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?”)
Ooh, I’m looking forward to this. As a layman, one thing I’d love to see covered is the measurement problem. It seems like a lot of scientists that know better can fall into lazy language that makes it sound to the layman like consciousness itself changes reality. But some lucid, repeatable-by-someone-like-me explanation of why that’s definitely not the case would be very well appreciated.
Also, some discussion of the different “interpretations” of QM would be great. I’ve never gotten my head around the Copenhagen interpretation.
(if this is not a question about QM, then please ignore.)
So-called “wave-particle duality” is sometimes described as a paradox. Is it?
What about decoherence? How many physicists today believe it is correct? Does it solve the so-called “measurement problem”? Can we finally be sure that Schroedinger’s Cat is either alive (yay!) or dead (Oh, dear!), and not in a state of superposition?
I’m looking forward to this, Sean, but honestly what I want to know is if there is a book about quantum mechanics written for laymen that actually explains the theory in detail _without_ using analogies that oversimplify and _without_ constantly resorting to poetic language? I wouldn’t mind having to think and work a bit to understand what’s being said, I wouldn’t mind having to read a few equations, I wouldn’t even mind having to learn some math I don’t know. I’d just like to read a book from an author that really tries to make me understand QM, as opposed to giving us a rough idea of what it might be like if I understood it. I want The Blind Watchmaker of quantum mechanics.
I would like to hear more about what a Hilbert space is vs vector space?
Also, maybe you can start with the double slit experiment and what it means for a photon to interfere with itself leading to path integral representation of QM.
either way can’t wait!
I’m a complete layperson, but I’ve always been intrigued by the inability to directly see quantum states (maybe what Matt means by ‘the measurement problem’?). Why is it that they can’t be directly measured? Pretend I’m the grandmother. Without the physics degree.
Here are a few ideas. I’m a somewhat educated layperson, so probably the right target audience for Bloggingheads.
1. Do any of the proposed solutions to the measurement problem (or “interpretations” of QM more generally) have consequences that would allow us to distinguish them empirically, or get evidence one way or another?
2. What is the math behind quantum mechanics like? I’m not asking for a math class, of course — maybe just an impressionistic sense of the major pieces & how they fit together. The heart of Newtonian mechanics seems to be calculus and differential equations — are those equally central to QM or is there something more? Is group theory involved? (The point isn’t to understand the math — it’s that knowing what mathematical tools a theory involves can help give a sense of what the theory itself is like.)
3. Is it accurate or inaccurate to say that the “quantum” in QM means that the world is digital rather than analog?
4. Quantum computing. Revolutionary or overhyped, and why?
And finally, an anti-request. There are so many good discussions out there of the basics on the uncertainty principle, wave-particle duality, and the two-slit experiment that you won’t be doing much of a service if you spend a lot of time on yet another layman’s version of those things. Explain them of course, but don’t get too bogged down in the details!
I’d like a cogent discussion of Aspect’s experiment and it’s relationship to Bell’s inequality. Did it firmly prove Bell’s inequality?
Bell’s inequality itself is a pretty intereting topic.
e.
I also like some discussion of Bell and very recent experiments - Leggett, Zeilinger, realism vs locality, etc.
Perhaps you can work in the “church of the larger Hilbert space” into the discussion about the wavefunction of the universe.
For those outside quantum information theory, the church of the larger Hilbert space is a joking reference to the fact that any quantum states which are ‘mixed’ in the sense of being a probabilistic combination of two or more ‘pure’ states (pure states being those which live in the Hilbert space) can always be thought of as part of an even larger system which is pure. So one speaks of “purifying” a mixed state, or, in cases of extreme need, going to the church of the larger Hilbert space. So it’s relevant to the discussion of the wavefunction of the universe. sorta.
When one talks about a quantum of stuff, you are talking about the smallest possible piece of something. Quantum Mechanics basically states that there IS a smallest piece of energy. Quantum Mechanics also makes the assertion that one can say that there is a smallest distance, measure of time, velocity, force, etc. When one is talking about the smallest pieces of space does it make sense to think about these piece of space as having a shape?
From a layman, perhaps you could discuss the practical applications of QM. For example, I conceptually understand why GPS clocks have to be adjusted for Special and General Relativity. Yet, my understanding of the only detectable effect of QM in the “macro” world occurs with a precision device like a comb 1/10th as thick as a human hair at a temperature of about 78 above absolute 0.
I would enjoy being enlightened.
Thanks,
Sam
Question 1: Does “measurement” of a particle mean the same thing as “the world line of some particle interacts irreversibly with the world line of another particle”?
