Friday, December 02, 2016

Einstein's boxes: measurements don't occur "spontaneously"

Florin Moldoveanu discusses a thought experiment – but be sure that it may be turned into a real experiment – called "Einstein's box" (although it's not "the" famous Einstein's box setup from the Bohr-Einstein debates).
Are Einstein's Boxes an argument for nonlocality?
It seems unbelievable that an undergraduate problem that is so rudimentary is being "solved" incorrectly by the author of a book on foundations quantum mechanics as well as Moldoveanu himself.

The problem is the following: A quantum mechanical particle is located in a box. The wave function \(\psi(x,y,z)\) isn't specified and both men pretend that they don't need to talk about it at all. But let's suppose that it's the ground state of a potential well – a wave function that is real and positive inside, e.g.\[

\sin x \cdot \sin y \cdot \sin z

\] assuming that the box is defined by \(\{x,y,z\}\in (0,\pi)\). I emphasize that the probability distribution \(\abs{\psi}^2\) for the particle's position is in no way uniform – when it's in the ground state (lowest energy eigenstate), the particle is unlikely to be very close to the 6 walls (and much more unlikely to be close to the 12 edges and super-unlikely to be close to the 8 corners). Now, a barrier is inserted in the middle of the box, e.g. at \(x=\pi/2\) in my conventions. The question is what is the wave function after this insertion and whether the wave function loses the ability to interfere – loses the information about the relative phase of the part of the wave function in the \(x\lt \pi/2\) "B1" half-box and in the \(x\gt \pi/2\) "B2" half-box.

Moldoveanu correctly disagrees with a totally wrong book promoting Bohmism and written by Jean Bricmont. Most of Bricmont's "arguments" are meant to "prove" that there's nonlocality in Nature and concerning Einstein's box, he promotes the following dilemma:
Here is the dilemma: either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete because\[

\frac{\ket{B_1}+\ket{B_2}}{\sqrt 2}

\] does not describe what is going on.
This claim is clearly wrong. Quantum mechanics is complete and there is no nonlocality. In fact, once the box is divided (let's assume that we do so quickly, much faster than the timescale needed by the particle to travel from one side of the box to the other one), the wave function remains exactly the same.

Moldoveanu correctly says, using a bold face font:
My take on this is that the dilemma is incorrect.
So far so good. However, the following sentence is already complete rubbish:
Splitting the box amounts to a measurement regardless if you look inside the boxes or not and the particle will be in either B1 or B2.
What? This is just so stupid. A process can't "amount" to a measurement if it is not a measurement. And a measurement is a process that leads an observer to learn some actual information that the observer might use if he wanted. When a barrier is just moving from one place or another, no one has to learn anything. It's simply not a damn measurement.

An intelligent undergraduate student of physics should know how to describe the simple problem after the first 5 lectures in the first course on quantum mechanics. We may treat the barrier inside the box as a classical object. The single particle in the box is conserved and its propagation in the environment of the changing box may be captured by a simple time-dependent Hamiltonian.

So at the beginning, the Hamiltonian was\[

H(t) = \frac{p^2}{2m} + V(x,y,z;t)

\] where \(V(x,y,z;t)=0\) inside the box (\(\pi \times \pi\times \pi\)) or \(\infty\) outside the box. To insert the box means to gradually but quickly enough add another term to \(V(x,y,z;t)\) which is also \(\infty\) somewhere on a "plate" near \(x=\pi /2\), in the middle of the box. (You would have to decide whether the barrier gradually strengthens everywhere at once or whether you are moving it from one side of the box. The qualitative conclusions won't be affected.) Let's assume that all the walls including the barrier are reflective.

When you do it, what happens to the particle? It is simply described by a wave function that obeys Schrödinger's equation with the time-dependent Hamiltonian. So immediately after the wall arises, the particle has the same wave function that I mentioned at the top. With the wall, the wave function will evolve and won't be quite stationary. Note that the ground state of the divided-box Hamiltonian (for the box with the barrier) will be something like\[

\left|\sin 2x\right| \cdot \sin y \cdot \sin z

\] inside both parts of the box. However, the initial wave function – which was a ground state of the old, undivided Hamiltonian – won't evolve to the ground state of the new Hamiltonian. There's no reason for such a simple rule. Such an evolution "ground state to ground state" would only be a sensible assumption if the wall were inserted slowly, adiabatically. I made the opposite assumption.

Well, if the wall is inserted slowly, the adiabatic approximation is OK and the original ground state with the factor \(\sin x\) evolves to the divided-box ground state with the factor \(\left|\sin 2x\right|\). These wave functions aren't identical – another "detail" that all those "interpreters" completely obscure. But the function with \(\left|\sin 2x\right|\) may be considered as the "actual" form of the wave function that Bricmont and Moldoveanu sloppily call\[

\frac{\ket{B_1}+\ket{B_2}}{\sqrt 2}

\] So has the "which half-box" information been measured? Not at all. No photon etc. that would be sensitive to the particle's position has been sent anywhere, so this information couldn't have been measured. Moldoveanu's claim to the contrary is absolutely idiotic.

He wrote three segments or sentences in the bold face. I have already quoted the first segment. The second one says:
Do you get interference or not? I say you will not get any interference because by weighing the boxes before releasing the particle inside the interferometer gives you the which way information.
What? Of course, no interference disappears just because a box was divided to two parts. The interference phenomena result from the information about the relative quantum phase between the two parts of the wave function, the part inside B1 and the part inside B2. At least at the beginning, the information about this relative phase is perfectly known and calculable.

What may happen is that after some time, the phases of both parts of the wave function evolve quickly and independently and the two half-boxes' sizes differ a bit. Consequently, the relative phase may quickly change with time as well and become unknown some time later. In practice, our ability to draw an interference pattern may get harder because of that.

But if the half-boxes are exactly the same in size and if we do the experiments quickly enough, no measurement has been done, no collapse has taken place, and of course the relative phase is there.

Moldoveanu repeats the totally wrong claim about the collapse – which hasn't taken place – in several forms and the last bold face comment says:
Nature and quantum mechanics is contextual: when we do introduce the divider the experimental context changes.
This is an extremely vague sentence incorporating some buzzwords they like. "Contextuality" is one of them. OK, is Nature and quantum mechanics contextual? It depends what you exactly mean by this word. If you mean it as a stupid postmodern, ideologically distorted synonym for Bohr's complementarity, then yes, Nature and quantum mechanics are "contextual" (because they're complementary in Bohr's sense).

However, that's clearly not what Moldoveanu and similar would-be researchers of quantum mechanics mean. They always have some totally wrong classical model in mind. They always think that Nature follows the general laws of classical physics well before it does – their supposed measurement (and the loss of the information about relative quantum phases) always takes place prematurely which is why most of their predictions (e.g. Moldoveanu's prediction that the interference patterns must disappear) are just wrong.

In particular, Moldoveanu believes that observations take place spontaneously and without observers – and that's how the experimental context changes. No, observations can't take place without observations i.e. without observers who learn some information. So if the existence of these obviously forbidden processes (by quantum mechanics) is an inseparable part of your understanding of the word "contextuality" or any other buzzword, then these buzzwords are simply and fundamentally wrong.

The experimental context changes when you make the Hamiltonian time-dependent (by moving a barrier) but the character and implication of this change is completely different than what Moldoveanu envisions.

Can we please agree that Mr Bricmont and Mr Moldoveanu don't understand the basic quantum mechanics that good undergraduate students should learn after the first five lectures if not earlier? I am so offended by these self-evident crackpots who don't have a clue even about the basics but who try to market themselves as experts, anyway.

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