- #1

snoopies622

- 791

- 22

Suppose I have some sinusoidal electromagnetic waves, plane-shaped, moving in a vacuum with frequency f and wavelength [itex] \lambda [/itex]. I want to consider the energy content of a certain volume of space through which they are passing. Let's say it is a cube with side lengths [itex] \Delta x [/itex] where [itex] \Delta x << \lambda [/itex] .

According to classical physics, I can find the energy inside the cube by integrating the sum of the square of the electric field strength and the square of the magnetic field strength over the volume. Let's call the result of that computation [itex] E_1 [/itex] .

According to elementary quantum mechanics, the energy of these waves is being carried by photons, each with an energy content of hf. Let's define [itex] E_2 [/itex] to be that quantity, hf.

I'm wondering what will happen if I measure the energy quantity in my cube and [itex] E_1/E_2 [/itex] is not an integer.

When I look into the subject of the quantization of the electromagnetic field, one thing I keep finding mentioned is the quantum harmonic oscillator (QHO), and the idea that an EM field is like a big collection of QHO's.

I know that for any particular frequency f, a QHO can have only the energies

hf(n+1/2), where n=0,1,2...

So an oscillator corresponding to the frequency of my light waves can have several different energy values, but none of which is necessarily [itex] E_1 [/itex] .

I also know that in quantum mechanics a system at any given moment is represented by a vector, and this vector does not have to be an eigenvector of an energy operator, but can instead be a linear combination of them, and so for a given system and state one could have an energy expectation value of any real quantity in between two energy eigenvalues.

Therefore, there may be a QHO corresponding to frequency f - the frequency of my light waves - and a vector for it that has an energy expectation value of [itex] E_1 [/itex], as long as the coefficients are chosen correctly. (By coefficients I mean the coefficients of the energy eigenvectors such that their linear combination makes the desired state vector here.)

So - if my thinking isn't way off - that leaves only the question: how does one find the coefficients? I assume that since the EM waves are already sinusoidal, one can't use Fourier analysis to break them down any further, since in that sense each sine wave is itself already a basis vector.

According to classical physics, I can find the energy inside the cube by integrating the sum of the square of the electric field strength and the square of the magnetic field strength over the volume. Let's call the result of that computation [itex] E_1 [/itex] .

According to elementary quantum mechanics, the energy of these waves is being carried by photons, each with an energy content of hf. Let's define [itex] E_2 [/itex] to be that quantity, hf.

I'm wondering what will happen if I measure the energy quantity in my cube and [itex] E_1/E_2 [/itex] is not an integer.

When I look into the subject of the quantization of the electromagnetic field, one thing I keep finding mentioned is the quantum harmonic oscillator (QHO), and the idea that an EM field is like a big collection of QHO's.

I know that for any particular frequency f, a QHO can have only the energies

hf(n+1/2), where n=0,1,2...

So an oscillator corresponding to the frequency of my light waves can have several different energy values, but none of which is necessarily [itex] E_1 [/itex] .

I also know that in quantum mechanics a system at any given moment is represented by a vector, and this vector does not have to be an eigenvector of an energy operator, but can instead be a linear combination of them, and so for a given system and state one could have an energy expectation value of any real quantity in between two energy eigenvalues.

Therefore, there may be a QHO corresponding to frequency f - the frequency of my light waves - and a vector for it that has an energy expectation value of [itex] E_1 [/itex], as long as the coefficients are chosen correctly. (By coefficients I mean the coefficients of the energy eigenvectors such that their linear combination makes the desired state vector here.)

So - if my thinking isn't way off - that leaves only the question: how does one find the coefficients? I assume that since the EM waves are already sinusoidal, one can't use Fourier analysis to break them down any further, since in that sense each sine wave is itself already a basis vector.

Last edited: