Talk:Quantum mechanics/Archive 1

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First discussion

Quantum mechanics has provoked a strong philosophical debate. The fundamental problem is that causality and determinism is lost: while the probability distributions evolve according to a well established deterministic law, the values of the observables themselves do not. Because of this, Albert Einstein held that quantum mechanics must be incomplete.

It would be helpful to try to give some basic explanation of why Einstein's view is widely held to be incorrect--his view seems like common sense, but common sense is often wrong, as theoretical physicists enjoy pointing out. So, why is it wrong, in this case? By the way, please don't answer this question on the /Talk page--please put the answer on the QM page. Thanks in advance! --LMS

It's not entirely clear that Einstein was wrong on all counts, just wrong on at least one of them. :-) The Bell's-inequality experiments of Aspect prove beyond any doubt that either (1) Observable effects exist that cannot be deterministic results of inherent properties of matter; or (2) The universe is non-local; i.e., physical effects can propogate faster than light. Nobody knows which. --LDC

It proves neither, since neither is the case in the multi-universe interpretation. --JG

I'll put a discussion of these issues on the Copenhagen interpretation page. --AxelBoldt

In the "Description of the theory" section, it states that the view of the electron circling a protron was replaced by the view of a static "probability cloud". However, this gives the misleading interpretation of the electron as being "smeared" in some distribution around the proton, which isn't really case. The electron does in fact move around the proton, we are just unable to predict the nature of this movement and can only predict the probability of finding the electron in any given location.

That the electron is not "smeared" throughout the distribution, but occupies unique points in space is evidenced by the relative inaccuracy of the Hartree-Fock approximation compared to Density Functional Theory or other methods that attempt to account for the "correlation energy" that arises due to the interactions between moving electrons. --Matt Stoker

Current understanding is that the electron is smeared. If a wave function was just a probability distribution for a particle that actually had a given position, the double-slit experiment wouldn't work, since the electron would have to travel through one slit or the other. I'll admit to not knowing precisely how correlation energy works, but absolutely none of the fundamental quantum mechanics I have seen treats orbitals as anything other than stationary states. --Josh Grosse

The electron cannot be smeared over the orbitial, since if it were then the electric field would also be smeared. This smeared view of the electric field is the limiting assumption in the Hartree-Fock approximation and is the reason for it's limited applicability to multi-electron systems. The electric field cannot be treated in an average or smeared fashion because electrons repulse each other, this repulsion results in an instantaneous distortion of the probability distribution for a given electron that depends on the instantaneous position of the other electrons at any given point in time. The difference between the exact energy and the Hartree-Fock energy due to these instantaneous electron interactions is called the "correlation energy".

My understanding is that modifications of the double-slit experiment were performed in which the researchers attempted to detect which slit the electron passed through. In these experiments, they were able to determine which slit each electron passed through, but the measurements perturbed the system, such that the typical diffraction patern did not occur. In other words, when the electron was detected as a particle with a definite position, the system behaved as if the electron were a particle. The wave function collapsed, because the electron position was detected. I would imagine that the interactions between electrons are similar. Instantaneous interactions between electrons cause the wave functions to collapse, so at any given time the electrons "see" the other electrons with a unique position and the electric field corresponds to specific electron positions. --Matt Stoker

But electrons are always interacting with other electrons, so if this was enough to collapse the wave function, you could never have an electron in two places at once. What happens when two electrons interact is that their wave functions become entangled, and that's where the correlation comes from, although I don't know the mathematical details. In the copenhagen interpretation, you need a real observer to cause collapse, while in the many-worlds interpretation, collapse does not occur at all (only entanglement between the system and observer). --JG

When I wrote the sentence about static electron probability clouds, I did not have "smeared out electrons" in mind; rather, I thought about a probability distribution that tells you how likely it is to find the electron at a given point, the electron being a particle, not a cloud. Maybe I should clarify that somehow? Any suggestions?

Also, we have some treatment of the double-slit experiment in Wave-Particle duality. Let me know if that is inaccurate. --AxelBoldt

Request to leave the sentence alone. In copenhagen electrons are clouds until observed - that's what wave-particle duality is all about - and in other interpretations they are clouds period.

