Re: [Phys-L] Photons vs. electrons in excitation

[John Denker <jsd@av8n.com>]

On 10/26/2016 11:27 PM, Antti Savinainen wrote:
> it is well known that the excitation of an electron in the ground
> state (e.g. in hydrogen) requires a photon with a certain energy. The
> energy must match with the energy difference between some excited state and
> the ground state. If the incoming photon does not have the "right" energy
> it does not cause excitation; instead, scattering takes place.
>
> However, if the incoming particles are electrons with a certain
> kinetic energy  which does not match the energy difference between any
> excited state and the ground state, they nevertheless cause excitation as
> long as the electrons have enough energy to do so. Why is it so that the
> bound electron is so "picky" with photons but when it comes to electrons,
> anything goes as long as there is enough energy? This means that the
> electron loses part of its kinetic energy in the interaction with the
> atom and the "lost" energy appears in the excitation.

Consider the following analogy to classical waves and classical resonance:

Suppose we have a flagpole with a high-Q resonance.  You can drive
it with any signal you like, not necessarily a resonant signal.  A
particularly interesting case is a sudden impulse:  just kick the
flagpole, at a point not too close to the ground.  This corresponds
to striking a piano string with a hammer.  The resonant system will
respond.  In fact the response to a delta-function excitation is
just the Green function of the system.  You have to kick a flagpole
pretty hard to get a response that is big enough to be easily observable.

Another option is to apply a much more gentle force with a time
dependence that matches the resonant frequency sufficiently closely.
Then you can over time build up a large response.

The flagpole parable is relevant to atomic physics as follows:

The incident electron is like a sudden kick.  The electrostatic force
is negligible when the electron is far away, then it becomes large when
the electron whizzes past the atom, and then it becomes negligible again.

  If you want an even more extreme example of the same idea, consider
  bombarding the atom with an electrically neutral incident particle.
  Then the force is even smaller when the particle is far away, and
  the impulse is even more sudden.

  You can get the same result using non-resonant light.  You can
  have essentially a delta function of EM radiation, using a
  high-intensity femtosecond pulse.  It looks like half a cycle
  of a sine wave.  It can be produced by a mode-locked laser.

In contrast, prosaic monochromatic light exerts a gentle periodic
force on the atom.  If it's on resonance, this can accumulate over
time to produce a large effect.  If it's off resonance, the effect
is much smaller.

=========

You can look at the same physics another way, leading to all the same
conclusions, by considering things on a frequency-by-frequency basis.

The Fourier transform of a sudden kick (i.e. delta function) has
components at all possible frequencies.  The tiny part of this that
overlaps with the resonance of the target atom will produce an effect,
and the rest more-or-less goes to waste.

In contrast, the weak periodic interaction has essentially only one
frequency.  It produces a big result if it's on resonance, and almost
no result otherwise.

=================

Quantum mechanics introduces some additional subtleties, but I
suspect they are not relevant to the question that was asked,
unless I misunderstand the question.  Understanding the basic
wave mechanics is a valuable springboard to understanding the QM.

Also:  Beware that the Q of an atomic resonance is not infinite.
Typical Q values are on the order of a million for an "allowed"
transition.  That's important, because it means that being "on
resonance" is not a binary yes/no proposition.  It is entirely
possible to have a gentle periodic interaction that is only
slightly off resonance.  The response will have a nontrivial
phase and amplitude.

  Note that a stabilized single-mode laser can be tuned with
  a precision much better than one part per million.  So it
  is entirely feasible to drive an atom only slightly off
  resonance.

  Also with macroscopic resonators (flagpoles, tuning forks,
  RC circuits, etc.) it is straightforward to drive them
  only slightly off resonance.