What is the theoretical proof of the quantization of the energy of the EM field?

[Abu Abdallah]

Hi,
The energy of the quantum mechanical harmonic oscillator is proved to
be quantized after solving the Schrodingers equation which leads to Hermite equation and discovering that normalizable solutions of the wavefunction exist only for a discrete spectrum of energy. When the electromagnetic field is quantized in the beginning of any textbook on Quantum Optics, (see for example Zubairy and Scully), the field is supposed to be inside a bounded cavity and is decomposed into the normal modes of this cavity. The conjugate coordinates and momentum that comprise the field Lagrangian are
converted into operators and the commutation relations between qi and
pi, namely: [qi,pj]=i hbar.delta ij are imposed on the generalized coordinate
and momentum. The operators a, a+ are directly produced from these
generalized coordinates and momenta and by writing the Hamiltonian of
the field we discover that it's of the same form of the hamiltonian of
the mechanical quantum harmonic oscillator. We then jump to the
conclusion that the energy of the EM field is also quantized and the
operators a, a+ are creation and annihilation operators of photons!
Is this a sound proof for the quantization of the energy of
electromagnetic field? I don't think so...

 

[Abu Abdallah]

I received this answer from Dr. Daniel Steck :
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Hi (...)
That book isn't my preferred one for thorough treatments, but the
basic idea is that there is no proof for quantizing the
electromagnetic field, you *postulate* that there is a quantum
description of the EM field. The quantization procedure (where you
identify cononical coordinates and promote them to operators) is the
standard way of constructing the quantum model from the classical
model. The normal-mode decomposition is a trick to make the
quantization easier, otherwise you have to deal with field operators
as the fundamental object, which are much more complicated.

Hope that helps.

Daniel A. Steck
Oregon Center for Optics and Department of Physics
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Isn't it strange that we can't prove the quantization of EM energy ?!

 

[StatMechGuy]

You can't really "prove" any of quantum mechanics, you just go through the canonical quantization procedure. To do the same thing for the EM field, you just write down the classical field lagrangian, find the conjugate momentum, write down the hamiltonian, and use the commutation relations. I'm not sure why this is terribly peculiar. Mark Srednicki does this very early in his field theory book, which you can find on-line for free.