Actually there's no conflict between having an electron "smeared out" and it having a velocity (~momentum). Since the electron is described by a wave(function) it can (and will) do both at the same time. The prime example here is a free electron. If a free electron has an exactly determined momentum its wavefunction will be spread out evenly over whole space (maybe a rather theoretical example..). Actually the electrons around a nucleus also both have a momentum and are delocalized (and you can solve the Shrödinger equation for Hydrogen exactly). Even if you go from Hartree-Fock to Full CI (Which is a method to solve the Schrödinger equation exactly within a finite basis-set) you still get delocalized electrons. -- Ulf Ekström

Axelbolt: the electron "probability" clouds are not actual abstract probability in respect to determining where an "electron particle" is, but rather a physical reality as shown by the double slit experiment. The method of observation determines the outcome. Which also brings me to another point, why is not more serious attention given to that besides a description of the multi-worlds theory? Can you say, bias? --Brendan Stork

Having reviewed more of the literature on this topic, I concede that I was incorrect, so I'm removing the discussion regarding electron clouds and acknowledging that the current description in the article is correct -- Matt Stoker

Note: I moved it back from Talk:Quantum Mechanics. It would have been hard to find otherwise, and needs to be part of the historical record. Graham87 02:39, 29 March 2010 (UTC)

Perhaps some mention of the problem that inspired Planck to invent Quantum Mechanics is in order. IIRC, physicists were trying to figure out what electromagnetic waves were in an oven that had a certain amount of heat in it. They knew that an integer multiple of the wavelength of the light in the oven would have to equal one of the dimensions of the oven, but every time they tried to figure it out, they ended up concluding that the oven had infinite energy in it. Planck was able to find the answer by assume that the energy in an electromagnetic wave was quantised such that E ∝ f. This went directly counter to the classical mechanics assumtion that E ∝ Amplitude.

not quite, in classical mechanics it's proportional to both the square of the amplitude and the square of the frequency 05:12, 8 Jan 2004 (UTC)

I would like to request that we start an article "Mathematical content of quantum mechanics" and move most of the math material that is right now in the main article there. Two reasons:

  1. This article was meant as a general introduction, accessible without a deep understanding of the math or specifically devised notations like bra-ket.
  2. The math treatment right now is incomplete (it doesn't mention that the operators don't have to be defined everywhere, it doesn't mention which operators belong to which observables, it doesn't mention the possibility that an operator may not have eigenvalues, it doesn't mention the importance of the spectral theorem in dealing with operators that don't have a point spectrum.

Correcting the problems in 2 would compound problem 1.


Good idea. How much of the current material do you think should be left in the article, and how much moved to the new page? The old section on "Mathematical Formulism" I found very difficult to read, which was why I expanded it. -- CYD

I can write a very simple version, basically saying that states are elements of Hilbert spaces, observables are operators, and the states evolve according to the Schrodinger equation, and then link to the math article. --AxelBoldt

I've correct some mistakes here,

1. ``three things that QM explains was misleading. Quantum mechanics explains (literally) thousands of things classical theory cannot. I changed the wording to reflect this.

2. There are classical systems where vairables can only take on discrete values (vibrational modes for example) so I took that part out

Those are all wave's however, whats new is that particles are waves too, and thus this happens for particles; Pretty much all of quantum mechanics predictions occur in classical wave mechanics 05:12, 8 Jan 2004 (UTC)

I put it back in, since arguably the explanation of energy quanta is the major achievement of QM and gave it its name. The fact that classical theory can explain some quantizations doesn't take away from the fact that most quantizations are explained only by QM. --AxelBoldt
It's not a matter of ``most. The point is that quantum theory explains the right quantizations (energy levels in atoms for example). But the wording seems better now. --Matthew Nobes

3. The idea that quantum mechanics ``omits quantum field theory is a bit strong. I changed the wording of that paragraph to more accurately reflect things.

4. The history of quantum field theory was off. I corrected the QCD part and added about about the electroweak force.

5. To the primary author, why such an emphasis on the many-worlds interpretation? It's really not widely accepted...

I hear that Hawkings and with him all cosmologists believe in it (since you can't really talk about a measurement apparatus separate from the system if your system is the whole universe) and Feynman also seems to have been a many-worlder. --AxelBoldt
Just because he wrote a pop-sci book does not mean that Hawking speaks for most physicists (or cosmologists). My main point is that there is no particular reason to emphasize many-worlds over (say) Bohmian mechanics. -- Matthew

6. Also, the quotes section seems really silly to me. It gives the impression that QM is somehow wrong, misguided and/or incompherensible. It is none of these things. I would strongly encourge dropping it.

