Re: splitting wave energy into KE and PE

[John Denker <jsd@MONMOUTH.COM>]

At 09:31 AM 3/18/00 -0500, David Bowman wrote:
> [at low temperature] each term in the Hamiltonian is of the form:
>     h*f*[a^dagger]*[a]
> (where the quantities in the brackets are creation/destruction operators

I agree that's a perfectly reasonable way to write the Hamiltonian.
OTOH it is not the only reasonable way.  Suppose we choose
        Q = (a^dagger + a)/2
and call it our canonical coordinate.  The dynamically conjugate variable is
        P = (a^dagger - a)/2i
which we can think of as momentum.

Notes:
 1a) Physically, for electromagnetic waves, Q is the electric field
strength in some appropriate units, but formally we don't even need to know
that.
 1b) For transverse waves on a slinky, Q is the displacement;  the
equations are formally identical.
 2a) Physically, for EM waves, P is essentially the magnetic field
strength, but formally we don't need to know that.
 2b) For transverse waves on a slinky, P is the transverse momentum.
 **) In any case, all we really need to know is that P is dynamically
conjugate to Q.  When in doubt, look at the Lagrangian.  It knows.

We can write the Hamitonian as
        H = [sum over modes] hbar omega (a^dagger a + 1/2)
or equivalently as
        H = [sum over modes] hbar omega (Q^2 + P^2)

(Just grind out the expansion for P and Q to show the equivalence.)

Now ISTM that one could quite reasonably call the Q^2 piece a potential
energy, and the P^2 piece a kinetic energy.  Averaged over a full cycle
these two terms are equal in magnitude.  According to this way of looking
at things, one could say the field energy is half potential and half kinetic.

(Note the validity of this viewpoint does not depend on any assumptions
about temperature;  the aforementioned average does not need to be a
thermal average.)

David also wrote:
> E = |p|*c.... by this way of looking at things, all the photon
> energy is kinetic.

I don't find this argument very convincing.  One could make an analogous
statement about an ordinary mass-on-a-spring oscillator:  The mean energy
is proportional to the mean-square displacement, so by this way of looking
at things, all the oscillator energy is potential.  The fallacy is that the
average PE is proportional to the average KE, so an expression for one can
always be dressed up to look like an expression for the other.  (And to
complete the analogy, I think E=|p|c only holds when averaged over a cycle.)

===========
Bottom line:

1) AFAIK, it is always safe to think of the wave energy as half kinetic and
half potential.  If anybody has an example of a situation where this leads
to trouble, please explain.

2) In some situations the PE/KE split doesn't matter, in which case the
question of how to do the split need not be asked and need not be answered.