Let us continue reading the quantum paradox book of Yakir Aharonov and Daniel Rohrlich (2005).
There is the following thought experiment in the book. A particle is in a cylinder of a length L, closed with a piston. The particle is in an energy eigenstate, so that
L = (N / 2) λ
for some natural number N > 0 and the de Broglie wavelength λ of the particle. The piston is quickly pulled a length ΔL less than λ, so that L increases.
A. Projecting eigenfunctions
The discrete Fourier transform of the old wave function in wavelengths of the form
(L + ΔL) / (n / 2),
where n > 0, will contain Fourier components whose wavelength is smaller than λ. This suggests that the particle may end up in a higher energy eigenstate. Is this really possible?
B. Interpretation as a scattering experiment
Let us look at this as a kind of a scattering experiment of the particle from the piston. Then it is clear that the particle cannot pick up speed if it bounces back from a receding piston.
The "wave function" of the particle inside the cylinder really is a path integral of particle paths. Once the piston is moved, these paths will form in a new way. Energy conservation makes sure that any new stationary state has energy less or equal to the old stationary state. Thus, projecting the old wave function to new eigenfunctions is not the right way to calculate the new wave function.
C. Classical coherent wave
If we instead of a particle, confine a classical resonant coherent electromagnetic wave inside the cylinder, and pull the piston quickly, then the new waveform will probably contain Fourier components which have a shorter wavelength than the original wave.
It might be that projecting the old wave to new resonant waves (eigenfunctions) is a good approximate way to calculate the new wave.
Classically, the bouncing wave inside the cylinder pushes the cylinder. If we pull the piston, then the wave loses energy. Classically, high-frequency waves must drain their energy from the original wave.
What if we have just a single photon bouncing in the cylinder? Can it end up in a higher energy state, somehow draining energy from the receding piston?
Suppose that it is the photon itself which moves the piston farther. Energy conservation says that the photon cannot end up in a higher energy state.
D. Electron gas inside the cylinder
We cannot make a coherent electron wave. Let us assume that we have a gas of electrons bouncing inside the cylinder.
A path integral of N electrons might be the right way to calculate (though not practical). The piston cannot increase the energy of an electron in the path integral.
A possible resolution of the problem
If there is a classical electromagnetic wave confined in the cylinder, then we believe that classical physics is correct. When the piston is pulled, the wave loses energy. The resulting new wave will contain some higher frequencies than the original wave. The photons in the higher frequencies drained their energy from photons of the original wave.
If there is just a single particle in the cylinder, then we believe that a scattering experiment is the way to model the behavior. The wave function of the particle really is a path integral of various paths. The particle cannot end up having higher energy than it originally had.
Chiara Marletto and Vlatko Vedral in their 2020 paper The quantum totalitarian property and exact symmetries recommend using a path integral approach.
How does a high-energy photon drain energy from low-energy photons?
The piston must be an electric conductor to reflect the electromagnetic wave inside the cylinder. An electron in the piston can be accelerated through absorbing many low-energy photons. It is possible that the frequency of the electron oscillation is much higher than the frequency of the any of the absorbed photons. Then the electron will radiate high-energy photons.
It is an individual electron which converts energy in a bunch of low-energy photons into energy of high-energy photons.
Can we assume that a classical electromagnetic wave has a fixed number of photons?
Could it be that we, after all, can assume a fixed number of photons in a classical wave? It is the measuring device which converts energy in low-energy photons to a high energy photon?
A challenge in assuming a fixed number of photons is what happens if we measure a photon from an accelerating frame of reference. Suppose that a laser falling freely on Earth sends a single photon to space.
A measuring device floating freely in space would see the wave as a "chirp". The measuring device may absorb a photon from the chirp. There may be soft photons left over from the absorption, and the soft photons will escape to space.
In summary, the number of photons may be fixed in the laser beam in the inertial frame of the laser. But if one wants to absorb the energy of these photons in an accelerating frame, the number and energy of absorbed photons is not predetermined.
If the laser sends a full classical wave, then the measuring device in space will see a classical chirp wave. The measuring device will absorb photons of various frequencies. The measuring device cannot be fully resonant with a chirp. The Fourier decomposition of the remaining wave, after the measuring device, will be very complex. We may interpret that the remnant contains soft photons (and also some very high-frequency photons).
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