Do collapse interpretations respect conservation laws?

[greypilgrim]

Hi.

As far as I know, during the unitary evolution of quantum states, conservation laws are respected. Obviously this can't be true for the measurement process, if we only look at the system and exclude the observer. Now the simple explanation I've heard about this is that in quantum mechanics you can't observe a system without interacting with it, and this interaction causes the transfer of energy, momentum and so on such that those quantities are conserved in the bigger system including the observer.

While this of course makes sense, is it possible to prove this rigorously in the collapse formalism? Or do we need to assume that the bigger system including the observer evolves unitarily, which (I guess) essentially means refusing collapse interpretations?

 

[Dr_Nate]

I've wondered something similar. For example, in the double slit experiment the electron, or whatever, can end up far to the left or right even though it was fired from a centred electron gun with no lateral momentum. We know it was localized in the electron gun and now we know it is in the sensor. Therefore, it moved laterally, which suggests (does it?) it needed momentum to get there. Did it impart lateral momentum to the atoms of the sensor? If so, where is momentum that is conserved in the other direction?

 

[PeterDonis]

As far as I know, during the unitary evolution of quantum states, conservation laws are respected.

Only in the sense that if the total quantum state happens to be an eigenstate of an observable that satisfies a conservation law, unitary evolution will preserve it. For example, if the total quantum state is an eigenstate of the Hamiltonian and the Hamiltonian is time independent (so it satisfies an energy conservation law), unitary evolution will preserve the energy (the system will stay in that eigenstate of the Hamiltonian).

However, if a system is not in an eigenstate of an observable, even if that observable satisfies a conservation law, unitary evolution will not necessarily preserve the state and future measurements of that observable might not satisfy the conservation law. For example, if the total quantum state is not an eigenstate of the Hamiltonian, then unitary evolution might result in a state with different energy, even if the Hamiltonian itself is time-independent.

this interaction causes the transfer of energy, momentum and so on such that those quantities are conserved in the bigger system including the observer.

If the bigger system is in an eigenstate of momentum or energy, yes. Most discussions of this topic appear to assume that the bigger system is in such an eigenstate, but that doesn't mean it will always be true in practice.

is it possible to prove this rigorously in the collapse formalism?

Since there is no formalized definition of "measurement" in QM, I don't see how there could be a formalized proof that any measurement interaction will act as you describe. At any rate I'm not aware of one.

 

[kith]

For example, in the double slit experiment the electron, or whatever, can end up far to the left or right even though it was fired from a centred electron gun with no lateral momentum. We know it was localized in the electron gun and now we know it is in the sensor. Therefore, it moved laterally, which suggests (does it?) it needed momentum to get there. Did it impart lateral momentum to the atoms of the sensor? If so, where is momentum that is conserved in the other direction?

The electron state changes due to the interaction with the slit (or more precisely, the piece of matter which defines the slit). If the initial state of the electron and the slit was a momentum eigenstate with no lateral momentum and the final electron state after the measurement is a state with lateral momentum, the slit carries an equal amount of momentum in the opposite direction.

 

[greypilgrim]

If the initial state of the electron and the slit was a momentum eigenstate with no lateral momentum and the final electron state after the measurement is a state with lateral momentum, the slit carries an equal amount of momentum in the opposite direction.

Can you prove that within the formalism of QM?

 

[AlexCaledin]

The wavefunction at the sensors position can be considered a snapshot of classical histories probabilities, the collapse being a history choice, each history implementing a classical action minimum, thus conserving the momentum, right?

 

[mfb]

For example, if the total quantum state is not an eigenstate of the Hamiltonian, then unitary evolution might result in a state with different energy, even if the Hamiltonian itself is time-independent.

If you consider the whole system, i.e. include the preparation of the state, you'll still find that energy is conserved. The classical example is the preparation of an atom in a superposition of two energy levels, entangled with a photon that was either absorbed or not. If you throw away the photon (if present) and measure the energy of the atom you'll get one of two results, but that doesn't mean the overall energy would change.

 

[PeterDonis]

If you consider the whole system, i.e. include the preparation of the state, you'll still find that energy is conserved.

If the whole system is in an eigenstate of the Hamiltonian, yes. For example, the state you describe consisting of an atom entangled with the EM field is an eigenstate of the combined Hamiltonian (i.e., including the free atom, free field, and interaction terms). This will be true of pretty much any fairly common example, which is why the part about being an eigenstate of the total Hamiltonian is not usually explicitly mentioned.