CTS-QED-68-1

THE PHOTOELECTRIC EFFECT WIT

Willis E. Lamb, Jr.
Department of Physics, Yale Universq and currently
Center for Theoretical Studies, University of Miami
and
Marlan 0. Scully
Department of Physics and Materials Science Center Massachusetts Institute of Technology

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February 1968

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Center For Theoretical Studies
University of Miami/Coral Gables, Florida 33124

The Photoelectric Effect Without; Photons Willis E. Lamb, J r . - x
Department of Physics, Yale University and currently
Center f o r Theoretical Studies, University of Miami and
Marlan 0. Scully**Department of Physics and Materials Science Center
Massachusetts Institute of Technology
* Work supported in part by the National Aeronautics
and Space Administration and in part by the U. S. Air Force Office of Scientific Research.
** Work supported by the Advanced Research Projects Agency
Contract SD-90.

c
1
A misconception which most physicists acquire in their formative years is that the photoelectric effect requires the quantization of the electromagnetic field for its explanation. The following quotation taken from a widely used physics text book' illustrates the point.
"Einstein's photoelectric equation played an enormous part in the development of the modern quantum theory. But in spite of its generality and of the many successful applications that have been made of it in physical theories, the equation
is, as we shall see presently,based on a concept
of radiation - the concept of 'light quanta! -
completely at variance with the most fundamental concepts of the classical electromagnetic theory of
. radiation"
In fact we shall see that the photoelectric effect may be com-
pletely explained without invoking the concept of "light quanta". To be sure, certain aspects of nature require quantization of the electromagnetic field for their explanation, for example:
1. Planck distribution law for black body radiation
(1900) 9
2. Compton effect (1926),

2
3. Spontaneous emission (Dirac, 1927),
4. Electrodynamic level shifts (1947).
The photoelectric effect is definitely not included in the foregoing list. It is an historical accident that the photon concept should have acquired its strongest early support from Einstein's considerations on the photoelectric effect.
The physicists of the early years of this century deserve great credit for realizing that 11 something" had to be quantized.
The very existence of atoms, (Bohr, 1913), requires the
finiteness of Planck's constant to be taken into account for the behavior of the atomic electrons. However, once granted the existence of atoms, we shall see that all of the experimental photoelectric phenomena are described by a theory in which the electromagnetic field is treated classically while only the matter is treated quantum mechanically. These phenomena include (a) the Einstein photoelectric relation (l), (b) the linear relationship between photocurrent and light intensity and (c) observations at low radiation levels where photoelectrons are obtained after times of illumination insufficient for "accumulation" in an atom of enough energy to liberate an
electron, i.e., > G o
A more detailed account of the theory of photoelectron emission, in which both the detector (atomic electrons) and the electromagnetic field (photons) are quantized, is published elsewhere. 2

3

Our model for a photodetector consists of a collection of completely independent atoms. Each atom has one electron3 and is bathed in a classical electromagnetic field, for which the electric field is
. E(t) = E COS vt 0
As indicated in Fig. 1, the atom has a ground state
I,)g and a quasi-continuum of excited states Ik) which
are normalized in a length L very large compared to atomic dimensions, so that the k levels will approximate a continium when L is eventually allowed to become infinite. Photoelectron emission involves a transition from lg) to any of the (k) states. The Hamiltonian4 for the electron as it interacts with the field is

H = H 0 - eE(t)x,

(3)

where H is the Hamiltonian of the unperturbed atom, x is the
0
atomic coordinate operator and e is the negative electronic
charge. It is convenient to go into the interaction picture
so that the interaction Hamiltonian becomes

V(t) = - eE(t)x(t),

(4)

where

4

c,'.i 3 . x(t) = exp iHot/h x ex -iHot/h

(5)

The nonvanishing matrix elements of (5) are of the form

where 1.k is the energy of the kth excited state measured relative to the ground state. We ignore transitions between excited states, i.e., we are interested in knowing if the atom is ionized, not in what happens to the electron once it is in the quasi-continuum. The photoelectron density matrix obeys the equation of motion
which has the formal solution
It is convenient to substitute Eq. (8) into (7) to obtain the
alternative form for the time rate of change of the electronic density matrix

5
as i n r e f e r e n c e (2). Let us now proceed t o d e r i v e t h e r e l a t i o n
where E i s t h e k i n e t i c energy o f t h e l i b e r a t e d e l e c t r o n and @ t h e work f u n c t i o n of t h e d e t e c t o r . The energy d i s t r i b u t i o n of t h e e j e c t e d e l e c t r o n i s given by t h e diagonal elements of t h e e l e c t r o n d e n s i t y matrix, p k Y k ( t ) . The lowest order c o n t r i b u t i o n t o p ( t ) i s obtained by p l a c i n g p ( 0 ) i n t h e r i g h t
hand s i d e of Eq. (8), using t h e V ( t ) given by ( 3 ) , and n o t i n g
t h a t t h e only nonvanishing m a t r i x element of p ( 0 ) i s pg, g ( 0 ) = 1, we f i n d