Question 2: Can any other type of physics be formulated with this weird-ass probabilities-in-Hilbert-space formalism? Could you, for example, rework the Kepler problem with the position and momentum observables as the eigenvalues of some Hamiltonian? Would there be any advantage to it?
Question 3: In a delayed choice two-slit experiment, a particle knows when it’s emitted whether its path is going to be through one of two slits or a superposition of both paths based on how it’s going to be measured, even if the measurement happens 50,000,000 years later. HOW THE HELL … ahem. Excuse me. How does it GOD DAMN KNOW HOW … ahem excuse me again.
If that damn thing can somehow look 50,000,000 years into the future and see the laboratory it’s going to wind up in and see the scientist with his finger on a button and it makes its decision on how to propagate based on that, then the universe is rigidly deterministic to an extent that makes me want to just go and slit my wrists.
Like Clive Bruckman says in the X-Files: “How can I see the future if it doesn’t already exist?”
Is it reasonable to say; Wave=analog, particle=digital?
Is a photon, as a quanta of light, analogous to a drop of water, in that its consistency of size is a function of factors operating on it, much like the surface tension of water vs. gravity makes drips of water similarly sized. Or is it a fundamental…I wouldn’t say structure, but possibly internal constraint?
To further Phil’s question, is it that these past these smallest units of measurement, is it simply too blurred to distinguish one unit from the next, or are there clear units which cannot be further subdivided?
I have to protest (again…) that I don’t see the linearity of the Schrodinger equation (i.e the fact that you can add wavefunctions) as a distinguishing property of QM. In both classical and quantum mechanics the observable quantities (say, expectation value of operators) obey non-linear equations, and there is something describing the “state” of the system (wavefunction, or phase space distribution function), which is interpreted as probability and obeys a linear equation (Liouville or Schrodinger equation).
It so happens that in classical mechanics you typically deal with the former set of objects, and in QM (but not QFT…) you usually deals with the latter. That is a matter of pedagogy, nothing else.
1) What does it really mean that there is a measurement problem? (Looking for good speculation that has some depth to it, not textbook recitation…)
2) Where does ‘empirical data’ stop and ’speculation’ begin when it comes to the problem of state selection?
3) What’s up with the quantum Zeno effect? (Please relate answer to answers to one or both of above questions.)
4) Can the equations of QM be modified to accomodate a)probabilities that change with time, and/or b) bi-directional flows of time with non-equivalent influences?
5) Why default to equivalent starting probabilities when using QM to make predictions? Are there instances when previous data allows for fore-knowledge of a bias?
6) What’s the point of quantum field theory? Why/How was it developed? What does it allow you to do, exactly? Why has it not been reconciled with gravity? What are its limitations - ie, it’s good for predicting behavior of a large set of data, but not an individual observation, etc.?
This is fun! I’ll be thinking of some more questions…
I’m a biochemist so my understanding of quantum mechanics is limited so hopefully this doesn’t stray too far from what you were hoping to see:
1. What are virtual photons? Why can virtual photons cause a force of attraction between + and - charges but regular photons cannot do the same?
2. If we have, say, 8 qubits of entangled electrons trapped in holes on a chip of some sort and we also have a way of measuring the array of quibits then how is any information transmitted and stored? For instance, if the first and third electrons were excited to be spin-up but all the rest were untouched and spin-down then what would the superposition that resulted be and how would you use that to calculate something like the factorization of the number 15? Oh so confusing!
3. Half silvered mirrors are interesting. If a single wavelength of light is emitted and half of the photons will pass through the silvered mirror and continue on until they reach another silvered mirror, in which half again will pass through, then a detector on the other side will read 1/4 the number of photons as were presumably emitted. Now if a full reflective mirror is used to reflect the photons that were reflected off the first half slivered mirror and then reflected again to hit the second half silvered mirror such that the path length is equal to 1/2n times the wavelength of the light (where n is odd) then all of a sudden the detector will read 0 because the photons have canceled each other out. Does the size of n matter? If n were absolutely huge then the time it takes for the photon to get to the detector along the direct path is much much shorter than the time it takes the photon to get around the more circuitous path to detector, but yet they are still able to cancel each other out. (I don’t even know if this is true the way I have written it, so if anyone knows what I am talking about please correct my question to make more sense than I assume it does in this form). Therefore the photon’s path is determined at the time it is emitted. Could this be used for faster than light communication? For instance, if an emitter on a stationary satellite near earth passed light through two silvered mirrors as explained above and there were two mirrors on a satellite millions of miles away that could interrupt the signal by moving one of the mirrors into place then suddenly there would be a signal way back at the earth satellite instantly…again, this is assuming I know at all what I am talking about.
