Undergrad Classical vs. Quantum Defintion of Energy in Field Theory

[LarryS]

Classical fields are usually constructed using a collection of classical harmonic oscillators, e.g. masses connected to springs. The energy of a classical harmonic oscillator is proportional to the amplitude squared. QFT uses quantized versions of those same classical fields. But, in the quantum domain energy is proportional to frequency, not amplitude squared. Is there any mathematical connection between this classical definition of energy and the quantum definition (frequency)? Or is the classical definition of energy simply discarded and arbitrarily replaced with the quantum definition?

As always, thanks in advance.

 

[Simon Bridge]

The frequency of a HO is a characteristic of the system in both the classical and quantum regimes.
The energy stored in the quantum HO is proportional to "n" - the principle quantum number.
The quantization is not arbitrary - the value of h was determined from experiiment, and the quantization was demonstrated the same way.

Note: for the classical HO: $E= m\omega^2A^2$ ... so now it is in terms of both amplitude and frequency.
... for a quantized SHO, the amplitude A is related to the energy level n as: $\frac{1}{2}kA^2 = (n+\frac{1}{2})\hbar\omega$ ... that is to say, if you somehow had a physical mass on a spring that could only have quantized energy, then the amplitude would also be quantized.

None of this is arbitrary - it was not just pulled from the air.
The quantization is demonstrated in Nature.

Both the classical and quantum descriptions are modeled through hamiltonian mechanics.
The classical version is what you get on average over the quantum versions.

 

[LarryS]

The frequency of a HO is a characteristic of the system in both the classical and quantum regimes.
The energy stored in the quantum HO is proportional to "n" - the principle quantum number.
The quantization is not arbitrary - the value of h was determined from experiiment, and the quantization was demonstrated the same way.

Note: for the classical HO: $E= m\omega^2A^2$ ... so now it is in terms of both amplitude and frequency.
... for a quantized SHO, the amplitude A is related to the energy level n as: $\frac{1}{2}kA^2 = (n+\frac{1}{2})\hbar\omega$ ... that is to say, if you somehow had a physical mass on a spring that could only have quantized energy, then the amplitude would also be quantized.

None of this is arbitrary - it was not just pulled from the air.
The quantization is demonstrated in Nature.

Both the classical and quantum descriptions are modeled through hamiltonian mechanics.
The classical version is what you get on average over the quantum versions.

Thanks. I forgot about the "spring constant" k. That is the mathematical connection between the amplitude (classical) version of energy and the frequency (quantum) version. Also, I did not mean to imply that the quantum version of energy was completely arbitrary, only that I could not be mathematically derived from the classical/amplitude version.

 

[Simon Bridge]

I'm not being clear: classical and quantum use the same definition of energy.
[edit] Also ... No quantum theory is derived from the classical. If it was possible to do this, then quantum theory would be a subset of classical theory and we wouldn't actually need it for what we use it for.
What you need to look for is a derivation of the classical result from the quantum theory.
Classical physics is what happens on average.