Collapses against conservation laws

[jostpuur]

In the time evolution defined by the Shrodinger's equation, the expectation value of the energy is conserved. However, in a collapse of the state, where state vector gets projected onto some subspace of physical states, the expectation value is not conserved. Does this mean that through successive collapses the energy of some system (even closed system, if it contains macroscopic beings that can cause the collapses) can violate conservation law arbitrarily much?

 

[genneth]

One way to understand this is to remember that energy conservation only holds in a closed system. Collapse only happens when you interact with the system -- and then it's not closed any more.

 

[Demystifier]

The interaction with the system also conserves total energy. If the initial state before the interaction is an eigenstate of the total Hamiltonian, then so is the final state after the interaction, with the same eigenvalue. So far, I am talking about unitary interaction described by the Schrödinger equation. The total state after the interaction can be written as a superposition in which each term is also an eigenstate of the Hamiltonian with the same eigenvalue. These terms differ only in how this energy is distributed among different subsystems. Thus, energy of each subsystem is undetermined, but the total energy is determined. When you finally measure energy of one of the subsystems (i.e., when the collapse occurs), you cannot predict with certainty what that energy will be. Nevertheless, you know with certainty that this energy will be correlated with energies of other subsystems, such that the total energy will be conserved.

 

[genneth]

The interaction with the system also conserves total energy. If the initial state before the interaction is an eigenstate of the total Hamiltonian, then so is the final state after the interaction, with the same eigenvalue. So far, I am talking about unitary interaction described by the Schrödinger equation. The total state after the interaction can be written as a superposition in which each term is also an eigenstate of the Hamiltonian with the same eigenvalue. These terms differ only in how this energy is distributed among different subsystems. Thus, energy of each subsystem is undetermined, but the total energy is determined. When you finally measure energy of one of the subsystems (i.e., when the collapse occurs), you cannot predict with certainty what that energy will be. Nevertheless, you know with certainty that this energy will be correlated with energies of other subsystems, such that the total energy will be conserved.

Indeed -- just to be clear, it's the original system that's no longer closed when the act of measurement occurs; however, the combined system including the observer (and her equipment) is still closed.

 

[jostpuur]

Why is the total energy of the closed system conserved? Its time evolution is not given by Shrodinger's equation (which would conserve the energy expectation value) alone, but projections occur in it too.

You say that the system must interact with something, so that it could collapse. Okey, let's add that something to the system, and then we have a closed system. A closed system, in which projections and collapses occur!

 

[Demystifier]

Why is the total energy of the closed system conserved? Its time evolution is not given by Shrodinger's equation (which would conserve the energy expectation value) alone, but projections occur in it too.

What do you mean by "projections"?

By the way, how did you solved your problems with loging?

 

[genneth]

Why is the total energy of the closed system conserved? Its time evolution is not given by Shrodinger's equation (which would conserve the energy expectation value) alone, but projections occur in it too.

You say that the system must interact with something, so that it could collapse. Okey, let's add that something to the system, and then we have a closed system. A closed system, in which projections and collapses occur!

Projections *always* involve a system that is not accounted for by the state vector. That's why it can be non-unitary. If you like, projections are just a shorthand for interacting the system with a measuring ancillary and reading off the ancillary. You can't simply have a system that's going about its time evolution, and then suddenly decide to project itself wrt some operator.

 

[jostpuur]

What do you mean by "projections"?

Events where the state does not evolve according to Shrodinger's equation, but some of the components of the state vector vanish. This of course comes back to the problem of what is measurement, as usual.

By the way, how did you solved your problems with loging?

By waiting. https://www.physicsforums.com/showthread.php?t=180386 There's something about it.

 

[jostpuur]

Projections *always* involve a system that is not accounted for by the state vector. That's why it can be non-unitary. If you like, projections are just a shorthand for interacting the system with a measuring ancillary and reading off the ancillary. You can't simply have a system that's going about its time evolution, and then suddenly decide to project itself wrt some operator.

It could be I'm agreeing with this. I of course didn't start the thread with my own opinion about the correct solution, but instead just threw the problem there. But are you sure this is mainstream? That's like saying that the copenhagenian collapses never occur really, if you look the whole system from outside, but instead it only appears to happen when we look some parts of the system.

 

[genneth]

It could be I'm agreeing with this. I of course didn't start the thread with my own opinion about the correct solution, but instead just threw the problem there. But are you sure this is mainstream? That's like saying that the copenhagenian collapses never occur really, if you look the whole system from outside, but instead it only appears to happen when we look some parts of the system.

I believe so... I can't cite any original reference, but almost all textbooks (and university courses) on quantum information theory covers this. It's usually left out in physics textbooks, though I don't quite understand why -- it seems to be so very natural.

It's certainly true, and accepted everywhere like Mastercard, that in describing quantum systems, the boundary between system and observer can be shifted about, and a consistent view of things can be found.

 

[Demystifier]

Why is the total energy of the closed system conserved? Its time evolution is not given by Shrodinger's equation (which would conserve the energy expectation value) alone, but projections occur in it too.

So, projection is the same as collapse.
Then, as I explained above, energy conservation before the collapse implies also energy conservation with collapse. The assumption is that everything, including measurement devices, is described by quantum mechanics.

 

[meopemuk]

In the time evolution defined by the Shrodinger's equation, the expectation value of the energy is conserved. However, in a collapse of the state, where state vector gets projected onto some subspace of physical states, the expectation value is not conserved. Does this mean that through successive collapses the energy of some system (even closed system, if it contains macroscopic beings that can cause the collapses) can violate conservation law arbitrarily much?

No, there is no violation of the energy conservation law. Consider two different cases. In the first case the system has a definite value of energy E, i.e., it is in an eigenstate of the energy operator. Then by measuring energy, we would always collapse the system state to the same eigenstate and measure the same energy E. The conservation law holds.

In the second case the system is not in an eigenestate of the energy operator. So, its energy is not well-defined and is characterized by some probabilistic spread. Then, individual measurements are not predictable, and we can obtain different energy values each time we measure. But we can't say that the conservation law has been violated, because the system didn't have a definite energy in the first place. The energy conservation law remains valid in the sense of expectation values.

Eugene.

 

[jostpuur]

But we can't say that the conservation law has been violated, because the system didn't have a definite energy in the first place.

$$D_t\big(\langle\Psi|H|\Psi\rangle\big) = 0$$

I would interpret this as the conservation law, since it is the best you can say about the conservation of energy in the quantum theory. But this is equation is violated when the state gets projected. In this sense the conservation law is being violated, even though definite energy wouldn't exist.

 

[meopemuk]

$$D_t\big(\langle\Psi|H|\Psi\rangle\big) = 0$$

I would interpret this as the conservation law, since it is the best you can say about the conservation of energy in the quantum theory. But this is equation is violated when the state gets projected. In this sense the conservation law is being violated, even though definite energy wouldn't exist.

No, it is not violated. Because in order to get the expectation value $\langle\Psi|H|\Psi\rangle$ one measurement (collapse) is not enough. You need to perform measurements on a large ensemble of identically prepared systems. Then you'll see that the expectation value does not change with time.

Eugene.