What I find very odd in QM is that we can make a reliable specific superposition, like for a photon, but no one is supposed to be able to find out the details later. IOW, we can make a photon equivalent to elliptical polarization, given the wave function: A |R> + Be^(i theta) |R> and e.g. 0.6 for A and 0.8 for B. The phase then provides an angle, not just an ellipse shape, so we can be sure a filter tuned to that wave would let the photon through (as it would also have ideal 100% transmission for the equivalent classical polarized light beam.) If I am a confidante of the photon’s creator, I can know just how to orient the right filter (say, combo of QWP and LPF) to get all “hits” etc. But if I don’t already know, I can’t find out for sure: all I can do is try a filter and orientation and I might get transmission or not. Either way, the photon is “ruined” by either being absorbed or changed into the new filter’s base. (Projection postulate? I wonder why that doesn’t have its own Wikipedia article.) All I really know is, that photon couldn’t have been the orthogonal to the filter base if it went through. But if the trait is “real” (unlike the literal contradiction in Fourier analysis of exact momentum and exact position), why can’t I find out? (I know, doing so might lead to weird effects in entangled states, like FTL communication, but suppose it didn’t?)
This seems silly, like a kid saying “If you don’t know, I’m not going to tell you!” In some other comments around, I explained how we might circumvent that restriction by using the accumulation of angular momentum: Keep reflecting a photon around with mirrors, sending it through the same half-wave plate over and over (re-flipped by a second HWP if needed.) The HWP reverses the rotational sense (spin) but maintains the specific proportions of A and B (you may be surprised, but it does - known fact, and to be consistent with the affect on the classical wave.) If we did it enough, all those transits would build up detectable angular momentum in the HWP. It would be along a range, not an either/or because the result needs to be consistent with sending many many “separate” photons through (indistinguishability.) IOW, if the photon came out of a linear polarizer, the many transits wouldn’t build up net spin since the average effect is no rotation. Maybe it wouldn’t work, but it’s worth mulling over. It seems to resemble “weak measurements” as propounded by Yakir Aharonov.
What are the implications of QM on the arrow of time? When reality splits under Everett’s Many Worlds interpretation into a “parallel universe” does this constitute a third direction of time and, subsequently with another split, a fourth, and then a fifth … (ad infinitum)?
Correction, I mean the HWP swaps the values of A and B, so the proportion is in a sense “preserved” but saying it that way is confusing. Hence 0.6 for A and 0.8 for B turns into 0.8 for A and 0.6 for B, etc., which is then restored by a second transit through a HWP (and so on, hence the intuitively expected but legally challenging result …)
For Matt and LoCut, asking about the measurement problem, I think you should look up and read about the Stern-Gerlach experiment. I tend to use this experiment whenever a non-physicist friend/family member asks me about QM, simply because I think its an experiment that is fairly transparent and can be well understood from a good explanation about the actual setup and results, without needing to analogize about quantum mice traveling through a cheese field; it also provides an excellent experimental demonstration of many important and fundamental principles of QM.
1) quantization of particle properties (angular momentum in this case.)
2) How, after measurement, we don’t know all the information about the initial state (only magnitudes of constants, no phases), and
3) Multiple Stern-Gerlach apparati in a row demonstrate measurement changing a state. There’s also a lot of information here about how a state can be a combination of two states when described in a particular way (basis), but be conveniently described as a pure state in another basis.
Collapse of the Wave Function. This is related to the measurement problem (in fact it is fundamental to it!). As a physicist, I am mostly comfortable with QM, but I can’t shake the overwhelming feeling of guilt that I get whenever I refer to wavefunction collapse! It is the only example I can think of where a fundamental physical process is described by a non-unitary operation.
By the way - you are clearly going to have to expand this episode to a full semester…
Yeah, there’s clearly no way we’re getting to all of this in an hour. If we’re lucky, we might be able to agree on a definition of “quantum mechanics.”
But it’s good stuff, and we’ll try. Certainly we’ll talk a lot about the measurement problem and related issues. Bell’s inequality, maybe — that’s another thing for which I don’t yet have a good 30-second explanation.
Moshe, I’m not sure I get the heart of your objection. I wouldn’t think that we should draw an analogy between wave functions in QM and classical distribution functions; wave functions are analogous to points in phase space, and density matrices are analogous to distribution functions. But I suspect this isn’t the place to resolve that.
Janus, David himself has written a book which might be just what you are looking for. A couple of equations at the level of algebra, but most of the focus is on concepts, and it’s unfailingly precise.
