When you learned that the results of measurements in quantum mechanics are random, it may have raised a question in your mind: What about conservation laws? Do they only hold on average? For example, if you measure the energy of an atom, you might end up with a different amount of energy than the average, right? If there are random fluctuations in 'conserved' quantities, could the effect be used to violate conservation laws in a systematic way?
For example, consider a spin measurement for spin-1/2 particles. Each particle's spin carries an amount of angular momentum equal to hbar/2 in the direction it points. The particles are prepared so that their spins point in the +Z direction, and then sent into a Stern-Gerlach (SG) device, which we can rotate to measure spin along any direction. If we measure a spin in the X direction, the result is that the spin ends up in either the +X or -X direction. So it looks like we are violating conservation of angular momentum in a systematic way, destroying the +Z direction angular momentum we prepared the particles with. If that were true and the experiment is done in an isolated satellite, we could use it to build up a net angular momentum in the -Z direction.
If conservation laws mean anything, there must be something wrong with the above picture. Perhaps, one might think, there must be some back-action of the particles on the Stern-Gerlach device. That is, the missing angular momentum is being transferred into the SG device, as the particles exert torques on it with their magnetic moments as they come through.
The problem we run into next is that this seems to violate linearity: A +Z spin can be written as a superposition of a +X term and a -X term. After going through the SG device, there is decoherence (or as some people wrongly assume, wavefunction collapse), and what is observed is just a +X result or a -X result. Since QM is linear, the final wavefunction is a linear superposition of the terms that would have resulted if the original spins had been +X or -X. Such terms do not take the original +Z spin into account. So at least as far as an observer within such a term is concerned, there is no residual effect of the original spin direction, such as we would need if the SG device had received angular momentum that depended on that direction.
The solution to this puzzle, naturally, is to treat the measuring device as a fully quantum-mechanical system. That means that its angular orientation can not be precisely known, due to its finite uncertainty in angular momentum. (The uncertainty principle applies, limiting how small the product of the uncertainties of angle and angular momentum can get.) As a result, there will be very small 'error' terms in which the wrong spin outcome is measured, i.e. -X instead of +X, or an incoming spin is flipped.
This effect may seem negligible, but it is enough to allow the information about the original direction of the particle spin to be encoded in the final state of the SG device. It works out to be exactly enough of an effect to enforce the conservation law. The uncertainty in the SG device's angular momentum allows a sort of selection effect; in effect, the 'lost' angular momentum does end up in the SG device. The same kind of effect holds for all conservation laws. This is explained in detail in my eprint "There is No Violation of Conservation Laws in Quantum Measurement". It was first studied by Wigner in 1952, and is related to the Wigner-Araki-Yanase theorem (1960).
See also
"WAY beyond conservation laws"
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