Tag: HTTAQM

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How to think about Quantum Mechanics—Part 8: The quantum-classical limit as music

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

On microscopic scales, sound is air pressure f(t) fluctuating in time t. Taking the Fourier transform of f(t) gives the frequency distribution \hat{f}(\omega), but in an eternal way, applying to the entire time interval for t\in [-\infty,\infty].

Yet on macroscopic scales, sound is described as having a frequency distribution as a function of time, i.e., a note has both a pitch and a duration. There are many formalisms for describing this (e.g., wavelets), but a well-known limitation is that the frequency \omega of a note is only well-defined up to an uncertainty that is inversely proportional to its duration \Delta t.

At the mathematical level, a given wavefunction \psi(x) is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction \psi. The mapping of the analogyI am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal f(t) the “musical score” of f.a   is \{t,\omega,f\} \to \{x,p,\psi\}. Wavefunctions can of course be complex, but we can restrict ourself to a real-valued wavefunction without any trouble; we are not worrying about the dynamics of wavefunctions, so you can pretend the Hamiltonian vanishes if you like.

In order to get the acoustic analog of Planck’s constant \hbar, it helps to imagine going back to a time when the pitch of a note was measured with a unit that did not have a known connection to absolute frequency, i.e.,… [continue reading]

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How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well.… [continue reading]

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

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[This post was originally “Part 0”, but it’s been moved. Other parts in this series: 1,2,3,4,5,6,7,8.]

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. To be abstract, you could say it measures a commuting subalgebra, with the nondegenerate case corresponding to the subalgebra having maximum dimensionality (i.e., the same number of dimensions as the Hilbert space). Commuting subalgebras with maximum dimension are in one-to-one correspondence with orthonormal bases, modulo multiplying the vectors by pure phases.a   You can probably start to see this by just noting that there’s no actual, physical difference between measuring X and X^3; the apparatus that would perform the two measurements are identical.… [continue reading]

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

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

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.

Second possible objection: “If the universe starts out with some finite spread in energy, what happens if it then ‘branches’ into multiple worlds, some of which overlap with energy eigenstates outside that energy spread?” Or, another phrasing: “What happens if the basis in which the universe decoheres doesn’t commute with energy basis? Is it then possible to create energy, at least in some branches?”… [continue reading]

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

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

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 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.

So where do these preferred bases and subsystem structure come from? Why is it so useful to talk about these things as resources when their very existence seems to be dependent on our mathematical formalism? Generally it is because these preferred structures are determined by certain aspects of the dynamics out in the real world (as encoded in the Hamiltonian) that make certain physical operations possible and others completely infeasible.… [continue reading]

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

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

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). As \hbar \to 0, the exponent containing the action spins wildly and averages to zero for all paths not in the immediate vicinity of the classical path that make the action stationary.

Other times this takes the form of Ehrenfest’s theorem, which shows that the expectation values of functions of position and momentum follow the classical equations of motion.… [continue reading]

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How to think about Quantum Mechanics—Part 3: The pointer and Schmidt bases

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

A common mistake made by folks newly exposed to the concept of decoherence is to conflate the Schmidt basis with the pointer basis induced by decoherence.

[Show refresher on Schmidt decompsition]
Given any two quantum systems \mathcal{S} and \mathcal{E} and a pure joint state \vert \psi (t) \rangle \in \mathcal{H} = \mathcal{S} \otimes \mathcal{E}, there always exists a Schmidt decomposition of the form

(1)   \begin{align*} \vert \psi (t) \rangle = \sum_k c_k \vert S_k (t) \rangle \vert E_k (t) \rangle \end{align*}

where \vert S_k (t) \rangle and \vert E_k (t) \rangle are local orthonormal Schmidt bases on \mathcal{S} and \mathcal{E}, respectively.

Now, any state in such a joint Hilbert space can be expressed as \vert \psi \rangle = \sum_{i,j} d_{i,j} \vert S_i \rangle \vert E_j \rangle for arbitrary fixed orthonormal bases \vert S_i \rangle and \vert E_j \rangle. What makes the Schmidt decomposition non-trivial is that it has only a single index k rather than two indices i and j. (In particular, this means that the Schmidt decomposition constains at most \mathrm{min}(\mathrm{dim}\,\mathcal{S},\mathrm{dim}\,\mathcal{E}) non-vanishing terms, even if \mathrm{dim}\,\mathcal{E} \gg \mathrm{dim}\,\mathcal{S}.) The price paid is that the Schmidt bases, \vert S_k \rangle and \vert E_k \rangle, depend on the state \vert \psi \rangle.

When the values \vert c_i \vert in the Schmidt decomposition are non-degenerate, the local bases are unique up to a phase. As \vert \psi (t) \rangle evolves in time, this decomposition is defined for each time t. The bases \vert S_i (t) \rangle and \vert E_i (t) \rangle evolve along with it, and can be considered to be a property of the state \vert \psi (t) \rangle. In fact, they correspond to the eigenvectors of the respective reduced density matrices of \mathcal{S} and \mathcal{E}.

In the ideal case of so-called pure decoherence, the environment \mathcal{E} begins in an initial state \vert E_0 \rangle and is coupled to the system \mathcal{S} through a unitary of the form

(2)   \begin{align*} U(t) = \sum_k \vert S_k \rangle \langle S_k \vert \otimes U^{\mathcal{E}}_k(t) \end{align*}

with \langle E_k(t) \vert E_l(t) \rangle \to \delta_{k,l} as t \to \infty, where U^{\mathcal{E}}_k(t) is a conditional unitary on \mathcal{E} and \vert E_k(t) \rangle \equiv U^{\mathcal{E}}_k(t) \vert E_0 \rangle. The elements of the density matrix \rho of the system evolve as \rho_{k,l}(t) = \langle E_k(t) \vert E_l(t) \rangle \rho_{k,l}(0), i.e.… [continue reading]

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

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

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. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160]b  
  3. “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
  4. A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.
… [continue reading]