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
Andy S said…
“Question 3: In a delayed choice two-slit experiment, a particle knows when it’s emitted whether its path is going to be through one of two slits or a superposition of both paths based on how it’s going to be measured, even if the measurement happens 50,000,000 years later. HOW THE HELL … ahem. Excuse me. How does it GOD DAMN KNOW HOW … ahem excuse me again.
If that damn thing can somehow look 50,000,000 years into the future and see the laboratory it’s going to wind up in and see the scientist with his finger on a button and it makes its decision on how to propagate based on that, then the universe is rigidly deterministic to an extent that makes me want to just go and slit my wrists.”
…all of which ties in with Seans brief comments on the wave function of the universe!
This is going to be a very interesting discussion. I get the impression we are already in a postion to draw some (at least) preliminary conclusions!
QM, like SR and GR, is awesome!
Sean has plenty of material for the upcoming discussion!
How about giant magnetoresistance, without which no hard-disk iPod would be complete? Although arguably many bits of modern technology rely on quantum mechanics, GMR is a nice one to use as an exemplar, because it is a blatantly quantum-mechanical phenomenon that went astoundingly rapidly from discovery (1988) to universal industrial adoption and Nobel prize (2007)
It is also a good one to wave under the noses of those idiot politicians who think all money should go on “applicable” science rather than “useless” blue skies science (MRI is another good one there, co-invented in my department on the basis of an entirely pointless experiment, and now worth a billion or so a year).
Well, if the movie “What the Bleep…” is any indication, you can use it to sink free throws and to make yourself a great photographer or something.
wow! more than 50 comments! Usually the science stuff get 10 or 15, tops. Good going, Sean.
Re: the wave function of the universe. Wouldn’t this idea only make sense from the perspective of an observer standing outside the universe (which doesn’t make sense)?
Thanks.
Re: Debbie #38
That’s always bugged me, too. To date, I’ve got 4 Intro level Quantum texts in my library. Not one of them derives the equation.
The only one who makes a stab at it, is –you guessed it– Feynman in one of his Lectures on physics volumes.
It’s something along these lines: (note - there’s no preview, so my Latex’d equations might not be legible).
(also note: I’m only a B.S., so my equations might also be gibberish).
meaning the wave function at a future t’ is some operator times the wave function at t - the operator depends on t and t’.
Obviously,
So in the infinitesimal region around t, U is probably something like,
That makes
Getting the derivative gives you:
Big whoop right? We had some operator U that we didn’t know anything about, now we have some operator
that we don’t know anything about. But hit it on the left with 
and you get:
Adding the complex conjugate,
Which implies that
.
is anti-hermitian.
Cool. So
But we like Hermitian operators, so let’s multiply both sides by i. This gets us
That makes the operator on the right side Hermitian. That’s where the “i” in the Schroedinger equation comes from.
Next is the units.
always has some weird units like length^-1/2, and naturally 
would be length^-1/2 s^-1. That means that 
will have units of s^-1 or frequency.
then.
Lucky we called it
So
is Hermitian and has units of frequency, so it must be some kind of frequency observable. Well, the energy levels of atoms are always associated with absorption and emission frequencies so we can get the operator to have energy units by multiplying both sides by 
to get:
Which is just the Schroedinger equation, with
identified with the Hamiltonian.
That’s a great example, thanks! I really wasn’t aware of that, I assumed the modern computer was a product of classical physics (except for the laser in the optical drive), amazing that something as crucial as the hard drive depends on this “mysterious” science!
Basically, everytime you access a database, anywhere in the world, the results returned require Quantum Mechanics to be correct.
http://www.research.ibm.com/research/gmr.html
Just glancing through some of the comments here, Sean, I would recommend that you take at least a moment during BloggingHeads to mention that the Periodic Table of the Elements is QM at its finest. Everybody seems so interested in the off-the-deep-end questions that nobody has looked closer to home–how do we know the shape of water or CO2, or one of a hundred thousand important biomolecules? Basic QM and the Periodic table gave us all the necessary tools. I say this because the Periodic Table is something that nearly every Layman will have passing familiarity with and because it’s one of the greatest, most impactful, most tangible victories of quantum theory. Moreover, while I know it’s not as sexy as Quantum information theory and entanglement, nor as hardcore as QED or QCD or the Standard Model, it is accessible.
andy.s,
What you describe is interesting as far as it goes, but there are some problems in taking it as an explanation of the Schrödinger equation:
- Schrödinger developed his theory in the context of solutions to partial differential equations, not state vectors in Hilbert space. What you are describing seems to be closer to Dirac’s presentation of QM.
