Energy conservation+superpositions=entanglement?

[Scott Hill]

A particle in a quantum harmonic oscillator can be in a superposition of energy eigenstates, and so the energy is not well-defined. However, energy is still conserved, so if I understand it correctly the "uncertainty" in the superposition's energy must be matched by uncertainty elsewhere in the Universe. is this entanglement we're talking about here, or is there another explanation for how energy conservation works here?

 

[kith]

A particle in a quantum harmonic oscillator can be in a superposition of energy eigenstates, and so the energy is not well-defined. However, energy is still conserved, so if I understand it correctly the "uncertainty" in the superposition's energy must be matched by uncertainty elsewhere in the Universe.

The uncertainty associated with a superposition state of a certain system isn't related to other systems in any way.

On the other hand, if you have two systems in an entangled state, you cannot assign definite state vectors to the individual systems in the first place. Look up the difference between "pure" and "mixed" states if you are interested in this.

or is there another explanation for how energy conservation works here?

If you don't have a definite energy, energy conservation refers to the expectation value of energy.

 

[Scott Hill]

The uncertainty associated with a superposition state of a certain system isn't related to other systems in any way.

On the other hand, if you have two systems in an entangled state, you cannot assign definite state vectors to the individual systems in the first place. Look up the difference between "pure" and "mixed" states if you are interested in this.If you don't have a definite energy, energy conservation refers to the expectation value of energy.

OK. Collapsing the wavefunction can cause a dramatic change in the expectation value of the energy, though; how is that energy accounted for?

Thanks.

 

[kith]

OK. Collapsing the wavefunction can cause a dramatic change in the expectation value of the energy, though; how is that energy accounted for?

First of all, bringing your system into contact with a measurement apparatus makes it an open system, so its energy need not be conserved. Naturally, one would try to use a full description including the apparatus and see what happens there. But then, you unfortunately run into all the well-known problems of the foundations of QM.

Also that the expectation value changes dramatically when you perform a measurement happens already in classical statistical mechanics. I'm not saying that QM is completely analogous but if the state somehow encodes subjective information, this behaviour is not so surprising.

I'm afraid I don't have a clearer answer to your question.

 

[Scott Hill]

OK, I've got it I think. It reminded me of the classic entanglement problem where an atom emits two circularly-polarized photons in opposite directions: angular momentum conservation forces the two particles to have opposite polarizations. I think there's still an "entanglement" argument to be made in there somewhere, but I'll think about it some more. Thanks!

 

[kith]

If you use a full quantum description for the system and the apparatus, you do get entanglement between the two which is relevant for your question. The problem lies in how this entangled state should be interpreted. It doesn't connect well with the pragmatical Copenhagen point of view on QM.

 

[Scott Hill]

Ah good, that's what I was thinking. But in *practice* we can just assume that it's the energy expectation value of the system that's conserved, and that measurement can exchange energy with the system.

Great! It's funny the questions that only occur to me once I have to teach a subject. :)