Well, if you are not worried by QM, maybe you don't really understand it... :-) Isn't it remarkable that almost all the major researchers in the field expressed their unease in some way? Why should we suppress these sentiments? Instead, maybe we should add some positive quotes. --AxelBoldt
Umm did you add the new quote? If so you made my point about ``big names versus people who nobody has ever heard of. --Matthew
Rest assured I understand QM just fine :) If read in context the writings of the founders of QM often appear less negative, further by including quotes from before the 1980's you throw out all of the interpretive work that has been done since. Things have been improved by leaps and bounds. Of course problems still remain. A quotes section cannot do justice this vast amount of work. As for positive quotes that doesn't really work becuase the average reader would assume that the Einstein quote is somehow ``better then the (say) Mermin quote on the basis of name recognition, despite the fact the the latter physicist understands *modern* quantum physics. In the end do what you wish, I'm only giving my perspective as a practicing physicist. --Matthew
Could you add some information to the Philosophy part of the article (or maybe create a new Philosophy of quantum mechanics) about the newest improvements in interpretating the theory? Is Copenhagen not the state of the art anymore?
Umm the philosophy/interpretation of QM is really not my field, so I'd be extremely reluctent to write anything about it. Ideas like dechoerence play a large role. It would also be importent to stress the crucial role experiments are now playing in resolving some of the mysteries. --Matthew
Also, what I never understood and maybe you could clear up on mathematical formulation of quantum mechanics: does the theory give a general (hopefully axiomatic) rule about which operator belongs to which observable and which unit vector belongs to which state of the system? In other words, if you encounter a completely new "black box" system with a couple of observables, which Hilbert space and which operators do you use? --AxelBoldt
This seems like two (or more) separate questions. The first is asking how one assigns the various operators to the observable. This is typically done with symmetry areguements a la Wigner. There is a different (practical) problem of mesurments on an unknown state. This is done by applying known measurement tools to the unknown state (i.e. a Stern Gerlach device) --Matthew
I guess the Wigner arguments are the ones I'm looking for. What would be a good reference? --AxelBoldt
Try a graduate level QM textbook. They all treat this stuff, with more or less detail. I learned it from the very clear presentation in L. Ballentine's ``Quantum Mechnics: A Modern Development. Sakurai or Merzbacher might also be good places to look. There's a new book out by Schwinger (posthumously published, entitled Quantum Mechanics) which looks like it treats things very clearly, but I've not had a chance to go through it. Also, the first volume of Wienberg's QFT textbook has a very clear and concise discussion, except he uses the Poincare group instead of the Gallian group. If you can understand his discussion then the application to non relativistic systems will be clear. --Matthew

I'm not the primary author, but I don't see any special emphasis on the many-worlds interpretation in the article. Could you elaborate?

Also, I believe the many-worlds interpretation has become more widely accepted amongst physicists than the Copenhagen interpretation, especially given the work in the past couple of decades on decoherence. Correct me if I'm wrong, though.

The quotes section is silly, I agree.

The philosophy section is currently rather poor. I ripped out a long description of the Cophenagen interpretation and the many-world interpretation, because those points are duplicated in the respective dedicated articles, and don't explain the philosophical issues. It needs to explain why Einstein felt that a probabilistic theory sucks, the philosophical problems with Copenhagen, etc.

-- CYD

I took out the following passage:

Quantum mechanics consists of 3 basic principles:

    1. All matter and energy is quantized. In other words, everthing comes in packets, or bundles. Therefore, one cannot have any amount of material. Instead, one must have a multiple of the smallest unit.
    2. All matter and energy exhibit has wave-particle duality. Matter and energy exhibit the properties of waves in some instances, and the properties of particles in others, depending on the experiment set up to observe it.
    3. Measurements of physical quantities are probabilistic. This is Heisenberg's Uncertainty Principle, which states that the position and velocity of a particle is not absolutely fixed at any given instant and can only be described in a range of probablities. Furthermore, any object does not have a definite position or velocity until it is measured.

I believe these are very misleading comments, and do not add anything to the article, but I'm willing to discuss it.

In (1), it is unclear what it means for "matter" or "material" to be quantized; less sloppy language is required. Furthermore, many quantum mechanical observables have continuous, not discrete spectra.

(2) had been mentioned earlier in the article, and Wave-Particle duality had already been linked to.

(3) had already been mentioned in the preceding paragraphs, and is much less clear than the prior explanations. Furthermore, it is momentum that is the conjugate observable to position, not velocity. Furthermore, the Heisenberg uncertainty principle works for observable pairs other than position and momentum.

I have not seen these three "principles" in any quantum mechanics textbooks. The postulates of quantum mechanics are described in the article mathematical formulation of quantum mechanics, which admittedly still needs work.