- The logical predecessor to the Schrödinger equation was of course Heisenberg’s theory. My one-minute summary of that invention was that Heisenberg was thinking about focusing on the Fourier components of the dynamical variables, and got the idea of limiting the frequencies in the spectrum of components to the allowed frequencies in the energy spectrum. This is described in Max Born’s book, The Dynamical Theory of Crystal Lattices, based on lectures given very shortly after Heisenberg’s invention. (The first part of the book is on lattices, the second is on Heisenberg’s brand-new QM.)
So why doesn’t any textbook ever derive it?
I’ve got several introductory texts and they all talk around it in different ways. It gets kind of annoying after a while.
@49 This one is actually not that hard! You’d have no chemistry at all, because the coulomb solutions don’t have stable orbits. I believe without a stable classical orbit the quantum business still couldn’t conspire to give you any chemistry.
I’m also not a quantum chemist, but I’m not sure you really need one here.
Following up on Doug’s answer (60), I think I also remember from partial diff-e, that any linear extra dimensions making the total number of linear dimensions an even number (4, in the case of the original question) results in every event echoing into infinity. Or something like that. I couldn’t even recognize a differential equation to save my life any more.
I’m guessing that if someone has a fairly clear-cut idea of what Hilbert Space is they are not in your target audience. I like #39 Cecil’s idea of using the two-slit experiment as a point of departure (a la Feynman). The explication thereof would give you the opptunity to explore any number of streams to any desired depth. I’d also second the suggestions those who have asked for some discussion of practical applications.
andy s.,
Are you equally bothered that textbooks don’t derive Maxwell’s equations? Or F=ma?
Can someone please explain bra-ket notation to me? The explanation on wikipedia just goews around in circles. I get as far as “Every key has a dual bra”, and follow the link on dual, and it’s off into chaosland.
How do you get from bra-ket to actual numbers? I see lost of psis and thetas, but at what point do you plug actual numbers into these things and get some sort of result?
Those spherical harmonics that are the orbitals of the hydrogen atom - does that apply to other atoms? What about molecules? I mean … as I understand it, the state of an electron is basically a complex number field which extends through all space, except that it’s very nearly zero almost everywhere except where the “location” of the electron is. If you have two electrons, then you multiply their wave functions together and integrate it, and the square modulus of the result gives you the probablility that they will interact. If they interact (exchange a virtual photon), the the momentum transfer means that they move apart, and that’s what electrical repulsion is. Or something. I’m still not sure how this interacts with time - the probability that they interact has got to be a “chance per second”, and all this stuff has got to be symmetrical WRT relativistic boosts.
Now, was that kind of right? Or is the field some sort of thing where at each “point” in space there is actually a matrix of complex numbers?
If an electron hits an antielectron and is anihalated, does the wave equation field thing dissapear everywhere simultaneously? Or does the dissapearance of it sort of propagate outward at the speed of light?
Anyway.
How did they figure out that Buckminsterfullerene was going to be yellow? I mean - what numbers to you plug in, and where do you plug them? In order to come up with that result, it couldn’t all have been algebra - they’d have to have dome some adding and multiplying to come up with the absorption spectrum. Sorry to carp on about it, but from the wikipedia page on bra-keyt notation, I can’t see how any of this stuff gets from math to reality.
Why does a water molecule have that mickey-mouse shape? How do you get from “The hydrogens have one electron, the oxygen has 8″ to that particular shape?
That’ll do.
48. TimG:
Why does [measurement] 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.
Yes, we see it does happen, but that is all from examples of traditional “one shot” measurements where a single interaction results in either an eigenstate left over, or destruction of the particle (e.g., photon either passing or being absorbed by a polarizing filter. But that is not what happens when a photon passes through a half-wave plate! (Or, similar birefringent element.) Like I said, the photon gets its RH and LH bases swapped, but not collapsed. Of course, one pass through a HWP doesn’t measure anything anyway. Yet many passes should build up detectable angular momentum along a range, not just binary results, as I explained above. I can’t say for sure and must be humble about something that pushes the envelope, just asking for it to be considered.
weichi
My copy of Reitz and Milford on E&M has a quite extensive discussion on Maxwell’s equations and I think most students would be uncomfortable with a text that stated them right off the bat with no discussion of their roots in Coulomb’s law, Ampere’s law, etc.
As for F=ma it is, considerably more intuitive than
.
Intuition is a fairly useful faculty. It certainly leads me to suspect that you don’t have a lot of friends.
