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maline
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A measurement of an observable that does not commute with energy will generally cause a change in the expectation value of the energy. Is there a clear formalism to describe how energy is conserved overall?
maline said:Is there a clear formalism to describe how energy is conserved overall?
If you're looking at just the state of O during measurement, that's not a closed system so there's no reason to expect that energy will be conserved. That's true even in classical physics; solutions for systems under observation that conserve energy by ignoring interactions with the observing devices are idealizations, not exact.maline said:A measurement of an observable that does not commute with energy will generally cause a change in the expectation value of the energy. Is there a clear formalism to describe how energy is conserved overall?
But unitary evolution of the combined Hamiltonian does not lead to a well-defined measurement result! It leads precisely to a MWI-type state; a weighed superposition of decohered "branches" that include varying measurement results. I am trying to consider the result as final- this should be legitimate post-decoherence- and describe conservation laws that can be followed through the nonunitary measurement process.Nugatory said:If you're considering the entire system made up of D and O, the Hamiltonian of that system will include terms for the interaction between D and O. I believe that's the formalism you're looking for; it works across all interpretations and unitary evolution of that Hamiltonian will conserve energy.
I don't see why this should be a problem. I am interested in conserving the energy expectation value; there should be no need to perform another measurement.Nugatory said:if you're going to measure D+O we'll need another measuring device for that system, and we'll find ourselves in an infinite regress.
The crucial question is - energy of what? Energy of the measured subsystem, or energy of the whole universe including the measuring apparatus? The former does not need to be conserved. The latter does.maline said:A measurement of an observable that does not commute with energy will generally cause a change in the expectation value of the energy. Is there a clear formalism to describe how energy is conserved overall?
Yes, but each particular branch will have the same total energy, equal to the initial total energy. So whatever interpretation you use to explain the appearance of single measurement outcomes, the total energy will be conserved.maline said:But unitary evolution of the combined Hamiltonian does not lead to a well-defined measurement result! It leads precisely to a MWI-type state; a weighed superposition of decohered "branches" that include varying measurement results.
Sure, that's the result I want to know how to derive! How does the formalism guarantee this?Demystifier said:Yes, but each particular branch will have the same total energy, equal to the initial total energy
This is actually part of what I want to know: Is it only the expectation for the total energy that is conserved, or can we say more? Specifically, suppose start with a device and an particle each in pure energy states, and far apart. They then approach each other and the device measures the particles's position, after which they separate again. Now we perform ideal energy measurements on both the device and the particle. COE should at least imply that the expectation for the sum equals the original sum. But is it true, as Khashishi is suggesting, that the sum of the measurements will definitely equal the the original sum, through an entanglement?Khashishi said:D and O will be entangled after the interaction, such that the total energy is conserved
You are switching between considering them as two open systems and one closed system. If we're expecting energy to be conserved, we must start with the latter viewpoint and stick with it - we have one system with one Hamiltonian that contains an interaction term that is negligible when they are far apart. After the interaction, the system will be entangled in a way that correlates the observables "energy of O" and "energy of D".maline said:This is actually part of what I want to know: Is it only the expectation for the total energy that is conserved, or can we say more? Specifically, suppose start with a device and an particle each in pure energy states, and far apart. They then approach each other and the device measures the particles's position, after which they separate again.
What do you mean by an "ideal" measurement? Any measurement is going to interact with the system being measured and therefore will perturb it in some way.Now we perform ideal energy measurements on both the device and the particle. COE should at least imply that the expectation for the sum equals the original sum.
He's saying that the energy of the entangled system is the same before and after the interaction, which is tantamount to saying that if the energy of O and of D are separate observables, they must be entangled. That's not the same thing as saying that the sum of the measurements of these two observables must equal the original energy - in general it will not because we had to introduce an additional measuring device D' for measuring the energy of D, so we're in the infinite regress I mentioned earlier.But is it true, as Khashishi is suggesting, that the sum of the measurements will definitely equal the the original sum, through an entanglement?
