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?”)
Comments
165 responses to “Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask”
When it comes to Planck scale physics I think that there is a need for a totally different manner of thinking. The Simon & Garfunkel song modified to “When I think back on grad school, it’s a wonder I can think at all,” almost comes to mind. In many ways it requires some form of “unlearning,” or acknowledging that our theories are basically symbolic forms which as best have a tentative sort of truth.
Lawrence B. Crowell
James Robson, the uncertainty principle (at least, the position/momentum version) isn’t just an ad hoc rule without basis in something even more fundamental. I’ve seen it explained in terms of the Fourier analysis of the wave function of a particle. If the momentum is well defined, the frequency spread is tiny and the wave “packet” created by interference of all those waves is very long. If the position is well-defined (small packet) then the Fourier analysis requires a large spread in the frequency composition. Hence the momentum is wide spread and thus uncertain (since frequency is related to kinetic energy and thus velocity, given a mass.) This is an unavoidable consequence of the fundamental maths of waves; I suppose it “couldn’t be otherwise” given the wave nature of matter/energy to begin with. Contrast this with the projection postulate, that “measurement” must convert the uncertainties in the WF into a basis state. AFAIK the PP doesn’t have to be true in the same sense as the UP, and I offered a possible way around it in #s 20 and 22.
However there is one oddity, a sort of paradox, I thought of about the whole idea of giving a wavelength to “an” object using lambda = h/mv [m is relativistic mass here.] That seems OK at first glance, but consider an objection Galileo made to Aristotelian mechanics: A-mech said that the speed of an “object’s” fall was proportional to its mass. Galileo objected, that it is ambiguous what constitutes “an” object or several, etc. For what if we tie two rocks together with a string. Now are they “a” more massive object or still “two”? What if the connection was more substantial? It is ambiguous “how many objects” there are, but the total mass is a coherent concept. This lent weight (pardon the pun) to Galileo’s notion that rate of fall was independent of “object” properties, ultimately enshrined in GR as the Equivalence Principle.
Hmmm… But the matter wavelength rule (related to the UP) specifies a mass of a specific object! So, what if we “tied two neutrons together” with a piece of string, etc? What de Broglie wavelength and thus uncertainties should apply? Sure, fundamental particles are not like rocks and have some special identity, etc. But the problem is real as a matter of principle, and QM is supposed to apply “to anything.” So, what about molecules tied together with little strands of CH2 polymer, etc? What wavelength applies to this complex, does it matter how well connected they are? I already suspect the dynamical coordination is crucial to the answer, which may not be a perfectly neat answer either. It probably hashes out in the interference of the different possible waves, or such, given the interactive situation. I still wonder how measurement and “collapse” is affected by collective connections.
Sam,
Which would be where inflation theory comes in and that seems to get more nebulous by the day
Erk! I can’t resist… Wouldn’t it have to take light 13.7 bly’s to cross that ‘tiny quantum entity,’ otherwise it would seem to be expanding volume in a stable dimension of space, as measured by lightspeed? If space expands, should the normal measure of it increase proportionally?
Neil B.: Yes, space has expanded since the “Big Bang”…of course, to the extent that a tiny quantum entity now has an observed radius of 13.7 Billion light years.
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The SNI data and WMAP tells us otherwise. The spatial manifold on the Hubble frame was rapidly expanded. As such the distance to far galaxies begins to increase then the further out they do so faster than expected by simple motion. As one looks further out these galaxies are on different local frames. The light cones are “turned” away from the direction of our local light cone. Of course if one were to transform to this distant frame the light cone coincident with our galaxy would appear turned away. So the universe is homogeneous and isotropic. This also means that while the spatial surface on the Hubble frame might be a three dimensional R^3 flat space it is also embedded in a spacetime that is not flat. This means that the spatial surface evolves into successive spatial surfaces by being stretched, or where spatial points slide away from each other. As a result the CMB exists somewhere out to 70-80 billion light years away.
Cosmology is a strange science. The one thing which makes it fundamentally different is that there is only one system we have to work with. This gets really strange if one considers quantum cosmology. The outcome of the quantum wave function, proximally thinking of this as similar to the so called wave function collapse, is the one system we observe. Yet QM is a statistical theory — but we have no experimental statistics! There is nothing else out there to compare things to and no way to obtain a “scatter plot” of quantum outcomes. It is a standard of science that experiments and observations must consider an ensemble — observations of many quasars, many particle scattering events, many ant colonies of the same species, and so forth.