I think this is probably off-topic, because I guess it probably has more to do with relativistic theories than with QM per se. But I have yet to read a really good explanation of how time dimension(s) and spatial dimensions are different. The whole “time as a fourth dimension” thing has made intuitive sense to me since I was a little kid. I can picture the whole of spacetime as the 4-D analogue of a 3-D block with world-lines running through it, a block that one can slice on different angles into different relativistic viewpoints on the world. But I’ve never quite understood, okay, time is a dimension like the left-right axis, but then why does it seem so different?
One explanation I could make sense of would just say, well it’s only different in that the universe had this very low-entropy state at one end (the beginning) of its time dimension, so we have the “arrow of time” and thus the notion of experience, the “course” of our lives, etc. But I’ve also read in several places that, no, it’s not that easy, temporal and spatial dimensions really are just different.
I don’t expect solid answers; I imagine this might be an area where different theories (general relativity, string theory, QM?) have different things to say (whether or not their answers are mutually contradictory). But is there a good discussion of “time as a dimension, but not a spatial dimension” anywhere?
It certainly leads me to suspect that you don’t have a lot of friends.
Sometimes I wish that I could enjoy the whole game of internet put-downs and flame-throwing, because it seems like something that would be a lot of fun if you were the kind of person who could enjoy it. Other times, I’m glad that it just seems ridiculous.
Talk about the classical analog of the Heisenberg Uncertainty Principle in classical waves- it makes sense when you’re thinking of water and not electrons, and the HUP is one of the things that makes QM so mysterious to the layman.
On the wave function of the universe I love Gell-Mann’s comment to Jim Hartle about the Hartle-Hawking wavefunction. “Hey Jim, if you know the wavefunction of the universe how come you’re not rich ?” or words to that effect.
I watched Jim Hartle’s talk at Oxford for the Everett 50 years of MW meeting on the web. It was very interesting, I can’t remember the URL of the back of my head but its well worth a view together with the accompanying slides.
Anyway as a chemist my interests in QM are more immediate and practical. Yesterday I compiled GAMESS (General Atomic and Molecular Electronic Structure System) on my system. So I guess I should shut up and calculate.
As a non-physicist, I’d add a vote for David Albert’s book as a nice introduction to QM.
andy s,
I apologize if I struck a nerve - it wasn’t my intention.
Are you really looking for a derivation - i.e., logical deduction from some “more fundamental” foundation? I don’t think you are going to find it - Schrod eqn is just the god-given way that the world works (*). Sure, you can offer plausibility arguments like in your post 55 (and I agree they are helpful!) but that’s not what I think of as a “derivation”.
In particular, I don’t think it’s correct at all to think of Maxwell’s equations as derived from Coulomb & Ampere laws; in fact I prefer to think of Coulomb’s law as a consequence of Maxwell eqns (in the electrostatic case). The roots of maxwell in coulomb and ampere are historical, not logical.
Anyway, some places to look:
* have you looked at Shankar’s path integrals chapter? 8.6 in particular shows how path integral approach is equivalent to schrod eqn, and maybe you’ll find path integral a more intuitive starting point?
* Landau & Lifschitz QM (section 17) shows that the classical limit of Schrod is the Hamilton-Jacobi eqn.
* Schrodinger’s paper of 1926 “Quantization as a problem of proper values” (reproduced in the book below) explains how he come up with time-independent version. He starts with H-J eqn.
* The book “Probability and Schrodinger’s Mechanics” by David Cook promise a “variational derivation” of Schrod Eqn in 8.1. I think it’s an expansion of schrod’s original argument, but haven’t read it, so not sure.
Anyway, my view is that the Schrod eqn is not something that can be derived - it simply is.
(*) I guess you can derive schrod eqn from quantum field theory, but to me that just begs the question of how do you derive quantum field theory.
1 What does the many-worlds interpretation really mean ? And why do so many physicists believe it’s correct ?
2 How is many-worlds different from parallel universes or the multiverse ?
3 Given that quantum mechanics is the basic theory that explains how the world works, why don’t we see its bizarre aspects in everyday life ? (I think this gets at the heart of many laypeople’s interest in QM) Could you discuss decoherence vs the theory of Zeillinger’s group (referenced in the link in comment # 50 above)
4 The work of Vedral and Zeilinger’s group on entanglement in many-body systems or bulk materials (this gets at beginning to see one of QM bizarre aspects
in (closer to) everyday life situations)
Why do you have to limit yourself to one episode ? Sean, you do a great service by attempting to explain physics to non-professionals, and in the age of the Internet the ability (bandwidth) to communicate has been immensely increased.