Evaluation o f t h e i n t e g r a l (12) with n e g l e c t of s m a l l non-resonant terms gives

pk,k(t)

exk, g Eo/2h12 s i n 2

6
It is clear that an electron will be raised to the kth excited state with appreciable probability only if
\
k = hv, with an energy spread of order

which quickly becomes experimentally undetectable. We denote by @ = c1 the energy of the first state of the quasi-continuous spectrum. T h i s is the ionization energy of the atom, or work
function of the photodetector. Measuring the energy E > 0 of
the liberated electron from the ionization energy by setting

ck = @ + E ,

one has the desired Einstein photoelectric equation

hV = @ + E .

(17)

Equation (17) can only be satisfied if hv > @, and we see that
the energy dependence of the ejected photoelectrons obeys the Einstein relationship even for a classical radiation field illuminating quantized atoms. The concept of a photon is not needed.
The total probability P(t) for finding a photoelectron is given by

7
P ( t ) = kc pk,k(t)
This may be evaluated for the lowest order expressions
(13) by replacing the sum over k by an integral over e
s. ..., 2.. -+ dco(c).
k where o ( c ) is the number of states per unit range of c . In the familiar manner, (13) becomes effectively
and (18) takes the time proportional form
P(t) = Y t,
where the constant y is
This formula implies a "rate" of transitions y for each atom of a collection of independent atoms. This derivation provides a solution to an exercise proposed by Schiff5. It should be noted that (22) is proportional to the light intensity. Equation (21) certainly does not imply the "time delay" which some people used to expect for the photoelectrons produced by a classical e.m. field. The perturbation begins to mix the excited states into the electronic wave function as soon

8
as it is turned on. Of course, the calculation of P(t) refers to a probability f o r an ensemble in a typically quantum mechanical manner, and statistical fluctuations would be inevitable, but as shown elsewhere2, are fully and satisfactorily describable by the theory.
In most time dependent problems one is contented with a result like (21), correct to lowest order perturbation theory. Because of the simplicity of the model for our photodetector, it is possible to carry out the calculation to all orders in the perturbation, and hence to have an example where rate equations are rigorously justified.
We begin by noting that, from Eq. ( 8 ) , P(t) obeys the relations
d P ( t ) = ddtCkp k,k(t) dt
or writing out the matrix product explicitly, observing (6)

9

Since pg,g (t')

and all p (t')
k,k

are positive and

their sum is unity, it is clear that the p (t') have
k,k

an upper bound as functions of k. This may be estimated

in the following fashion. It was seen from Eq. (15)

that only the quasi-continuous states k in a range of

energy

(given by an uncertainty principle) are appreciably excited. The number N of these s t a t e s is about

At this stage in the calculation we might not be sure
that the k dependence of p (t')is given by the perturbation
k,k
result (13), but the exact distribution6 certainly cannot be appreciably sharper than given by (15). We may then
estimate an upper bound for the ~ ~ , ~by( t ' )

The density of states a(hv) is proportional to the
normalization length L, and hence as L +m, we have N +w

10
remains finite for any finite time t'. Then as L -fa
the terms in (24)involving ~ ~ , ~ ( t 'c)an be neglected compared to those involving p (t') so Eq. (24)
g,g becomes

f C.C.

(29)

Replacing the sum over k by an integral over excited state energies c, and using (28), we obtain

dt ed!=

lot
~ ( c ) dttlex ,g Eo /2h12

The integral over c leads to a delta function 2nh6(t-t') multiplied by some slowly varying factors evaluated at resonance c = hv, and after doing the t integration we find the simple differential equation
ddtP0 = y[l-P(t)] ,
where the rate constant y has the value (22) indicated by the perturbative theory. The solution of this differential equation P(t) = 1 - e-yt is intuitively obvious.

11
In conclusion, we understand the photoeffect as being the result of a classical field falling on a quantized atomic electron. The introduction of the photon concept is neither logically implied by nor necessary for the explanation of the photoelectric effect.

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FOOTNOTES
1. F. K. Richtmyer, E. H. Kennard, T. Lauritsen, Introduction to Modern Physics, 5th ed. (McGraw
Hill, New York 1955) p. 94.
2. M. 0. Scully and W. E. Lamb, Jr., Phys. Rev. (to be published, 1968.)
3 . For simplicity we consider the problem to be one dimensional with electric field and motion of the electron confined to the x axis.
4. Only the electric dipole part of the perturbation
is taken, i.e., retardation and magnetic effects are neglected. This approximation is made only to simplify the equations.
5. L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1955), p. 220, problem 2.
6. This supposition can eventually be confirmed by examining equation (30).
FIGURE CAPTION
Classical field falling on atom having ground state \g> and continuum of excited states \k>,lowest excited state energy is @.

b x
m
J
N