Lawrence B. Crowell
Lawrence, to avoid confusion note that ‘Yes, space has expanded since the “Big Bang”…’ is from Sam Cox at #125. As for the WF of the universe, what indeed is “collapse” supposed to mean for it as a whole? I have to wonder, about Sean’s initial statement that if we “consider” two electrons, we have one WF for two particles, not two WFs. But there can be “two WFs” existing because we can write up two (?) separate Schrödinger equations to evolve under the rules, it just can’t be done that way if we “consider” them together – ?! Just what is this “considering”, especially transitioned from “thought” to “reality”?
BTW, have you tried walking through the concept about measuring photon polarization I’ve been talking about?
Re #41, exactly, and I’d go further and suggest that in a sense everything of an appropriate nature can (with suitable probability) “observe” everything else that comes its way, like the ultimate police state where each citizen dutifully spies on everyone else!
Furthermore, if one applies the same principle even to small-scale vacuum fluctuations and states then so-called dark energy could perhaps be explained as a book keeping device to cancel or vitiate extra information/energy that would otherwise be manifested in violation of the “no cloning” QM result.
This isn’t a hobby horse BTW, little more than a passing fancy. All the same, although much more could be said, with the new rules recently stated for this blog, I daren’t elaborate further 😉
BTW, *completely* off topic footnote, but I can’t resist mentioning it for the benefit of all you helpful and interesting contributers, especially those whose native language is not English. There’s a wonderful free pop-up spell-checker and dictionary called WordWeb that works with any Windows app. See http://wordweb.info/. All you do is install it and type ctrl+alt+W to see the spelling and meaning(s) of the word your cursor is on. (Needless to say I’m not associated with them in any way.)
lee (#24) wrote:
Seems like http://arxiv.org/abs/0807.1544 might have a bearing on that.
Are s, px, py and pz orbitals described by a single wave function? What about sp, sp2 and sp3 orbitals, are they also described by one wave function? Is the Periodic Table and bonding in chemistry described by single wave functions?
Lawrence B. Crowell
most helpful and interesing, thank you.
I used a “Quantum” model lawnmower to cut my lawn this afternoon! (Really!) You can imagine how it works: You turn it on, then it starts to blur and soon covers the whole yard like a gray mist. Then here and there, “at random”, blades of grass will suddenly be snipped down. With 6.0 HP, quite a few blades go down each second but it takes a while to get 90% cut. (I say, that’s enough and to heck with the stragglers, since the chance of some (but the same chance per any one blade of course) remaining blade of grass being quantum mechanically whacked go down as fewer and fewer are left intact.) You have to get out of the yard of course, since it could slice your ankle if you hang around (I’m not taking any chances!) I love it, but don’t have much clue how it works.
Hey Sean, how do you account for the Periodic Table that is made up of hydrogen like orbitals? How could one wave function describe multi-electron atoms? That is entirely contrary to the structure of chemistry.
It is interesting that this discussion has turned to atomic bonds. Do the electrons in a molecule exist in a single wave function? The unequivocal answer is yes! This is certainly the case for simple molecules, and I suspect also for very complex molecules as well. By this I think that the electronic system for DNA and complex polypeptides may well form a single Fermi-electronic system. At this time the only electronic system which involves molecular biological systems that is well understood is the hydrogen bond between purines and pyramidines in DNA. Outside of that our knowledge literally falls off into a dark age.
The problem is that this involves some fine tuned quantum physics. With solid state physics the electronic state is defined largely according to a conduction band, usually with respect to a Fermi surface, and atomic levels. With a complex molecule, say a complex saccharide, polypeptide, or DNA the electrons are arranged in what I can only describe in more “artistic” systems. The words I use here betray our complete ignorance of this sort of problem.
To start to address this problem we might begin to examine some of the symmetires inherent in many body problems, here with Fermi statistics, where they are electromagnetically attracted by “classical” positive charges. The problem really has all the intellectual challenge of quantum cosmology. This might have much to do with the shape and functions of kinases. phophotases, transferases and other molecules involed with biological processes.
