Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.
So — what do you want to know? What are the really interesting questions about QM that we should be talking about?
One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.
Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?
(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)
Love physics… but not very knowledabe other than at the popular science publication level.
Not sure if this is directly related to QM, but I have always been perplexed by “virtual” particles.
What problem are they created to address? Are they “real”, or just a mathematical contruct to help in understanding and analzing? Do virtual particles and the forces they represent have any analogy to the wave/particle duality?
Speaking of questions and answers, Dennis Overbye of the New York Times is responding to questions by email throughout this week (7/7 – 7/11). One already answered is relevant to the topic of this post:
See Overbye’s thoughtful response.
Here’s my best effort to explain this to lay person, off the top of my head.
In classical mechanics, you can’t have a superposition of two different states. You can either say “The particle is over here”, or “the particle is over there”, but not “the particle is simultaneously somewhat over here and somewhat over there.”
However, with (classical) waves (such as sound or light), you [i]can[/i] have a mix of two different states. You can play two different notes on the piano at the same time, or mix two different colors of light. If I ask “What frequency is this sound (or light)?” the answer might very well be “It’s a little bit this, and a little bit that.”
In quantum mechanics, particles also have wave-like behavior, so it’s possible to get a mix of two states. Just like the mix of two notes (two frequency states) can’t be said to have a single definite frequency, in QM you can have a mix of multiple position states without a single definite position. The particle is somewhat in one place and somewhat in another.
The real weirdness is that when you observe the state (measuring the particle to have some particular position) you change the state so that the particle is [i]definitely[/i] in that position. (At least, that’s the Copenhagen view on what you’re doing.) So you can’t [i]first[/i] measure the particle to be in one position and then immediately afterwards measure it to be in some far away position.
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How well does that explanation hold up in terms of readability/accuracy?
I’m a non-laymen, but I still would like to throw my two cents into the hat and suggest two things that I’ve never seen covered in a “popular” exposition of QM that I think should be.
1) Since you mentioned the wave function of the universe, how about the fact that the measurement problem is no problem from that point of view? In other words, the wave function of the universe never collapses. (Right? And if I’m wrong, then what’s doing the measuring that causes the collapse?) I feel like a lot of philosophers would have avoided misusing quantum mechanics if they understood this aspect, which I think is not too hard to convey.
2) Quantum cryptography. Not *too* difficult to explain (although maybe too difficult for a 1-hour show), but also avoids all the hype and questions about the engineering reality of quantum computation. Quantum crypto is real, and has been done at surprisingly high speeds over (to me, at least) surprisingly long distances. Maybe this fact alone should be mentioned during a discussion of quantum computing.
That’s it. Best of luck to you. I’m looking forward to it.
Perhaps most important is linear algebra — the math of vector spaces and linear operators — which (for those who don’t know what they are) can be thought of as generalizations of the vectors and matrices from high school algebra. In particular, quantum mechanical states are vectors in a particular kind of vector space called a Hilbert space, and the “observables” (things you can measure like “where the particles are”) are represented by operators acting on these states.
Differential equations are also very important. Much of non-relativistic quantum mechanics boils down to solving a particular differential equation — Schrodinger’s equation.
I second the motion for Collapse of the Wave Function. I’m a former physicist (ABD), and I never got a handle on all the various (weird!) consequences of wave function collapse.
As near as I can tell, most of the weirdness of QM is in wave function collapse, not wave function evolution. That’s where all the observer effects (and the Deepak Chopra-esque misappropriations of QM) show up….
Thanks!
TimG,
I think you forgot to mention that operators in quantum mechanics are matrices.
JD,
If you’re interested in reading up on it, I’d suggest taking a look at quantum decoherence, as well as some of the derivative concepts like einselection. The basic idea is that wave function collapse is just an observer effect due to how quantum mechanical systems interact with one another. Specifically, it comes about because observers like ourselves are also described by quantum mechanics, and our interactions with our surroundings prevent us from observing any other states our overall wavefunction might be in (e.g. if we observe ourselves being in the state that observed outcome A, we can’t experience the state that observed outcome B: that state has decohered from us).
In Roger Penrose’ “The Road to Reality” is a nice chapter about the measurement problem. He gives about six different possible “solutions”. His own pov is, that a new theory incooperating strings, quantum fields, supersymmetry etc. will give an answer to the measurement problem.
You shouldn’t forget, that mathematical formalism called QM ist about 70 years old now. It’s already a classic theory, which is very constrained (non relativistic, flat spacetime, not a multiparticle theory and it has the measurement problem…).
