# Conservation of energy in quantum measurement

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## Main Question or Discussion Point

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?

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Is there a clear formalism to describe how energy is conserved overall?
when a measurement is done on a quantum mechanical system one confines the state of the system by some interaction from outside agency.
and the quantum mechanical state comes to the measured value out of all the possible values in which it can stay with finite probability .

therefore the measurement disturbs the system and due to interaction with the 'measurement' tools some energy can flow in the process , thereby the conservation of energy or the expectation value of energy will change.

even if the system and the agency employed for performing the measurement is taken together then also the conservation may not be seen in the process as the general description of the quantum mechanical state has changed.

though the quantum description does not challenge the principle of energy conservation - but by a measurement performed in the classical world/tools or say interaction with photons to fix up say position of a quantum particle -one has driven the system to a particular value of position out of all possible values it could have stayed ,thus the energy may not be same as before the measurement.

Let me be specific. Suppose we have a closed system consisting of a measuring device D and an object O to be measured. O is initially in a pure energy state- say the ground state. D then measures some other observable of O, say a position measurement, and displays the result P on a macroscopic dial. (Of course, noone can see the dial because the system is closed, but decoherence theory tells us that this is not relevant to the evolution, correct?) After the measurement, O is no longer in a pure energy state, and the expectation value for its energy is certainly higher than the ground-state energy. If energy is to be conserved, we must conclude that a corresponding value was subtracted from the expectation for D's internal energy.
What I'd like to know is, do we have any formalism for the principles that govern such interactions and "maintain" conservation laws? Treating the measurement as a black-box "collapse" does not address the changes in D, and so cannot ensure COE.
Treating the scenario as a unitary evolution of the combined system, as is done in MWI, will not necessarily help either: I am asking about conservation of the expectation for the total energy in the particular world or decohered branch where result P was obtained, after normalizing the state back to unity. After all, we observe energy conservation in our own "world", not just in the overall "multiverse".
"Collapse" and MWI are the two pictures of measurement that I have been exposed to. What other ways are there of treating measurement, and do any of them show how conservation laws will be kept?

Nugatory
Mentor
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?
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.

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. Nonetheless, the formalism is not fully satisfying because 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. Classical physics avoids this annoyance because it allows us to ascribe properties to an unobserved system.

Khashishi
D and O will be entangled after the interaction, such that the total energy is conserved.

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.
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.

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.
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.

Demystifier
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?
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.
To describe it by a formalism, you must introduce the Hamiltonian for the total system (measured system + measuring apparatus + environment that interacts with those two) and study the Schrodinger evolution for the total state.

Demystifier
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.
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.

Yes, but each particular branch will have the same total energy, equal to the initial total energy
Sure, that's the result I want to know how to derive! How does the formalism guarantee this?
D and O will be entangled after the interaction, such that the total energy is conserved
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?

Nugatory
Mentor
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.
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".
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.
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.
But is it true, as Khashishi is suggesting, that the sum of the measurements will definitely equal the the original sum, through an entanglement?
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.

What we can say is the time derivative of the expectation value of the energy of the closed D+O system is zero. That's conservation of energy in quantum mechanical terms; you can get this result from Ehrenfest's theorem by noting that the Hamiltonian commutes with itself.

Demystifier
Sure, that's the result I want to know how to derive! How does the formalism guarantee this?
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 is
$$|\Psi(t)\rangle=e^{-iEt}|E\rangle$$
Since $|E\rangle$ is an energy eigenstate, it cannot be written as a superposition of energies different from $E$. Therefore at any time $t$ the total state can be written as
$$|\Psi(t)\rangle=\sum_k e^{-iEt}c_k(t)|E,k\rangle$$
where $k$ is some degeneracy label distinguishing different states of the same energy $E$. In particular $|E,k\rangle$ may be macroscopically distinct states for different $k$, in which case the sum above is a decomposition into different macroscopic branches. Therefore each branch has the same energy $E$. Q.E.D.

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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.
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.

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. My bad for bringing it up. In the case that I started with, only O began with a well-defined energy, and we are considering conservation of the expectation value for the energy of D+O. For that case we certainly need additional derivation.

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.
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.
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.

Demystifier
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.
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.

Demystifier
Decoherence results from the near-orthogonality of the pointer states, which are basically positional configurations, not (in general) pure energy states.
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.

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.
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
By "final energy" you mean the expectation value within the branch?
Yes.

In other words, in our world which does include (the very mysterious) selection of particular decoherence states, CoE is true only probabilistically?!
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 of
http://lanl.arxiv.org/abs/1209.5196

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
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!

In MWI, there will at least be conservation over the multiverse as a whole. Small comfort...

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"?

Demystifier
There are certainly systems is our world with uncertain energies,
There are certainly sub-systems with uncertain energy. But I am not sure about total closed systems.

and when these decohere, even without any interaction with the enviroment,
How can something decohere without interaction with environment?

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"?
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.

How can something decohere without interaction with environment?
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
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?
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?

But D has a very large number of degrees of freedom, so how can you know that energy of D+O is uncertain?
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?

Also, would you mind elaborating on this statement:
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
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
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?
I guess not.

Also, would you mind elaborating on this statement:
Please see Appendix A that I mentioned there.

Didn't we agree that pure energy states do not evolve; that ψ*ψ is constant in time (for any basis)?
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.

Do you mean that the hidden variables of BM evolve in time?
Yes, that too.

Please see Appendix A that I mentioned there.
I tried, but I'm afraid I didn't understand much beyond what I already knew.

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 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!