for steady state”which includes thermal equilibrium as an impor-

tant special case”is T2’1 = W1’2 , so that

[A2’1 + B2’1 ρ (ω, T )] N2 = B1’2 ρ (ω, T ) N1 . (1.14)

Since the atoms and the radiation ¬eld are both in thermal equilibrium with the walls

of the cavity at temperature T , the atomic populations satisfy Boltzmann™s principle,

½ The quantum nature of light

e’β

N1 1

= eβ ω

= ’β , (1.15)

N2 e 2

where β = 1/kB T . Using this relation in eqn (1.14) leads to

A2’1

ρ (ω, T ) = . (1.16)

B1’2 exp (β ω) ’ B2’1

This solution has very striking consequences. In the limit of in¬nite temperature

(β ’ 0), the spectral function approaches a constant value:

A2’1

ρ (ω, T ) ’ . (1.17)

B1’2 ’ B2’1

On the other hand, it seems natural to expect that the energy density in any ¬nite

frequency interval should increase without bound in the limit of high temperatures.

The only way to avoid this contradiction is to impose

B1’2 = B2’1 = B , (1.18)

i.e. the rate of stimulated emission must exactly equal the rate of absorption for a

physically acceptable spectral function. This is an example of the principle of detailed

balance (Chandler, 1987, Sec. 8.3), which also follows from time-reversal symmetry.

Substituting eqn (1.18) into eqn (1.16) yields the new form

A 1

ρ (ω, T ) = , (1.19)

B exp (β ω) ’ 1

where we have further simpli¬ed the notation by setting A2’1 = A. In the low

temperature”or high energy”limit, ω kB T (β ω 1), the energy density is

A

ρ (ω, T ) = exp (’β ω) . (1.20)

B

This is Wien™s law, and it indeed agrees with experiment in the high energy limit.

By contrast, in the low energy limit, ω kB T ”i.e. the photon energy is small

compared to the average thermal energy”the classical Rayleigh“Jeans law is known

to be correct. This allows us to determine the ratio A/B by comparing eqn (1.19) to

eqn (1.4), with the result

ω3

A

= . (1.21)

π 2 c3

B

Thus the standard form for the Planck distribution,

ω3 1

ρ (ω, T ) = , (1.22)

exp (β ω) ’ 1

π 2 c3

is completely ¬xed by applying the powerful principles of thermodynamics to two-level

atoms in thermal equilibrium with the radiation ¬eld inside a cavity.

½

Photons

Einstein™s argument for the A and B coe¬cients correctly correlates an impressive

range of experimental results. On the other hand, it does not provide an explanation

for the quantum jumps involved in spontaneous emission, stimulated emission, and

absorption, nor does it give any way to relate the A and B coe¬cients to the micro-

scopic properties of atoms. These features will be explained in the full quantum theory

of light which is presented in the following chapters.

1.2.3 Applications

In addition to providing a framework for understanding the experiments discussed in

Section 1.1, the photon model can also be used for more practical applications. For

example, let us model an absorbing medium as a slab of thickness ∆z and area S

containing N = n∆zS two-level atoms, where n is the density of atoms. The energy

density of light in the frequency interval (ω, ω + ∆ω) at the entrance face is u (ω, z) =

ρ (z, ω) ∆ω, where ρ (z, ω) is the spectral function of the incident light. The incident

¬‚ux is then cu (ω, z), so energy enters and leaves the slab at the rates cu (ω, z) S and

cu (ω, z + ∆z) S, respectively, as pictured in Fig. 1.9.

By energy conservation, the di¬erence between these rates is the rate at which

energy is absorbed in the slab. In order to calculate this correctly, we must provide

a slightly more detailed model of the absorption process. So far, we have used an

all-or-nothing picture in which absorption occurs at the sharply de¬ned frequency

( 2 ’ 1 ) / . In reality, the atoms respond in a continuous way to light at frequency ω.

This is described by a line shape function L (ω), where L (ω) ∆ω is the fraction of

atoms for which ( 2 ’ 1 ) / lies in the interval (ω, ω + ∆ω). In succeeding chapters we

will encounter many mechanisms that contribute to the line shape, but in the spirit

of the photon model we simply assume that L (ω) is positive and normalized by

∞

dωL (ω) = 1 . (1.23)

0

We ¬rst consider the case that all of the atoms are in the ground state, then eqn (1.10)

yields

[cu (z + ∆z) ’ cu (z)] S = ’ ( ω) (Bρ (z, ω)) (L (ω) ∆ωn∆zS) . (1.24)

In the limit ∆z ’ 0 this becomes a di¬erential equation:

du (z, ω)

= ’ ωnBL (ω) u (z, ω) ,

c (1.25)

dz

c uz + ∆z

c uz

Fig. 1.9 Light in the frequency interval

(ω, ω + ∆ω) falls on a slab of thickness

∆z and area S. The incident ¬‚ux is

cu (z, ω) = cρ (z, ω) ∆ω, where ρ (z, ω) is the

∆z spectral function.

¾¼ The quantum nature of light

with the solution

nL (ω) B ω

u (z, ω) = u (0, ω) e’±(ω)z , where ±(ω) = . (1.26)

c

This is Beer™s law of absorption, and ±(ω) is the absorption coe¬cient.

In the opposite situation that all atoms are in the upper state, stimulated emission

replaces absorption, and the same kind of calculation leads to

du (z, ω)

c = ωnBL (ω) u (z, ω) , (1.27)

dz

with the solution

nL (ω) B ω

u (z, ω) = u (0, ω) e± (ω)z , ± (ω) = . (1.28)

c

In this case we get negative absorption, that is, the ampli¬cation of light.

If both levels are nondegenerate, the general case is described by densities n1 and

n2 for atoms in the lower and upper states respectively, with n1 + n2 = n. In the

previous results this means replacing n by n1 in the ¬rst case and n by n2 in the

second. In this situation,

(n2 ’ n1 ) L (ω) B ω

du (z, ω)

= g(ω)u (z, ω) , where g(ω) = . (1.29)

dz c

For thermal equilibrium n1 > n2 , so we get an absorbing medium, but with a popu-

lation inversion, n2 > n1 , we ¬nd instead a gain medium with gain g(ω) > 0. This

is the principle behind the laser (Schawlow and Townes, 1958).

1.3 Are photons necessary?

Now that we have established that the photon model is su¬cient for the interpretation

of the experiments described in Section 1.1, we ask if it is necessary. We investigate

this question by attempting to describe each of the principal experiments using a