From my attempts to understand the topic, it seems particles and waves of light are treated as two descriptions of the same state, but are they? Energy and matter are different but interchangeable states of energy, but while matter is gravitationally collapsing, energy, as radiation, is expanding. Since the ignition of matter is where matter turns to energy, where is it that energy turns to matter? Is it when radiation as a wave becomes a photon? Say the process of measuring this wave amounts to grounding it to the measuring device. So the energy wave collapses to this point of contact, like a miniature lightning bolt.
That way the wave collapses as a function of contact and measurement is only a potential aspect of that, while there is no universal collapse, as matter is also igniting and expanding back out as waves of radiation.
Janus (#5) if you’re comfortable with linear algebra, I reckon the best book on QM is “Principles of Quantum Mechanics” by Ramamurti Shankar (2nd ed, 1994).
Basic linear algebra is a doddle, once you get the hang of it, and there are plenty of books. But like almost everything in maths and science these days, even the simplest topics can be developed and elaborated to incredible depths. So I’d be careful not to get bogged down and discouraged by some huge algebra tome beyond your needs.
As one answer to Sean’s question, I’d like it confirmed (if true!) and emphasised that “observation” need not imply a sentient observer, as the word is conventionally understood. So a rock on Pluto could just as well be said to “observe” a photon collapsing onto it. This would eliminate a lot of mystical mumbo jumbo about the special role of sentient observers, for example bringing about their own existence by influencing past events.
For a nice layperson’s description of QM, I like Feynman’s book “QED”: no formulas, no semimystical mumbo-jumbo, frank acknowledgement of the puzzling aspects.
My questions are maybe a little more advanced:
* Why should observables be operators on Hilbert space, and why should they both yield an eigenvalue and leave the system in an eigenstate?
* Is there a tractable example of decoherence? That is, some system where you can work out the behaviour explicitly on paper, but in which there’s a tunable “peeking” parameter which you can turn from small (observable superposition effects) to large (classical behaviour)? For example a double-slit experiment with an electron where you have a charged test mass near one of the slits – close and you can “see” which slit the electron went through, so you get no discernible interference, far and you can’t “see” but you get superposition effects…
I suspect it’s not either. It’s very possible to think of density matrices as fundamental, and wave functions as a very useful special case. Consider
(a) density matrices already have the global phase symmetry “modded out”, and (b) the decomposition of a density matrix into pure states is not unique.
It’s Galilean relativistic, just not Einsteinian relativistic. There’s nothing better for understanding just how much the phase is mathematical bookkeeping than realizing that not only is the global phase irrelevant, but that changes in phase between one component of the wave function and another are observer dependent. In one sense this is obvious — different observers will naturally use different basis vectors. For spin and rotations these are connected by representations of the rotation group. For Galilean boosts, these are connected with representations of the translation(*) group, e^(ipx/hbar}. But I find this to be somewhat surprising.
It’s also a fine multiparticle theory. The distinction is that it’s a fixed particle-number and not a variable particle-number theory
(*) Ok, technically, not translation, but translation + boosts, the translation bit is near trivial, and the boost bit looks like translation, since it’s just addition of vectors.
Can I put in a request for at least a paragraph on how the Schroedinger equation was first formed? Or the other two variations. Whenever I took quantum courses the equation was just presented without any commentary on how he came up with the equation. I usually find that understanding the derivation of an equation is more useful in helping people understand what an equation means. In some ways it seems the equation existed before scientists had an interpretation of what the wave function means and that just struck me as wacky.
Sean:
I would suggest that you and David discuss in detail the two-slit experiment. This is not a passe topic. It involves all the essential characteristics of QM. Topics to be covered:
1. What do we really mean when say that we detect the entity in the two-slit experiment be it an electron, photon or other?
2. Is the entity destroyed in this detection process?
3. Do the apertures “detect” the entity? If not why not?
4. Do the apertures have an effect on the entities? If yes what effect. If not why not?
5. What constitutes a “real” entity?
6. Why cannot the “wave function” be considered real?
7. If the “wave function” is real then doesn’t this resolve all of the issues?
There is an interesting discussion of the “wave function” and other matters by Feynman from a conference on the role of gravity and the need for its quantization, held at the University of North Carolina in Chapel Hill in 1957:
http://arxiv.org/ftp/arxiv/papers/0804/0804.3348.pdf
Like #35, I am frustrated by people who use QM to support dualism by claiming that minds can alter physical reality by providing a conscious observer to create QM effects such as the results of the double-slit experiment. I believe, like #35, that a mechanical observer would get the same results, but of course it is difficult to prove this without a conscious observer being involved at some point.
The closest I can come to a counter-argument is to conceive of a adouble-slit experiment in which automated mechanical systems measure and record both which slit photons go through and the resulting interfrence or non-interference pattern. Then before a human reviews the latter results, the former results are physically wiped from computer memory without review.
Anyway, I would appreciate the thoughts of someone who is smarter and more more knowledgable on this – seconding #35’s request.