-- CYD

I agree, the first one is simply wrong, and the other two are already covered. And I also agree that we have to start getting serious about the math. forumulation :-) --AxelBoldt

Can AxelBoldt please justify his recent removal of the statement that QM explains and quantifies the particle nature of light? It seems to me that the particle nature of light is inherent in the definition of the wavefunction - one could dismiss it as a mere "postulate", but the same argument could be applied to the wave nature of matter. AFAIK the relationship between photon energy and frequency is derivable - say from the quantisation of angular momentum.

Actually, in my opinion that whole first section could be replaced by a description of the empirical observations leading to QM (currently relegated to the first paragraph of the "history" section). It seems strange to me to introduce QM with a randomly selected handful of theoretical results. -- Tim Starling

Quantum mechanics doesn't talk about light; Quantum electrodynamics does that. Quantum mechanics is strictly about masses moving around. A photon is a quantum of the electromagnetic field, so you need quantum field theory. AxelBoldt 01:59 Nov 11, 2002 (UTC)

That's all very well, but it leaves us with nowhere to discuss the historical development of QM in a balanced way. Planck's and Einstein's work involving light came well before QED was developed. Perhaps this page should be moved to Quantum physics (which I note is just a redirect to this page at the moment) - that way we can talk about the early development of quantum theory without guilt or confusion. We would also have a place to talk about popular thought experiments such as the single photon double slit interference pattern. -- Tim
Planck's and Einstein's work is mentioned in the History section without guilt or confusion: it preceded and lead to QM, so it belongs there. It is not part of QM however. The particle wave duality page discusses the double slit experiment. AxelBoldt
Okay, I give up. I guess I'm just used to the historical development being used as an intro.
Afterthought: you say QM doesn't talk about light, but what would you call the theory of atomic radiation based on time-dependent perturbation theory and classical electrodynamics?
I don't know what that is. AxelBoldt 04:14 Nov 11, 2002 (UTC)
Easily fixed - [1] -- Tim Starling

The particle nature of light goes back to Einstein's explanation of the photoelectric effect, which is one of the foundations of quantum mechanics and obviously predates quantum field theory. Quantum mechanics doesn't necessarily deal with masses; you're thinking about the Schrodinger equation for a massive particle. For example, the quantum mechanics of spins doesn't talk about masses. -- CYD

Yes, I think Axel would argue that "quantum mechanics" is different to "quantum physics" - "mechanics" referring only nuts and bolts and the like. After four years of studying the subject, this distinction is new to me, and it sounds like it's new to you too. Obviously mathematicians are better at nitpicking than mere physicists. Nitpicking aside, there's a certainly common usage of the term QM which encompasses QED and the other "extensions". We have to ask ourselves - do we want to use Axel's definition, or everyone else's definition, in this encyclopedia? -- Tim Starling

Well, if you guys agree that QED is a part of QM rather than an extension of QM, then this article definitely needs to be changed. What is the proper name for the theory "observables are self-adjoint operators"? AxelBoldt

I wasn't sure, so I emailled God. Okay, maybe it wasn't God, just a professor of theoretical physics from my university. Anyway, here's what he said:
Yes, I would say that by common usage "quantum mechanics" usually
refers to the (nonrelativistic) quantum mechanics described by the
Schroedinger equation, while the full relativistic theory involving
"second quantization" is referred to as "quantum field theory". But then
an alternative usage of "quantum mechanics" would be to refer to the whole
field of quantum phenomena. As you say, "quantum physics" would be a
better term here; but the second usage certainly does exist.
So what do you think, guys, do we
  1. Move the page to quantum physics
  2. Change to the popular science definition
  3. Stick with the technical definition
I'm going for (1) - it's obviously a less ambiguous term. We can redirect from QM to QP, and put a note in QP about usage of the term QM. -- Tim Starling

In your characterization of the Prof's answer, why did you relabel his "common usage" as "technical definition" and his "alternative usage" as "popular science definition"? This seems to be a distortion to me. AxelBoldt 00:32 Nov 15, 2002 (UTC)

If it's a distortion then I apologise - I only meant to summarise, with labels based on my own experience of the usage of those terms. I take it by your tone that you're voting for number 3? Tim Starling

Certain pairs of observables, for example the position and momentum of a particle, can never be simultaneously ascertained to arbitrary precision (see Heisenberg's uncertainty principle).

This is not an effect classical physics cannot account for. Rather, this is natural result of quantum theory.

