Yakir Aharonov, Sandu Popescu, and Daniel Rohrlich have an interesting paper On conservation laws in quantum mechanics (January 5, 2021, PNAS):
They have noticed a problem similar to the one which we in this blog have called the "length scale problem".
In our June 3, 2021 blog post we wrote that if a laser beam is reflected or refracted by a very small object, then a moving observer may see a very high energy photon when he moves past the object. That is because the electromagnetic waveform close to the object has very fine detail, and the Fourier decomposition of the wave then will contain very high frequencies.
photon
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box
Aharonov et al. accomplish the same by preparing a photon into a box, in a pure state which has carefully chosen Fourier components. Then in the middle of the box, the wave function locally seems to have a wavelength much shorter than any of the Fourier components.
Aharonov et al. open a small window in the box for a short time. The photon may escape through the window, having a frequency much higher than any of the Fourier components had. We have a paradox.
Aharonov et al. call for a new energy conservation principle in quantum mechanics.
Instead of a single photon, we could prepare many coherent photons in the Aharonov et al. box. Then we would have a standing classical wave in the box, whose apparent wavelength in the middle of the box would be very short. An observer could then measure a very high-frequency wave, and consequently, a very high-energy photon in the box, even though we only put low-energy photons inside. Classically, there is no paradox. The observed high-frequency wave draws its energy from the low-frequency waves in the box.
Aharonov et al. present a thought experiment where we have many boxes and a single photon in each of them. If we measure a high-energy photon in one of the boxes, could it be that it draws its energy from the photons in other boxes?
The interaction with the measuring apparatus destroys the high-frequency wave in the middle?
Could it be that the measuring apparatus disturbs the wave in the middle of the box so much that it destroys any high frequencies?
Probably not. If we put a classical electromagnetic wave in the box, we certainly can measure the the high-frequency wave in the middle. The apparatus does not destroy high frequencies.
Possible solutions for the paradox
In our February 1, 2021 blog post we suggested that the paradox can be solved with a particle model, where it is the path integral which introduces wavelike properties into the system. Then one can only observe low-energy photons in the Aharonov et al. box.
But the solution does not work if we have a classical coherent wave. Classically, we can certainly observe a high-frequency wave in the middle of the box. It would be very strange if the classical high-frequency wave does not consist of photons.
In the June 3, 2021 blog post we suggested that we must drop the notion of a fixed number of quanta in an electromagnetic wave. We gave the following motivation: in a quantum mechanical experiment, one should assume the minimum of things about a photon. For example, one must not assume any definite path for a photon. A step further is that in a coherent wave, one must not assume any fixed number of photons.
Let us try to outline a solution for the paradox:
If we have just a single low-energy photon in the box, then energy conservation dictates that we cannot observe a high-energy photon. A particle model with a path integral approach may be a suitable way to model this.
But if we have a whole laser beam of coherent photons bouncing around in the box, then we believe that classical physics is the correct way to model the process. Then one can observe a high-energy photon. Energy conservation has to be enforced in the classical way. One may observe a high-energy photon, but it draws its energy from a bunch of low-energy photons which were inserted into the box.
Does the energy come from the preparation of the experiment or from the measuring device?
Aharonov et al. discuss the possibility that an observed high-energy photon might draw its energy from the measurement operation. They conclude that it is not possible.
Let us analyze this further. How do we put a low-energy photon into a box? We might put an excited hydrogen atom into the box and let it decay.
To confine a wave function in a fixed location, we need to "cut off" the fringes of the wave function. Could it be that this cutting procedure introduces high frequencies into the wave function and explains the birth of a high-energy quantum? For example, the hydrogen atom may be accelerated to a very high speed, and the photon which it emits may have very high energy.
A measurement is like preparation with time reversed. Could it be that the measurement supplies the energy? In the case of a classical laser beam wave, it is hard to see how the measurement could supply the energy in high-frequency waves. Clearly, high frequencies draw their energy from low-frequency waves.
Question. If large-energy photons get their energy from low-energy photons, how many low-energy photons we must have in the box so that this can happen? One photon is not enough. How about ten?
Some further links:
Chiara Marletto and Vlatko Vedral (2020) discuss another thought experiment where a box contains a photon in an energy eigenstate, and the box is suddenly made longer.
On February 2, 2021 we wrote about the paper of Sean Carroll and Jackie Lodman where the expectation value of energy changes in a measurement.
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