To what extent do quantum phenomena need explanations beyond existing quantum theory?
I mean this question even apart from the measurement problem. Suppose (like Short et al) you can describe non-local correlation using a formalism that involves classical binary inputs and outputs, plus some randomness. Or suppose you can do the same thing using game theory.
Are such accounts “explanations” that teach us something new about the quantum world? Or are they just examples of clever formalism?
What is the quantum equation for consciousness? I frequently see physicists (e.g. Andre Linde) talk about how a conscious observer is needed for “observation’ of an event/particle, so I figure this must be well-characterized physical quantity.
I have a beef against “decoherence” as a supposed solution for semi-solving (even at best) the problem of the collapse of the wave function. Maybe or not I truly appreciate the concept, but in any case: regardless of whether the wave superpositions are coherent or incoherent, waves would just stay waves unless there was some additional influence or principle forcing the sudden localizations that we call “collapses.” There is nothing intrinsic to the math of waves that enables or even describes “collapses” (not to be confused with the coming and going of e.g. thick spots due to shuffling of frequencies and the resultant Fourier composition of peaks, etc. – that’s still just a pure wave effect per se.) The collapse thing has to be “put in by hand.” You can believe in multiple worlds (hypocritical for anyone otherwise professing positivism and “falsifiability, BTW), but even then collapse events are gratuitous: we are just making multiple examples of all possible ways the collapses could occur. But by contrast, a bunch of true waves would just stay waves and not collapse at all, whether in some possible way or all possible ways!
Maybe the question I asked in #20 is pertinent since half-wave plates interact with transiting photons in a peculiar way. Each transit must, on average, impart some spin (conservation of angular momentum) since the transit of many photons adds to measurable net spin. Yet oddly, transit does not “collapse” a photon, the photon’s circular base state composition is switched. Hence it stays a coherent superposition. You’d need to re/read that post to appreciate the full implications. In any case, this whole thing is perhaps the most challenging intellectual and metaphysical issue connected to “the universe” that the human mind has to ponder.
Neil,
I dunno. Every summit climbed only seems to leave us looking at a larger one in the distance.
What are the main “everyday” applications of Quamtum Mechanics? ie How useful is QM outside of academia?
I can think of the lasers used in optical storage, bar-code readers and medicine, but are there any other everyday applications? (AFAIK the transitors used in current computers can be explained by classical physics)
The thing that gets me is the “sums of history” idea. It sounds incredibly cool and every time I read about it, I think, “That can’t really be what they mean!”
What does it mean?
Anne writes:
I think there’s probably a lot of different levels on which this question can be answered. Here’s my attempt at a basic answer:
Why a Hilbert space? A Hilbert space is a particular kind of vector space, with some additional structure (particularly and inner product). A vector space is a set of elements (the vectors) with some operations defined on those elements. Specifically, you can add two vectors together, or you can multiply them by scalars (that is, numbers). The addition and multiplication also have to satisfy certain axioms.
In quantum mechanics, you can have a superposition of two states. That is, if you have a state A and another state B, then it is also possible to have a state that’s, say, 30% A and 70% B.
state = sqrt(0.3)*A + sqrt(0.7)*B
So clearly our states can be multiplied by numbers and added together. This leads us to vector spaces and, more specifically, Hilbert spaces (once we define an inner product and check that all the appropriate axioms hold.)
Why are observables operators? I guess the simplest answer is that the act of observing changes the system being observed, so the observables can’t merely be numbers, they have to operate on the state. Studying the properties off these observations one sees that they satisfy the necessary axioms to be linear operators, and, in fact, self-adjoint operators.
Why eigenvalues and eigenvectors (a.k.a. eigenstates)? Well, if you have a state with a 30% chance of X = 1 and a 70% chance of X = 2, the expectation value of X is 0.3*1 + 0.7*2 = 1.7 You’re multiplying the different components of the state by the value of the observable in that component, which is exactly what would happen if they’re eigenvectors of the X operator with the eigenvalues 1 and 2.
Why does it leave the system in an eigenstate? That’s an empirical observation: if you measure a particular observable, the state is changed so that further measurements yield the same value for the observable. Why that happens is a deep question, but we can see that it does happen and the math has to reflect that.
What would chemistry be like if there were four dimensions (with very little curvature, like the three we know so well)? Would there be four 2p orbitals? Could we have quadruple bonds? Everyone of whom I ask this question starts out “Well, I”m not a quantum chemist…”, and I haven’t met any actual Q-chemists yet…
These are all great questions. I’d be interested in your reaction to the new work by Zeilinger et al to test realism (following in the footsteps of Bell’s and Aspect’s work). From: http://www.seedmagazine.com/news/2008/06/the_reality_tests_1.php