Lawrence B. Crowell
Celestial mechanician wrote:
The short answer is that this is only the most basic approximation to the true state of the atom, ignoring the Coulomb force between electrons (but taking into account the effects of the Pauli exclusion principle — since otherwise you’d just have all the electrons in the same orbital.) For greater accuracy, there are various ways to approximate the effect of the interaction between electrons.
TimG, since you seem to know: I am aware that we can show how the electron etc. wave function responds to fields (I think, directly in terms of “potentials” and also affected by weird pseudofields like the vector potential A field as in Aharonov-Bohm effect.) But I wonder what sort of E field etc. comes from an electron wave/s. I intuitively expect, the E field is what you’d expect from a “cloud” of charge spread out according to the probability distribution given by the WF. That ties in with the calculation of upper orbitals being determined by the electrons “orbiting” around a screened nuclear charge, etc. Since the electron waves are spread out, it isn’t then as simple as e.g. the four outer electrons is carbon seeming to orbit a nuclear charge of only +4. But is it even as “simple” as using the entire WF/s to get an effective E field at different distances from the nucleus, did I get the idea right? I am not talking about how hard it is to solve analytically, just to get the physical picture right, tx and this should help the understanding of lots of readers.
A few recent comments asked about whether a single wavefuction could (or should, I suppose) be used to describe multi-part systems such as atoms and molecules. This relates (all the long way back in history – over 130+ blog entries!) to Sean’s motivational intro were the point was made that this will be the case according to conventional QM.
When considering a QM system, the supposedly individual parts are not assigned individual descriptions (wavefunctions if appropriate, but more generally some vector in some linear space where they might live). Instead, this description can only be applied to the whole system as supposedly isolated from us. This is where the real problems start if you (as we all do) approach this with classical intuition. Even if we imagine parts of this system should be regarded as separate (e.g. due to distance/time preventing interactions) then this does not necessarily allow us to split the description up as we might do classically: the EPR “paradox” case is the most famous example.
Taking this to its logical conclusion we arrive at the “wavefunction of the Universe” as we can’t really draw any dividing lines anywhere.
Looking forward to the video blog.
-James.
James Robson,
Wouldn’t be possible, at least in theory, to describe single wavefuctions as [I]intricated[/I] multi-part systems?
In GR there is E=MC^2, which is time dependant. What is the equivelant in QM?..and why must this equation, if triggered in/on, a quantum scale, have instant and far reaching consequnece for conversion process, ie from one local quantum horizon, planck scale?.. to say the edge of the observable Universe?
P.S I meant to ask for/if there exists the above process, then could this be the trigger (instantaneous) mechanicism for Big Bang? in relativity time causes interactions to have discrete locations for events to happen at different times, little bangs!.. but in QM all events are effectivly continueous?
Hi Layman_42,
I was kinda hoping one of the experts would have replied to your question by now and saved me from any embarrassment! My physics training was limited and a long time ago.
The answer to your question is “yes” as far as I am aware. I was just talking about the model offered by “conventional” quantum mechanics – but if you shop around you might find something more to your liking.
In fact, you might be able to do better than you suggest in your question, and continue with the idea that your particles have classical positions and momenta at all times – just like in the good old classical days. Of course, everything comes at a price. These “hidden variable” theories, as they are called, seem to be constrained by Bell’s theorem (based on his famous inequalities) and must abandon either locality (things can only affect their neighbours and these effects propagate outwards at finite speed – i.e. something can’t instantaneously affect something else at a distance) and/or “realism” (that a measurement only reveals what was objectively and definitely there in the first place (no casting of dice!)). The most famous of these approaches is due to David Bohm and maintains reality at the cost of locality.
However, when QM is combined with relativity (only the special theory as yet in established theory) the notion of distinct particles does seem to be become even more unlikely. Particles can now be created and destroyed (in fact this happens all the time), and so their independence is gone. It seems better to think of them as (quantised) vibrations of fields. In fact, when the interactions between these fields are taken into account then things get even worse thanks to Haag’s theorem.
This is the kind of stuff I personally would like to hear discussed by more knowledgeable people than myself people in the video blog.