It's very simple. Assume that initial state ##|\Psi(0)\rangle## of the total system is an energy eigenstate ##|E\rangle## satisfying ##H|E\rangle=E|E\rangle##, where ##H## is the Hamiltonian operator for the whole system. Then the Hamiltonian evolution implies that the total state as a function of time ismaline said:Sure, that's the result I want to know how to derive! How does the formalism guarantee this?
http://physics.stackexchange.com/questions/4047/energy-conservation-and-quantum-measurementmaline said:A measurement of an observable that does not commute with energy will generally cause a change in the expectation value of the energy. Is there a clear formalism to describe how energy is conserved overall?
I'm afraid I don't see why it should be true that these branches have energy E. Decoherence results from the near-orthogonality of the pointer states, which are basically positional configurations, not (in general) pure energy states. Although ψ(t) is an energy eigenstate, its decomposition in the pointer basis will include states that, when themselves decomposed in the energy basis, will include many possible energies.Demystifier said:Since |E⟩|E⟩|E\rangle is an energy eigenstate, it cannot be written as a superposition of energies different from EEE. Therefore at any time ttt the total state can be written as
|Ψ(t)⟩=∑ke−iEtck(t)|E,k⟩|Ψ(t)⟩=∑ke−iEtck(t)|E,k⟩|\Psi(t)\rangle=\sum_k e^{-iEt}c_k(t)|E,k\rangle
where kkk is some degeneracy label distinguishing different states of the same energy EEE. In particular |E,k⟩|E,k⟩|E,k\rangle may be macroscopically distinct states for different kkk, in which case the sum above is a decomposition into different macroscopic branches.
Nugatory said:What do you mean by an "ideal" measurement? Any measurement is going to interact with the system being measured and therefore will perturb it in some way.
An ideal measurement perturbs the system as little as possible. If after the interaction the system D+O is in an eigenstate of some observable that is conserved, and the corresponding operator for D alone commutes with the one for D+O, then it's theoretically possible to measure D, get a nondeterministic result, then measure O and be confident that the two results will sum to the original eigenvalue. I am trying to work out under what circumstances (obviously idealized) will conservation laws lead to such entanglements.Nugatory said:That's not the same thing as saying that the sum of the measurements of these two observables must equal the original energy - in general it will not because we had to introduce an additional measuring device D' for measuring the energy of D, so we're in the infinite regress I mentioned earlier.
You are right. To have any real evolution described by the Schrodinger equation, you need at least a small uncertainty of energy in the initial state. (Taking this into account, now I see that my post was misleading.) But it is not difficult to prove that, by Schrodinger evolution, the uncertainty of energy does not change with time. In practice, this means that final energy in a MWI branch may have a a value much different from the initial average energy, but (assuming that the initial uncertainty was small) the probability for this is very small. It is much more likely that the final energy will be very close to the initial average energy.maline said:Anyhow, the idea of starting off in an energy eigenstate of D+O is really kind of absurd, because then there would be no evolution and no interaction, only a meaningless phase rotation of the whole system.
Decoherence states are approximate position eigenstates and approximate energy eigenstates. The latter property is important because otherwise states would not be stable so decoherence would no irreversible.maline said:Decoherence results from the near-orthogonality of the pointer states, which are basically positional configurations, not (in general) pure energy states.
By "final energy" you mean the expectation value within the branch? In other words, in our world which does include (the very mysterious) selection of particular decoherence states, CoE is true only probabilistically?!Demystifier said:In practice, this means that final energy in a MWI branch may have a a value much different from the initial average energy, but (assuming that the initial uncertainty was small) the probability for this is very small. It is much more likely that the final energy will be very close to the initial average energy.
Yes.maline said:By "final energy" you mean the expectation value within the branch?