After all, you did inherit Feynman’s desk
This is the beauty of decoherence: it doesn’t have to be put in by hand at all. A really simplistic way of looking at it is as follows. Imagine that we start with a two-state system:
(normalization ignored for clarity)
Now, as we progress this state forward in time, the two states will oscillate between one another dependent upon the difference in energy. But what happens when we observe the state? Observation of the state is provided by an interaction with some other system, a system that happens to be vastly more complex (let’s call the other system
:
After this interaction, what happens is that the oscillation time between states 1 and 2 becomes huge. If the state
is complex enough, the time to oscillate can become effectively infinite. Due to the fact that we are complex, then, every time we interact with such systems we end up preventing further interference: the wave function decoheres, and only one result is ever visible. Nothing put in by hand, just simple wave function evolution.
P.S. I hope the tex came out okay.
I think you’re misunderstanding what is meant by energy. Energy and matter are not different sots of thing. You don’t convert from one to another. Instead, energy is a property of matter. Now, you can convert from one sort of energy into another. For example, if I collide an electron and a positron together, the two can annihilate to form a pair of photons. This isn’t a conversion between matter and energy: photons are still a form of matter. What happens, however, is that the mass-energy of the electrons gets converted into kinetic energy of the photons.
So far as we know, these interactions are point interactions. But this will probably be amended once we understand more about high energy physics.
Sean, your
compares quantum mechanics with classical particle physics, which is a straw man. Compare quantum field theory with “probability densities over classical field states” (aka random fields) — which, given that all our best theories are field theories, and given that probability is ever-present in Physics, seems less likely to be a straw man — we find that superposition is present in both. Indeed, a random field can be formulated as a commutative algebra of operators, making differences between quantum and classical only of the algebraic structure and interpretation of their measurement algebras. Bell inequalities become an essentially negligible issue (I argue that the conspiracy assumption is natural for a random field, whereas I agree with the conventional argument that it is unreasonable for classical particle models).
With apologies for grinding my axe, for “Bell inequalities for random fields”, see cond-mat/0403692, J. Phys. A: Math. Gen. 39 (2006) 7441-7455; for an algebraic formulation of interacting random fields, “Lie fields revisited”, see Arxiv:0704.3420, J. Math. Phys. 48, 122302(2007). These will not convince you of much, I have not yet developed my approach enough for it to be much more than speculative, but you might perhaps take away that superposition is not a good separator between classical and quantum. If you understand and start to implement in your thinking that classical particle physics is a straw man you will be ahead of the curve. Asking how much of this can be put into a discussion directed at the layman is an awkward question of course.
Here’s one for today…
Interesting discussion everyone. A question for the experts. I recently came across Ballentine’s quantum mechanics book and he makes a fairly convincing case to me that this “collapse of the wave function” business is not needed in the measurement problem if one uses the statistical point of view for the wave function. How generally accepted is this viewpoint ?
Thinking more about the question of “deriving” the schrod eqn, maybe an analogy would be to the “derivation” of maxwell’s eqns in, e.g., Schwartz’s Principles of Electrodynamics, where he takes coulomb’s law, invariance of charge under lorentz boosts, and lorentz invariance, and argues not only that a magnetic field is required, but gets the full maxwell equations (if i recall correctly). So he is starting with three very well-established experimental facts and shows that consistency requires this extra structure (loosely speaking).
(I’m not sure whether his argument can really be viewed as an air-tight logical deduction, but it certainly provides motivation and insight).
It would be nice if there were a similar set of well-established experimental facts that could similarly lead you to schrod eqn. My guess is that there *isn’t* such an argument, because it seems like the experimental facts would have to involve the wavefunction, but we dont’ really *have* any experimental facts about wavefunctions. The fact that the wavefunction - the very quantity whose evolution schrod eqn describes - is not an observable gives it a very different flavor than the electromagnetic field.
Jason Dick, thanks for the helpful attempt. However, I still don’t think you get the deep objection, which is that even that one resulting wave “that we observe” still has no reason to suddenly shrink into a tiny space, it should still be an extended wave anyway, etc. You and others are still taking the observation regime for granted and can’t seem to “get above it”, you are IMHO like fish who can’t appreciate what their being in water does.
In any case, the multiple transmission thought experiment I have been describing above doesn’t rely on any particular interpretation of decoherence, etc., it is based on the logical implications of what we already know about those particular interactions between photons and wave plates.
weichi, I think you’re missing andy s.’s point a bit. It’s all fine and good to say the Schrodinger eqn. “just is”, and we may not have some sort of derivation from first principles, but nevertheless Schrodinger himself must have got it from somewhere, right?
That is to say, Schrodinger didn’t simply write down random symbols until he found an equation that predicted the spectrum of the Hydrogen atom, did he?
Most intro-level text books do seem to just state Schrodinger’s equation without any explanation of where it comes from. I’ll go check my graduate QM book and see if it does any better.