-James
Hi Paul Valletta,
E=MC^2 which, of course, is the most famous equation ever devised, is, as you say, a result of special relativity, not QM, thinking (though you can find debates about the “true” nature of this assertion (SR or QM) on the web if you look). I vote for special relativity here – but still the basic SR geometry needs to be supplanted with something else to get the result (SR only places restrictions on the possible dynamical laws; it does not tell us what they are). Other symmetry and conservation law ideas must brought in to finish it off (these are usually the same thing as Emmy Noether pointed out). See Terrance Tao’s blog for a neat discussion of the derivation of E=MC^2. Also I seem to recall Rindler’s “Special Relativity” book was good for this fundamental stuff such as justifying (well, motivating) the linearity of the Lorentz transformations (this is probably out of print by now).
As for QM, well as far as I am aware all QM theories in the standard model are based on Lagrangians or Hamiltonians just as classical models usually are. The nature of energy is tied down there. If you take a classical system where the Hamiltonian represents the energy, and put a Newtonian expression in for that enery (just as Schroedinger did on his second attempt) then the QM version of your equations will inherit this same behaviour. Alternatively, if you put in the SR version of energy/momentum instead you will get a quantum version of SR energy/momentum (such as Schroedinger did originally. And, like he, you will find you hit a lot of problems (potentially interesting ones for the video blog. Anyway, of course, it took a Brit – Dirac – to sort this out 😉
You mention GR – but as I’m sure you’re aware, energy – particularly the conservation thereof – is a delicate topic in this context. Without some guarantees of particularly good behaviour from the universe it seems to be pretty awkward to work with it. This is largely because GR is “background independent” and the fixed space-time that you might use to organise things and do your energy bookkeeping is not there like it used to be – e.g. it can wave around in an energetically ambiguous way, for example.
So much for the first part of your enquiry. As for the second, it goes way above my head and I suspect that you may be breaking new ground. As such I can only refer you to the historical experts in these fields:
http://www.treknation.com/episodes/tng/season6/descent_part_one.shtml
-James
Sorry for the delay.
Neil,
No, decoherence explains this just fine.
Here’s the deal: the essential effect of performing a measurement is to force the state in question into a specific superposition of states. For example, if we pass a particle through a pair of slits, we force the particle into what is basically a two-component wave function: one state that passed through slit 1, the other state which passed through slit 2. When the particle hits the screen, it is forced into a superposition of position states given by the various locations at which we can measure the particle.
The key, then, is noting that the various states into which the particle is forced can no longer interfere if decoherence occurs. Thus the interaction in question goes as follows:
Initial superposition of states -> basis transformation to different superposition of states -> decoherence so only one of the latter states is observed.
Note that in conjunction with Anne’s previous question, this is also why observables are Hermition operators: observing is a process by which we transform the wave function into some particular sort of basis in the Hilbert space. A general invertible basis transformation is given by a Hermitian operator (invertible because we don’t want any components of the wave function to be simply lost).
Jason, thanks, but I still think you don’t realize, how the collapse-causing imposition from outside on the waves otherwise acting forever as waves, is being taken for granted unwittingly. Classically for example, a wave passing through two slights is just two wave fronts interfering, and stays that way. I mean, the waves could cause charges to jiggle and make other waves, but there wouldn’t be a “hit” from a given photon. It isn’t just about waves and interference, the “h” is put in by hand. Otherwise we’d just have classical physics, where an EM wave would jiggle electrons all around in the path of the wave and it would never zero in on a specific atom etc. if atoms could still exist, and come together right there.
All the talk of bases, Hilbert space, “observation” as a special event, etc., is only like that because QM imposes the collapse rule. You don’t realize IMHO you are taking the result as if it was an explanation of itself at the end of the chain. This is like pulling up by bootstraps in the bad, logically inappropriate sense (in case anyone thought there was a clever “good” way to do it.) The imposition from who knows what of collapse explains the apparent effectiveness of “decoherence”, not the other way around.
BTW, if you or anyone has answers to my other questions, I’d appreciate some answers even if brief, tx.
Funny how the blog conversation on a given topic, even one as inexhustable as this one, eventually collapses. I wonder if QM has any thing to do with it? Does the topic eventually get measured in some way?
The last QM book I read was “Deep Down” things by Bruce Schumm, in which he says, about quantum spin:
I should have read that before I read Tomonaga’s “The Story of Spin.” I think, as I sit here listening to Coltrane and Garner, the most important point that you can make, Sean, is that, as Einstein put, “It would be enough to understand the electron.” (or something to that effect.)