If the initial state had uncertain energy, then yes. But in most interpretations of QM time evolution is possible even without uncertain energy, so it is not at all obvious that uncertain energy is necessary. See Appendix A ofmaline said:In other words, in our world which does include (the very mysterious) selection of particular decoherence states, CoE is true only probabilistically?!
I don't understand. There are certainly systems is our world with uncertain energies, and when these decohere, even without any interaction with the enviroment, the energy expectation value will change. It even has a small probability of changing by a large amount. Thus energy conservation is no more than a statistical average!Demystifier said:If the initial state had uncertain energy, then yes. But in most interpretations of QM time evolution is possible even without uncertain energy, so it is not at all obvious that uncertain energy is necessary
There are certainly sub-systems with uncertain energy. But I am not sure about total closed systems.maline said:There are certainly systems is our world with uncertain energies,
How can something decohere without interaction with environment?maline said:and when these decohere, even without any interaction with the enviroment,
In BM, the energy of the particles (defined by instantaneous positions) is not necessarily conserved. But it is a hidden variable, so it does not imply that measured energy is not conserved.maline said:How about in BM? Is it possible to define a deterministically conserved energy as a function of the "hidden" instantaneous positions along with the objective "state"?
I am still thinking of a closed system D+O, where D measures O and thereby evolves into macroscopically distinct pointer states. My understanding is that these states will be decohered, with no environment necessarily involved. This was supposed to make "Schrodinger cat" states not physically relevant. Am I mistaken?Demystifier said:How can something decohere without interaction with environment?
Ah, I see. Often D is considered to be an environment of O. It's also OK to think of D+O as a closed system. But D has a very large number of degrees of freedom, so how can you know that energy of D+O is uncertain?maline said:I am still thinking of a closed system D+O, where D measures O and thereby evolves into macroscopically distinct pointer states. My understanding is that these states will be decohered, with no environment necessarily involved. This was supposed to make "Schrodinger cat" states not physically relevant. Am I mistaken?
All I am saying is that there are some closed macroscopic systems that are not in pure energy states. Is there any reason this would not be true?Demystifier said:But D has a very large number of degrees of freedom, so how can you know that energy of D+O is uncertain?
Didn't we agree that pure energy states do not evolve; that ψ*ψ is constant in time (for any basis)? Do you mean that the hidden variables of BM evolve in time?Demystifier said:But in most interpretations of QM time evolution is possible even without uncertain energy, so it is not at all obvious that uncertain energy is necessary
I guess not.maline said:All I am saying is that there are some closed macroscopic systems that are not in pure energy states. Is there any reason this would not be true?
Please see Appendix A that I mentioned there.maline said:Also, would you mind elaborating on this statement:
Yes, but most interpretations of QM (except MWI) assume, in one way or the other, that there is something beyond wave functions evolving according to the Schrodinger equation. Please see the Appendix A.maline said:Didn't we agree that pure energy states do not evolve; that ψ*ψ is constant in time (for any basis)?
Yes, that too.maline said:Do you mean that the hidden variables of BM evolve in time?
I tried, but I'm afraid I didn't understand much beyond what I already knew.Demystifier said:Please see Appendix A that I mentioned there.
In what interpretation other than BM is this the case? If you are referring to the nonunitary "collapse", am I wrong in saying that it is mathematically equivalent to selecting a decohered branch? If so, then it remains true that a pure energy state will not evolve- it will not develop any new decoherence, so there is no reason to invoke "collapse". So as long as we assume that there are closed systems that are dynamic (meaning that there is some change, unitary of not, in the state), then these have uncertain energy, and if they are macroscopic and undergo dehoherence then they have a probability of not conserving energy through the branch selection. Conclusion: CoE in a single world is only probabilistic!Demystifier said:Yes, but most interpretations of QM (except MWI) assume, in one way or the other, that there is something beyond wave functions evolving according to the Schrodinger equation.
In almost all of them, but let us not discuss them all at the moment.maline said:In what interpretation other than BM is this the case?
