The energy of a wave is proportional to its frequency. This means that if the wavefunction is a combination of multiple sine waves, the particles are in an “superposition” of energy. When measuring that energy, the wavefunction mysteriously appears to “collapse” into one of the superposition energies.
Popescu, Aharonov, and Rohrlich used thought experiments to uncover the paradox. Suppose you have a trapped photon in the box and the wavefunction of this photon has a superoscillation region. Quickly place the mirror in the path of the photon where the wavefunction vibrates, and hold the mirror there for a short time. In the meantime, if the photons are close enough to the mirror, the mirror will bounce the photons out of the box.
Remember that we are dealing with the wavefunction of photons here. The bounce does not make up the measurement, so the wavefunction does not collapse. Instead, it is split in two. Most of the wavefunction remains in the box, but a small, rapidly vibrating part near where the mirror is inserted leaves the box and heads for the detector.
This super-oscillating part has been extracted from the rest of the wavefunction and is now the same as a much higher energy photon. When this piece hits the detector, the entire wavefunction collapses. In that case, the detector is unlikely to record high-energy photons, but it is. It’s like gamma rays coming out of a box of red light. “This is shocking,” Popesque said.
A clever measurement scheme gives photons more energy than any component of the wavefunction. Where did the energy come from?
Mathematician Emmy Noether proved in 1915 that conserved quantities such as energy and momentum arise from natural symmetry. Energy is saved due to “time conversion symmetry”. That is, the rule is that the equations that govern particles sometimes remain the same. (Energy is a stable quantity that represents this identity.) Energy is not conserved, especially in situations where gravity distorts the structure of space-time. This is because this distortion changes physics in different places and times and is not preserved on a cosmic scale. Where the expansion of space introduces time dependence. But for something like light in a box, physicists agree: time-transformation symmetry (and thus energy savings) should hold.
However, applying Noether’s theorem to quantum mechanics equations is complicated.
In classical mechanics, you can always check the initial energy of a system, evolve it, and then check the final energy. Then you will find that the energy is kept constant. However, measuring the energy of a quantum system inevitably interferes with it by disrupting the wavefunction and prevents it from evolving as it would otherwise. Therefore, the only way to ensure that energy is saved with the evolution of quantum systems is to do it statistically. Run the experiment many times, checking the initial energy in half the time and the final energy in the other half. The statistical distribution of energy before and after the evolution of the system should be consistent.
Popesque says the thought experiment is embarrassing, but compatible with this version of the law of conservation of energy. Since the superoscillation region is only a small part of the wavefunction of photons, it is very unlikely that photons will be found there, and in rare cases “shocking” photons will emerge from the box. In the course of many executions, the energy balance is balanced. “… I don’t claim that the statistical version of energy saving is wrong,” he said. “But all we insist is that it’s not the end of the story.”