Amplitude-Frequency Relationship in Energy Calculation

[mraptor]

I know in classical physics we calculate the energy based on the Amplitude.
I also know we calculate the energy in QM based on Frequency.
I know also that it was experimentally deduced that it is the freq. based on the photo electric effect...and so on...

What I'm interested is in the connection ? Why ?
Energy: Does the quantum(freq) averages over the myriad of atoms to become classical effect (amplitude) ? OR there is no connection ?

Does the meaning of ENERGY in micro and macro world in those two cases have different meaning ?

 

[Sonderval]

Consider an em-wave to be specific.
In QM, the Energy-Frequency relation is the relation for a single photon. What is classically the amplitude becomes the number of photons in QM (which is then related to the field amplitude in quantum field theory).

 

[Darwin123]

I know in classical physics we calculate the energy based on the Amplitude.
I also know we calculate the energy in QM based on Frequency.
I know also that it was experimentally deduced that it is the freq. based on the photo electric effect...and so on...

What I'm interested is in the connection ? Why ?
Energy: Does the quantum(freq) averages over the myriad of atoms to become classical effect (amplitude) ? OR there is no connection ?

Does the meaning of ENERGY in micro and macro world in those two cases have different meaning ?

Here is something that used to confuse me. Quantum mechanics became clearer to me when I realized that it was amplitude that was actually being quantized, not frequency.
Frequency is a quantity that can be discrete even in classical physics. The fundamental and overtone frequencies of a plucked string with fixed ends are separated by finite quantities, for example. Furthermore, one could get discrete energy levels in an atoms by making the assumption that the electron is a coupled harmonic oscillator. One doesn't absolutely need quantum mechanics to explain energy levels in atoms.
Amplitude is a quantity that is always continuous in classical physics. The amplitude is discrete in quantum mechanics, not classical physics. The theory of quantum mechanics started with the realization that amplitudes of waves can be discrete. The word quantization was first applied not to frequencies, but to amplitudes.
The square of the amplitude of a wave is proportional to the energy density. Therefore, discrete values of energy imply discrete values of amplitude and vica versa. The discrete states of energy are separated by steps approximately equal to Planck's constant times the frequency. However, the frequency itself is determined by the parameters of the harmonic oscillator. The calculation of the frequency can be done using Newton's Laws, or maybe Einstein's relativity.
The wave particle duality has several equivalent ways to visualize it. One is what is taught in introductory courses. The square of the wave amplitude is proportional to the probability of detecting a particle there. However, there is another way to visualize the duality. One can visualize the quantized wave as a wave where the possible amplitudes are discrete instead of continuous. The discrete steps in amplitude are determined by the "de Broglie" relations.
One of the tricks in analyzing a quantum mechanical system is being able to separate the quantization of frequency from the quantization of amplitude.
Planck's analysis of the black body radiation spectrum used the assumption that the electrons in the atoms were attached to the nucleus as harmonic oscillators. It wasn't important what the frequency of that harmonic oscillator was. Up to this point, the problem was classical. The breakthrough occurred when Planck realized that the amplitude of this harmonic oscillator could only change in discrete steps. This was completely new. The assumption that the amplitude could change in discrete steps was the start of quantum mechanics.
Einstein made a contribution by the realization that quantized amplitudes were equivalent to particles. He found that by visualizing light as being a very weird type of particle (i.e., the photon), one could get the same results as if the electromagnetic wave changed in discrete steps. However, these photons were not like the corpuscles of Newton. They are constructs that came about due to the quantization of the light wave's amplitude.
Both ways to visualize quantum mechanics are counter-intuitive. However, physicists actually started developing quantum mechanics using the quantized amplitude picture. I think that the quantized amplitude picture is easier to use when phonons are involved. Phonons can be visualized as particles, but then one has to figure out what to do with the atoms that carry the vibrations. However, that is my preference. The two ways of visualizing it are equivalent.
To summarize: Quantization can be visualized two ways.
1) Wave-particle duality: One can imagine a particle that is somehow equivalent to a wave.
2) Amplitude quantization: One can imagine a wave with an amplitude that is only allowed to change in discrete steps.

 

[haael]

I know in classical physics we calculate the energy based on the Amplitude.
I also know we calculate the energy in QM based on Frequency.

What I'm interested is in the connection ? Why ?

It is true for any oscillating motion. The total energy of any wave (i.e. a sound wave) is proportional to the frequency and the squared amplitude.