That is an excellent question! With having a modern version of collapse in mind, you are right. But in the past collapse has been introduced before the concept of decoherence has been understood. So in some old-fashioned vague style it does make sense to assume that collapse is possible even without prior decoherence.maline said:If you are referring to the nonunitary "collapse", am I wrong in saying that it is mathematically equivalent to selecting a decohered branch?
Suppose that initial energy is uncertain, for instance that the initial energy has some unknown value in the interval [7,8]. Then suppose that the final measured energy attains a definite value, say 7.4. Could you claim that energy has not been conserved in this case?maline said:Conclusion: CoE in a single world is only probabilistic!
Why are you speaking as if the energy was a hidden variable with a well-defined but unknown value? Is there any consistent model in which this is true?Demystifier said:Suppose that initial energy is uncertain, for instance that the initial energy has some unknown value in the interval [7,8]
The conservation I was hoping to find was of the expectation for the energy, as is the case with unitary evolution. If for instance the expectation was 7.5, then "collapse" into a state with expectation 7.4 -including pure states- would be a violation. I was hoping that some constraint on decoherence within closed systems would force all the branches to have expectation 7.5. You have been telling me that this is not the case; thus conservation of the energy expectation holds only in the MWI multiverse, or as a statistical average in our world.Demystifier said:Suppose that initial energy is uncertain, for instance that the initial energy has some unknown value in the interval [7,8]. Then suppose that the final measured energy attains a definite value, say 7.4. Could you claim that energy has not been conserved in this case?
From looking a bit more at the article you posted, my impression is that you agree that in any interpretation without hidden variables, and in which a "quantum state" can be ascribed to macroscopic objects, a closed system that evolves is not in a pure energy state. You are attempting to restore the possibility of evolving pure states by defining a partial state conditional on the "hidden" Bohmian positions which do evolve. Fascinating concept, but certainly different from standard QM. Please correct me if I am misunderstanding.maline said:Didn't we agree that pure energy states do not evolve; that ψ*ψ is constant in time (for any basis)? Do you mean that the hidden variables of BM evolve in time?
Yes, Bohmian mechanics can be interpreted in this way.maline said:Why are you speaking as if the energy was a hidden variable with a well-defined but unknown value? Is there any consistent model in which this is true?
I don't know.maline said:Is there some other sense, in any interpretation, in which energy is conserved in a branch selection scenario?
I think you are right.maline said:From looking a bit more at the article you posted, my impression is that you agree that in any interpretation without hidden variables, and in which a "quantum state" can be ascribed to macroscopic objects, a closed system that evolves is not in a pure energy state. You are attempting to restore the possibility of evolving pure states by defining a partial state conditional on the "hidden" Bohmian positions which do evolve. Fascinating concept, but certainly different from standard QM. Please correct me if I am misunderstanding.
Well, I am in general more interested in idealized, pure physics results that in FAPP results (except that for some reason I am studying for an engineering degree...?)Demystifier said:But important point is this. What is at stake here is conservation of energy of the total closed system. But this is typically a macroscopic system, with macroscopically "large" total energy EEE. On the other hand, from the quantum uncertainty relations one can see that the uncertainty of energy ΔEΔE\Delta E is microscopically "small". In other words, the ratio ΔE/EΔE/E\Delta E/E is very small, in fact so small that it can be neglected for all practical purposes. This means that, for all practical purposes, energy can be considered conserved in the process of measurement.
Thanks for sharing this! In the case where the particle is measured with a high final energy, I think this shows the same effect we were discussing: the total energy after the measurement is higher than the initial energy, with the difference made up by a loss of energy in "branches" of the WF where the particle was not measured- if you believe in those.Demystifier said:A recent related paper:
http://lanl.arxiv.org/abs/1609.05041
It points out that the standard conservation law is only a statistical law, which, by itself, is not sufficient to understand conservation of energy at the individual level.