Common sense quantum mechanics

post by dvasya · 2014-05-15T20:10:11.534Z · LW · GW · Legacy · 43 comments

Contents

  Related to: Quantum physics sequence.
  1. Plausible reasoning and reproducibility
  4. Schrodinger equation
  5. What it all means
None
43 comments

Related to: Quantum physics sequence.

TLDR: Quantum mechanics can be derived from the rules of probabilistic reasoning. The wavefunction is a mathematical vehicle to transform a nonlinear problem into a linear one. The Born rule that is so puzzling for MWI results from the particular mathematical form of this functional substitution.

This is a brief overview a recent paper in Annals of Physics (recently mentioned in Discussion):

Quantum theory as the most robust description of reproducible experiments (arXiv)

by Hans De Raedt, Mikhail I. Katsnelson, and Kristel Michielsen. Abstract:

which is the well-known correlation of two spin-1/2 particles in the singlet state.

Needless to say, our derivation did not use any concepts of quantum theory. Only plain, rational reasoning strictly complying with the rules of logical inference and some elementary facts about the experiment were used

3. The Stern–Gerlach experiment

This case is analogous and simpler than the previous one. The setup contains a source emitting a particle with magnetic moment S, a magnet with field in the direction a, and two detectors D₊ and D_–.

Similarly to the previous section, P(x|θ,Z) = (1 + xE(θ) ) / 2, where E(θ) = P(+|θ,Z) – P(–|θ,Z) is an unknown periodic function. By complete analogy we seek the minimum of I_F and find that E(θ) = +–cos(θ), so that

In quantum theory, [this] equation is in essence just the postulate (Born’s rule) that the probability to observe the particle with spin up is given by the square of the absolute value of the amplitude of the wavefunction projected onto the spin-up state. Obviously, the variability of the conditions under which an experiment is carried out is not included in the quantum theoretical description. In contrast, in the logical inference approach, [equation] is not postulated but follows from the assumption that the (thought) experiment that is being performed yields the most reproducible results, revealing the conditions for an experiment to produce data which is described by quantum theory.

To repeat: there are no wavefunctions in the present approach. The only assumption is that a dependence of outcome on particle/magnet orientation is observed with robustness/reproducibility.

4. Schrodinger equation

A particle is located in unknown position θ on a line segment [–L, L]. Another line segment [–L, L] is uniformly covered with detectors. A source emits a signal and the particle's response is detected by one of the detectors.

After going to the continuum limit of infinitely many infinitely small detectors and accounting for translational invariance it is possible to show that the position of the particle θ and of the detector x can be interchanged so that dP(x|θ,Z)/dθ = –dP(x|θ,Z)/dx.

In exactly the same way as before we need to minimize Ev by minimizing the Fisher information, which is now

However, simply solving this minimization problem will not give us anything new because nothing so far accounted for the fact that the particle moves in a potential. This needs to be built into the problem. This can be done by requiring that the classical Hamilton-Jacobi equation holds on average. Using the Lagrange multiplier method, we now need to minimize the functional

Here S(x) is the action (Hamilton's principal function). This minimization yields solutions for the two functions P(x|θ,Z) and S(x). It is a difficult nonlinear minimization problem, but it is possible to find a matching solution in a tractable way using a mathematical "trick". It is known that standard variational minimization of the functional

yields the Schrodinger equation for its extrema. On the other hand, if one makes the substitution combining two real-valued functions P and S into a single complex-valued ψ,

Q is immediately transformed into F, concluding the derivation of the Schrodinger equation. Incidentally, ψ is constructed so that P(x|θ,Z) = |ψ(x|θ,Z)|², which is the Born rule.

Summing up the meaning of Schrodinger equation in the present context:

Of course, a priori there is no good reason to assume that on average there is agreement with Newtonian mechanics ... In other words, the time-independent Schrodinger equation describes the collective of repeated experiments ... subject to the condition that the averaged observations comply with Newtonian mechanics.

The authors then proceed to derive the time-dependent SE (independently from the stationary SE) in a largely similar fashion.

5. What it all means

Classical mechanics assumes that everything about the system's state and dynamics can be known (at least in principle). It starts from axioms and proceeds to derive its conclusions deductively (as opposed to inductive reasoning). In this respect quantum mechanics is to classical mechanics what probabilistic logic is to classical logic.

Quantum theory is viewed here not as a description of what really goes on at the microscopic level, but as an instance of logical inference:

in the logical inference approach, we take the point of view that a description of our knowledge of the phenomena at a certain level is independent of the description at a more detailed level.

and

quantum theory does not provide any insight into the motion of a particle but instead describes all what can be inferred (within the framework of logical inference) from or, using Bohr’s words, said about the observed data

Such a treatment of QM is similar in spirit to Jaynes' Information Theory and Statistical Mechanics papers (I, II). Traditionally statistical mechanics/thermodynamics is derived bottom-up from the microscopic mechanics and a series of postulates (such as ergodicity) that allow us to progressively ignore microscopic details under strictly defined conditions. In contrast, Jaynes starts with minimum possible assumptions:

"The quantity x is capable of assuming the discrete values x_i ... all we know is the expectation value of the function f(x) ... On the basis of this information, what is the expectation value of the function g(x)?"

and proceeds to derive the foundations of statistical physics from the maximum entropy principle. Of course, these papers deserve a separate post.

This community should be particularly interested in how this all aligns with the many-worlds interpretation. Obviously, any conclusions drawn from this work can only apply to the "quantum multiverse" level and cannot rule out or support any other many-worlds proposals.

In quantum physics, MWI does quite naturally resolve some difficult issues in the "wavefunction-centristic" view. However, we see that the concept wavefunction is not really central for quantum mechanics. This removes the whole problem of wavefunction collapse that MWI seeks to resolve.

The Born rule is arguably a big issue for MWI. But here it essentially boils down to "x is quadratic in t where t = sqrt(x)". Without the wavefunction (only probabilities) the problem simply does not appear.

Here is another interesting conclusion:

if it is difficult to engineer nanoscale devices which operate in a regime where the data is reproducible, it is also difficult to perform these experiments such that the data complies with quantum theory.

In particular, this relates to the decoherence of a system via random interactions with the environment. Thus decoherence becomes not as a physical intrinsically-quantum phenomenon of "worlds drifting apart", but a property of experiments that are not well-isolated from the influence of environment and therefore not reproducible. Well-isolated experiments are robust (and described by "quantum inference") and poorly-isolated experiments are not (hence quantum inference does not apply).

In sum, it appears that quantum physics when viewed as inference does not require many-worlds any more than probability theory does.

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