Re: photoelectric effect classically

[John Denker <jsd@MONMOUTH.COM>]

At 06:01 PM 11/4/00 +0200, Savinainen Antti wrote:

> I have a question on the wave theory predictions for the photoelectric
> effect.  According to the wave theory only the intensity should affect
> maximum kinetic energy of the ejected electrons.

One should be careful when talking about "the" wave theory.  There are lots
of wave theories.  Some people use "wave mechanics" as a near-synonym for
"quantum mechanics".  Some people call Schrödinger's equation "the" wave
equation.  However based on context (e.g. the subject line) I will assume
the goal is a _classical_ description of the photoelectric effect.

That is a very peculiar question.  We believe that QM gives the correct
physical description of the photoelectric effect.  The word "classical" is
used to describe certain limiting cases.  However, there is no unique
prescription for passing to the classical limit.

One possible (albeit imperfect) way to characterize the classical limit is
to take a quantum system in which all the quantum numbers are large
compared to unity.  An example of this is the earth in its orbit.  The
principal quantum number (orbital angular momentum) is quite large.

> Increasing intensity means increasing the magnitude of electric field
> vector which increases the force exerted on an electron by the incident beam.

OK, I guess.

> Hence the greater KE.

Greater than what?  Are you starting with the QM result and then treating
the classicalization process as a small perturbation?  That's a really bad
idea.

> The frequency of the light should not affect the KE. Classically intensity
> and energy density of electromagnetic wave do not depend on frequency (am
> I right?).

Quantum mechanically, a _beam_ can have any energy and energy density you
want.  The same for a classical beam.  Now a _photon_ is quantized, but
that's a different question.  Classically you don't have photons, so the
question does not arise.

> The explanation above is frequently given in high school and introductory
> university books. But is it really correct?

The foregoing is so incomplete that it is meaningless.  I cannot imagine an
experiment that would provide evidence for or against it.  If you have such
an experiment in mind, please elaborate.

> I started digging this because some of my students insisted more specific
> information on the classical predictions. I’ll give another explanation
> which I found from a bit more advanced text.
>
> The force exerted on an electron can be expressed as F = e(E + v cross B)
> and in case of linearly polarized light E = E0sinwt and B = B0sinwt. The
> electron gains energy and starts to oscillate. This takes some time. When
> energy of the electron is equal or just greater than the work function it
> is immediately ejected from the metal. So electron does *not* gain
> significant KE and this leads to low kinetic energies no matter what
> intensity is used. If intensity is greater the electron is released
> faster. This prediction contradicts with the former because now greater
> intensity does not imply greater KE.
>
> Which prediction is consistent with Maxwell’s wave theory?

The second scenario is much more physical.  It has its problems, but one
can at least imagine patching up the problems.

Again, the analogy to the earth in its orbit may be helpful.  Or a marble
rolling in a dish with a 1/r potential for r>r0.

One problem with scenario #2 is that it speaks of "the" frequency (w).  But
as the electron (or marble) picks up energy, it goes into orbitals with
lower and lower frequency.  If you're not careful, it will eventually be
off-resonance relative to the applied field.

You can patch this up by using a range of frequencies.

In any case, you _can_ shake a marble out of such a dish.  The potential
has a finite binding energy after all.  The energy with which it leaves
depends on how much energy you can dump into it during the last half-cycle
or so.  Therefore intensity does matter.  (This assumes the applied field
doesn't have any DC component -- in which case all bets would be off and
the whole question would be even sillier than I thought.)

Note that a fully QM description of the process gives the same answer, as
it should.

> One more question. How would the oscillation of the electron change if
> unpolarized light was used?

Not much.  Just consider the projection onto the relevant polarization
basis vectors.  One component adds energy.  The other just re-orients the
plane of the orbit.