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Energy conservation holds in quantum mechanics

Sometimes (and increasingly often), totally wrong and dumb papers are accepted by arXiv.org. An example is a quant-ph paper

Energy Non-Conservation in Quantum Mechanics
by Sean M. Carroll and Jackie Lodman which should have prevented the authors from passing the most elementary undergraduate course of quantum mechanics but it didn't. Instead, we live in the era when the mass media are intensely misinforming and indoctrinating the readers so you may read a promotion of this pseudoscience at the moderately widely read blog of the male co-author.

OK, they claim that the only meaningful conservation of energy in quantum mechanics is the conservation of the expectation value of the Hamiltonian; that this is violated; that this violation can be arbitrarily large; that this violation cannot be attributed to the energy of the observer or the apparatus; and that the "many-worlds interpretation" makes all these questions more controllable. Each of the five statements is absolutely and fundamentally wrong.



So first, it is simply not true that the conservation of energy in quantum mechanics may be reduced to the expectation value \(\langle E \rangle\). Instead, it is the energy itself that is conserved (in quantum mechanical systems with time-independent Hamiltonians). In the Heisenberg picture which makes it extremely transparent, the time evolution of operators is given by\[ i\hbar \frac{d \hat L}{dt} = [\hat L,\hat H]. \] I added the hats to emphasize that this is an operator equation which means that an equation holds for every matrix entry of the operator \(d \hat L / dt\).



A consequence of the Heisenberg equation above is its expectation value\[ i\hbar \frac{d \langle \hat L \rangle}{dt} = \langle [\hat L,\hat H] \rangle \] (the Ehrenfest theorem is the well-known example of this equation) but the opposite implication doesn't hold. The equation for the expectation value is just one "scalar" equation while the full matrix-valued equation – roughly equivalent to \(N^2\) scalar equations where \(N\) is the dimension of the Hilbert space (which is almost certainly infinite) – tells us much more.

Now, the authors claim that the expectation value of the Hamiltonian "is" energy in quantum mechanics. But it is not. The expectation value of the Hamiltonian is the expectation value of energy. But the "expectation value of energy" isn't the same thing as "energy". Can you spot three differences between them? Yes, the three differences are "expectation", "value", and "of". These are words with a meaning that play a role, not just empty words designed to make a moron sound smarter (which is how it is apparently used by these two writers).

We may discuss not only the expectation value of the energy but the expectation value \(\langle H^2 \rangle\) of \(H^2\), the squared energy, or the expectation value of any function of energy. All of them are "equally fundamental" and they are not simple functions of each other. Instead, all of them may be deduced from the probability distribution for energy:\[ \rho(E) = \bra \psi \delta (\hat H - E) \ket \psi \] or, if you work with a density matrix, the obviously equivalent formula is\[ \rho(E) = {\rm Tr}[\rho \cdot \delta(\hat H - E) ]. \] The resulting function \(\rho(E)\) depends on the state vector \(\ket\psi\) or the density matrix \(\rho\) and it integrates to one (because the delta-function does; and because \({\rm Tr}\,\rho \) or \(\bra \psi \psi\rangle\) is equal to one). If the spectrum of the Hamiltonian is discrete, this function \(\rho(E)\) will really be a distribution which is a combination of the Dirac delta-functions at the points of the eigenvalues.

Of course, some of the readers of that blog know very well that this talk about the expectation values as if they were energy is completely wrong. For example:
SOMDATTA BHATTACHARYA wrote:
January 28, 2021 AT 9:16 AM
The trouble with all this is it doesn’t make sense to talk about the expectation value of the Hamiltonian with respect to a linear superposition as representing any kind of an energy of it, any more than it does to talk about the expectation value of the position operator for a state as representing its position.
Right. The expectation value of a thing is an expectation value, not the thing itself. The expectation value, as the name indicates, is some average extracted from the expectations i.e. from the subjective knowledge or beliefs (rigorously based on the previous observations, however, not on random guesswork) of the observer which is stored in the wave function. And the Heisenberg equation above (whose right hand side is \([H,H]=0\) in the case of the time-independent Hamiltonian) implies that the whole probability distribution \(\rho(E)\) is constant in time. The probability of each value of energy is the same as it was a minute ago.

Up to the moment of the measurement. If the observer measures energy, the wave function gets reduced to an energy eigenstate and the value of energy is therefore "sharply" equal to the corresponding chosen energy eigenvalue. This eigenvalue \(E\) had to have \(\rho(E)\neq 0\), otherwise it couldn't have occurred. This is why there is never any real contradiction with the conservation of energy: the value of \(E\) had to be possible (and probably rather likely) before the measurement, \(\rho(E)\neq 0\), and this measured value became a fact after the measurement. Before the measurement of the energy, the state vector "probably" wasn't an energy eigenstate, but after the energy measurement, it is an energy eigenstate.

If we later measure another observable \(L\) (such as the position or momentum) that doesn't commute with the Hamiltonian, we bring the system to an eigenstate of \(L\) and modify the distribution of energies because the eigenstates of \(L\) "probably" aren't eigenstates of \(H\) at the same moment. If we measure the energy after that again, we may get a different eigenvalue. This is the standard outcome of the measurement of non-commuting observables, in this case \(L,H\), as guaranteed by the uncertainty principle: the measurement of one changes the other (changes the probabilistic distribution for the other). This kind of apparent violation of the energy conservation was first described by Wigner and in the Wigner-Araki-Yanase theorem. These authors just try to rediscover some very elementary insights and they do so incorrectly.

But the collapse of the wave function is a subjective, observer-bound phenomenon. From the viewpoint of another, external observer Z, both the measured system and the original observer are parts of a physical system described by Z's wave function (which isn't quite the same as the first observer Y's wave function because what they consider measurements aren't equal events). According to Z, the measurement by Y is just a standard unitary process in quantum mechanics in which the wave function evolves to an entangled state (entangling the measured object and the apparatus) and \(\rho_Z(E)\) is conserved, so the conservation of the energy is perfect. From the viewpoint of Z, any non-conservation of energy that is apparent to Y may be described as a transfer of energy between Y and the object observed by Y – both of them are subsystems of the system measured by Z.

When the Hamiltonian is time-independent, there is no legitimate justification for the claim that "the energy is not conserved". The energy may be uncertain or modified by measurements of other, non-commuting observables (that's the description by an observer who observes and therefore modifies the measured object); or the energy may be transferred from the observed object to the apparatus (and the first observer) or vice versa (that's the description by a third party observer). But the energy isn't lost and it isn't created out of nothing.

Needless to say, the authors of this utterly wrong paper (which also includes some totally useless particular setups) derive all their incorrect conclusions from the completely invalid idea that the wave function is an objective information about the world, i.e. a classical field in a space, which it is not. A wave function is a complex generalization of the probability distributions which encodes the subjective knowledge of an observer. Their conflicting and therefore incorrect assumption, so omnipresent in atrocious quality popular books about (or against) physics, also leads them to say some incoherent things about the "many worlds interpretation". The statements above are well-defined and formulated in a language that is pretty much unavoidable for doing quantum mechanics, up to equivalent translations (including literal translations to other languages). My statements are right and they are not open to any "interpretations". The word "interpretation" is used purely as a justification for writing absolute gibberish and this wrong paper is a great example of that fact (originally stated by Werner Heisenberg when he haplessly recommended the word "interpretation" to the pseudoscientists).

There exists no coherent "many worlds interpretation" (or any other anti-Copenhagen interpretation) that would allow to replace the description that I gave you above by an equally or more coherent one. There have been dozens of blog posts on this website about that, I don't plan to add another one because it was clearly a waste of time.

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