Well, according to Merzbacher’s Quantum Mechanics textbook, Schrodinger wanted an equation that agreed with classical mechanics in the classical limit.
Taking Schrodinger’s equation and using psi = exp(i*S/hbar), we are led to the Hamilton-Jacobi equation (which is one formulation of classical mechanics, equivalent to Newton’s laws, Lagrangian mechanics, etc.) Except, this equation has one extra term proportionate to hbar.
In other words, it agrees with classical mechanics in the limit where hbar is negligibly small.
That’s a reverse derivation — what Schrodinger really did was probably used his knowledge of classical mechanics to constrain the form of the quantum mechanical equation, and then played around with it a bit until he got one that also reproduced known experimental data. At least, that’s my best guess without actually looking up Schrodinger’s original papers (or rather, translations of them to English).
To clarify the above post, I don’t mean that the Hamilton-Jacobi equation has an extra term compared to classical mechanics. I mean that the Schrodinger equation leads to the Hamilton-Jacobi equation plus an extra term. Hamilton-Jacobi by itself is exactly equivalent to classical mechanics.
I heard the story that Schrodinger got his idea from attending a lecture given by De Broglie on his thesis work
on wave-particle duality. Schrodinger and Kramers were sitting next to each other and at the end of the talk, Kramers said to Schrodinger that if matter had wave proporties, then there must be a wave equation. Supposingly this comment started Schrodinger thinking about the topic though of course it does not answer andy s.’s question.
Jason,
Thank you for responding to my handwaving.
Given my mental dyslexia I think QM makes more sense if this statement is reversed, that matter is a property of energy. We assume there is some underlaying monolithic property, but by all evidence so far, reality seems to be a function of relative interactions of opposing forces, with matter as a fairly stable manifestation of this. The search for this underlaying property leads to smaller and smaller points of observation, while the macrocosmic reality continues to balance itself between polarities. What if they don’t find the Higgs? Are they just going to keep looking? String theory is seemingly about the strings, but it’s only their fluctuation that really matters. What if they are simply vortices?
TimG,
Well, I did provide a pointer to Schrod’s original paper, so I don’t think I *totally* missed his point
Agreed that it would be nice if textbooks explained why schrod choose particular eqn.
Probably not historical, but I think the clearest way to look at Schroedinger’s equation is that energy is the generator of time-translation in exactly the same way that momentum is the generator of space-translation. Noether’s theorem, etc.
Agreeing with Aaron: once we represent the results of experiments by operators acting on a Hilbert space, time evolution gives a one-parameter group, so by Stone’s theorem there is a self-adjoint operator, which in the case of time evolution we call the Hamiltonian, that generates the group. It’s convenient to represent statistical measurement by operators, and to represent the impossibility of joint probability distributions, for certain combinations of measurements, by using noncommuting operators.
More on “deriving” Schrodinger’s equation:
Consider a plane wave: psi = e^[i (k x - w t)]
This has
k psi = -i d/dx psi
and
w psi = i d/dt psi
From the de Broglie relations we have p = hbar k and E = hbar w
For this reason, we define:
p = -i hbar d/dx
and
E = i hbar d/dt
Now take the classical equation E = p^2 / 2m + V
and replace with our expressions for E and p in terms of differential operators.
This (acting on psi) is the Schrodinger equation.
Now, starting from plane wave solutions doesn’t make sense if V varies over space. But let’s suppose the Schrodinger equation is still correct. From this we can see:
(1) The equation has the right classical limit (as I mentioned above)
(2) It correctly predicts the spectrum of the hydrogen atom.
So, it looks like the equation is right even for the case where V depends on x.
TimG,
I like it. So we can use electron diffraction experiments to justify both 1) treating electron as a wave (so we have wavefunction) and 2) p = h-bar k deBroglie relation. For E = h-bar omega I guess you can refer to planck blackbody, and make a leap that it will hold for matter waves as well (just like deBroglie did!)
So treating these as your the experimental “facts”, your simple argument leads to Schrodinger (and also, I believe, Klein-Gordon if you take the relativistic case). It’s not a derivation from first principles, but better than pulling things out of thin air.
Yep, it gives the Klein-Gordon equation if you start from:
E^2 = p^2 c^2 + m^2 c^4
I’ve read somewhere that Schrodinger actually first discovered the Klein-Gordon equation, but rejected it because it gave incorrect predictions for the hydrogen atom. (Of course we now know the equation is appropriate for spinless particles, not electrons.)
1. Why can’t QM predict, i.e. have an analytic solution, the line spectra of He or any multielectron atom?
2. How does QM account for the continuous spe