How to think about Quantum Mechanics—Part 1: Measurements are about bases

[This post was originally “Part 0”, but it’s been moved. Other parts in this series: 1,2,3,4,5,6.]

In an ideal world, the formalism that you use to describe a physical system is in a one-to-one correspondence with the physically distinct configurations of the system. But sometimes it can be useful to introduce additional descriptions, in which case it is very important to understand the unphysical over-counting (e.g., gauge freedom). A scalar potential V(x) is a very convenient way of representing the vector force field, F(x) = \partial V(x), but any constant shift in the potential, V(x) \to V(x) + V_0, yields forces and dynamics that are indistinguishable, and hence the value of the potential on an absolute scale is unphysical.

One often hears that a quantum experiment measures an observable, but this is wrong, or very misleading, because it vastly over-counts the physically distinct sorts of measurements that are possible. It is much more precise to say that a given apparatus, with a given setting, simultaneously measures all observables with the same eigenvectors. More compactly, an apparatus measures an orthogonal basis – not an observable.We can also allow for the measured observable to be degenerate, in which case the apparatus simultaneously measures all observables with the same degenerate eigenspaces.[continue reading]

How to think about Quantum Mechanics—Part 6: Energy conservation and wavefunction branches

[Other parts in this series: 1,2,3,4,5,6.]

In discussions of the many-worlds interpretation (MWI) and the process of wavefunction branching, folks sometimes ask whether the branching process conflicts with conservations laws like the conservation of energy.Here are some related questions from around the web, not addressing branching or MWI. None of them get answered particularly well. a   There are actually two completely different objections that people sometimes make, which have to be addressed separately.

First possible objection: “If the universe splits into two branches, doesn’t the total amount of energy have to double?” This is the question Frank Wilczek appears to be addressing at the end of these notes.

I think this question can only be asked by someone who believes that many worlds is an interpretation that is just like Copenhagen (including, in particular, the idea that measurement events are different than normal unitary evolution) except that it simply declares that new worlds are created following measurements. But this is a misunderstanding of many worlds. MWI dispenses with collapse or any sort of departure from unitary evolution. The wavefunction just evolves along, maintaining its energy distributions, and energy doesn’t double when you mathematically identify a decomposition of the wavefunction into two orthogonal components.… [continue reading]

How to think about Quantum Mechanics—Part 5: Superpositions and entanglement are relative concepts

[Other parts in this series: 1,2,3,4,5,6.]

People often talk about “creating entanglement” or “creating a superposition” in the laboratory, and quite rightly think about superpositions and entanglement as resources for things like quantum-enhanced measurements and quantum computing.

However, it’s often not made explicit that a superposition is only defined relative to a particular preferred basis for a Hilbert space. A superposition \vert \psi \rangle = \vert 1 \rangle + \vert 2 \rangle is implicitly a superposition relative to the preferred basis \{\vert 1 \rangle, \vert 2 \rangle\}. Schrödinger’s cat is a superposition relative to the preferred basis \{\vert \mathrm{Alive} \rangle, \vert \mathrm{Dead} \rangle\}. Without a there being something special about these bases, the state \vert \psi \rangle is no more or less a superposition than \vert 1 \rangle and \vert 2 \rangle individually. Indeed, for a spin-1/2 system there is a mapping between bases for the Hilbert space and vector directions in real space (as well illustrated by the Bloch sphere); unless one specifies a preferred direction in real space to break rotational symmetry, there is no useful sense of putting that spin in a superposition.

Likewise, entanglement is only defined relative to a particular tensor decomposition of the Hilbert space into subsystems, \mathcal{H} = \mathcal{A} \otimes \mathcal{B}. For any given (possibly mixed) state of \mathcal{H}, it’s always possible to write down an alternate decomposition \mathcal{H} = \mathcal{X} \otimes \mathcal{Y} relative to which the state has no entanglement.… [continue reading]

How to think about Quantum Mechanics—Part 4: Quantum indeterminism as an anomaly

[Other parts in this series: 1,2,3,4,5,6.]

I am firmly of the view…that all the sciences are compatible and that detailed links can be, and are being, forged between them. But of course the links are subtle… a mathematical aspect of theory reduction that I regard as central, but which cannot be captured by the purely verbal arguments commonly employed in philosophical discussions of reduction. My contention here will be that many difficulties associated with reduction arise because they involve singular limits….What nonclassical phenomena emerge as h → 0? This sounds like nonsense, and indeed if the limit were not singular the answer would be: no such phenomena.Michael Berry

One of the great crimes against humanity occurs each year in introductory quantum mechanics courses when students are introduced to an \hbar \to 0 limit, sometimes decorated with words involving “the correspondence principle”. The problem isn’t with the content per se, but with the suggestion that this somehow gives a satisfying answer to why quantum mechanics looks like classical mechanics on large scales.

Sometimes this limit takes the form of a path integral, where the transition probability for a particle to move from position x_1 to x_2 in a time T is

(1)   \begin{align*} P_{x_1 \to x_2} &= \langle x_1 \vert e^{-i H T} \vert x_2 \rangle \\ &\propto \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i S[x(t),x'(t)]/\hbar} = \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i \int_0^T \mathrm{d}t L(x(t),x'(t))/\hbar} \end{align*}

where \int_{x_1,x_2} \mathcal{D}[x(t)] is the integral over all paths from x_1 to x_2, and S[x(t),x'(t)]= \int_0^T \mathrm{d}t L(x(t),x'(t)) is the action for that path (L being the Lagrangian corresponding to the Hamiltonian H).… [continue reading]

How to think about Quantum Mechanics—Part 2: Vacuum fluctuations

[Other parts in this series: 1,2,3,4,5,6.]

Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding“Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper”, although I’m sure there’s another term for this used by philosophers of science.) a  .

Here is everything you need to know:

  1. A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
  2. Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement.
[continue reading]