James Robson,
So if I understand, you say that 1) yes we should be able to describe any single closed system as intricated multi-parts system 2) when QM is combined to relativity it is not obvious that closed systems do exist.
In other words, if I have a system with n dimensions, I can rewrite it. If this system is only well approximated using n dimension, and “really” needs more than n D, then the cut off causes problem… and I don’t think there is any way to determine for sure the number of dimensions for a system. But maybe I’m wrong?
Anyway, thanks for your answer 😉
I can think of a few ways to try to answer this . . . First I’d say that one has to remember that electromagnetic fields, like atoms, are subject to quantum mechanics. Really you have to quantize the electromagnetic field, and then the interaction between charged particles is governed by the exchange of photons, which are excitations of the EM field. The theory that describes this is quantum electrodynamics.
(That said, there are still many applications for which a semiclassical picture — treating the electromagnetic field as basically classical, but the atom as quantum mechanical — will suffice. You don’t need to quantize the electromagnetic field to calculate the force on an atom due to a laser beam, for instance.)
Electromagnetic interactions as described by photons don’t always correspond to classical descriptions of EM fields. In particular, if you have a state with a particular number of photons (a “Fock state”), it basically corresponds to an EM wave with a totally random phase, giving a large uncertainty in the field amplitude. To get something more like classical EM, you can take a certain kind of state (called a “coherent state”) that is a superposition of states with different numbers of photons, so the expected number of photons has a Poisson distribution.
Let me bring things down to a less technical level and try to give you a simple reason why thinking of things in terms of EM fields doesn’t always make sense. Classically, the EM field at a given position due to one electron is proportional to the force another charged particle (say, another electron) would feel if placed at that position. But in quantum mechanics, you can have a situation where the electrons are correlated — that is, whether we have a second electron (our “test charge”) at a given position effects the probability of having the first electron (our “source charge”) at some position. If source and test charge aren’t independent, then how can we define an electric field due to the first charge?
Suppose we ignore the question of “What is the electromagnetic field?”, and just try to answer the question of what is the energy due to the interaction of a bunch of electrons orbiting an atomic nucleus. If we knew the full wave function of all the electrons (which includes correlations like I described in the previous paragraph), then we could take the probability (really probability density) of having the electrons at any particular positions, calulate the contribution to the energy (which goes like 1 over the distance between them, as in classical EM), and integrate this over all possible coordinates. Unfortunately we in general don’t know the full many-electron wave function.
So instead you can (as an approximation) ignore the correlations between the electrons, and treat the full wavefunction as being an antisymmetrized product of single electron wavefunctions. For instance, for problems involving an atom we could initially assume the electron states are just the orbitals that we get from solving the hydrogen atom (except with the nuclear charge adjusted to the actual nuclear charge of the atom we’re interested in.) Then you can construct an operator (called the “Fock operator”), which gives the energy of an electron in any one state due to Coulomb interactions with atoms in the other states, as well as including Coulomb interactions with the nucleus and kinetic energy terms. These Coulomb interactions are calculated basically as you suggest — take the norm squared of the single electron states to find the probability of having two electrons at any two positions, and then the energy is proportional to that times 1/r (with r the distance between those positions). This is integrated over all positions for the full energy.
The Fock operator (represented as a matrix) also contains some off diagonal elements, corresponding to the fact that you can swap the electrons in any two states without really changing anything (except the overall sign of the wave function), due to the fact that the full multi-electron state must be antisymmetric. So we can diagonalize this matrix and get new single-electron eigenstates, replacing the orbitals we started with. We can then start from the beginning with our new single electron states, and repeat the whole procedure again and again until it converges.
What I’ve described here is the “Hartree-Fock method” as applied to a single atom. As I said, it’s an approximation that ignores correlations between the electrons, but it still gives good results in many cases. For molecules, the procedure is similar, except you have to first separate out the motion of the nuclei (Born-Oppenheimer approximation), solve the elctron problem for fixed nuclear positions, and then find the nuclear positions that minimize the energy.
This all sounds very complicated, but this is a well-established technique and there are software packages available that do a lot of the work for you. Much current research is focussed on the case where this sort of approach fails — that is, systems where electron correlations play a crucial role.