Also, I think it would be better and easier to follow to move the section on the history and development of Quantum theory to the beginning of this article rather than jumping right into the theory itself. But the historical section has a lot of content relating to the philosophy, so it really must be redone. When I get a chance! MattH 10:36 25 Jul 2003 (UTC)

If you mean that the momentum-position uncertainty relation is not directly observable, I tend to agree. I've boldly moved the bullet point to another spot in the article. (Note, by the way, that the energy-time uncertainty relation is observable, in the form of spectral line broadening.) -- CYD

Very good! But the state is represented by a vector (or rather, by a ray) in a hilbert space only as long as it's not interacting. (A photon from a faraway star is a good approximation of it). A precise description is the density matrix, and that should in my opinion be mentioned.


Regarding the "Schrödinger equation doesn't contain gravity" quote from Feynman:

The quote was deleted because it is, firstly, a fairly inconsequential statement about quantum mechanics, even if spoken by Feynman; and secondly because it leads to some confusion over what is meant by "Schrödinger equation". As explained in the Schrödinger equation article, the term has two slightly different meanings. Modern quantum mechanics textbooks refer to the following equation as the Schrödinger equation:

where the quantum Hamiltonian H is not necessarily specified. Let's call this equation 1. It is obeyed by any quantum mechanical system, provided H is correctly given. In contrast, when older texts refer to the "Schrödinger equation", they also mean a specific form for the Hamiltonian, i.e.

This second equation, which we'll call equation 2, does not describe some quantum mechanical systems. For example, the equation for a charged particle in an electromagnetic field is

Tim Starling's remark on my talk page, that the "Schrödinger equation" (equation 2) doesn't contain electromagnetism, is therefore not incorrect. However, it is slightly misleading, since electromagnetic systems can be described quantum mechanically, i.e. by equation 1. (We can even formulate quantum Hamiltonian for the electromagnetic field, as is done in the theory of quantum electrodynamics.)

As for gravity, equation 2 can be used to describe the motion of a particle in a gravitational field, simply by making V the gravitational potential. It cannot, however, be used to model gravitational phenomena, not even the production of a gravitational field by a particle. Whether gravity can be described quantum mechanically, i.e. by equation 1, remains an open question. -- CYD

I am suprised to see no mention of Bohmian Mechanics in the entry. I will preface by saying that I myself am no quantum physicist. But, as I understand, Bohmian Mechanics constitutes a coherent interpretation of quantum phenomena without the need for surmising any of the no less than bizarre accounts of reality such as 'there is no fundamental reality' or 'there are infinately many realites', the essense of the Copenhagen Interpretation and Multiple Universes respectively. From the little that I understand, Bohmian Mechanics says that it is a particle 'riding' on a wive (a pilot wave of sorts) that correctly explains experiments like the two-slit experiment (which by the way is also worth mentioning in the entry as it is the simplest and most fundamental experiment that encapsulates the basic paradox of quantum reality - that is, particle/wave duality) I see the earlier attempts at explaining quantum mechanics by way of positing 'new' or 'modified' realities as rather vein attempts to account for the paradox which had every physicist thoroughly befuddled. This is also what gave rise to those all too famous quotes, and they too should be understood in the context of that initial struggle by the scientific community to come to terms with the nature of the paradox they were faced with. But as I mentioned, I was under the impression that BM solved the paradoxical behavior of the wave/particle. - Mike C

There is no paradox in QM except for the paradox of why physicists continually try to confuse the general public by using Copenhagen language to highlight the exact scenarios where the Copenhagen interpretation breaks down, then declaring "that's quantum mechanics for you". Schrödinger's equation:
is not ambiguous, so shut up and calculate.
yeah, who needs to use the reasoning centers of your brain to contemplate the ramifications of quantum mechanics when its much more comforting and easier to "shut up and calculate". Who needs to know whats really going on? --Brendan Stork

I added links to interpretation of quantum mechanics and Bohm interpretation. A short description of BM was copied from its article. -- Tim Starling 07:36, Dec 4, 2003 (UTC)

--I only meant to say that originally there was a paradox, namely, how in the world can a particle exhibit wave-like properties? This was what had everyone thoroughly baffled by the two-slit experiment. That initial bafflement and wave/particle paradox is an integral part of the history of QM and should be presented in such a context in the entry.

-secondly, Bell's theorum proved that any proposed model of quantum reality must be non-local. I feel that mentioning Bohms interpretation as non-local and as attracting little support from physicists in the same sentence will confuse the reader into the mistake that these two things are somehow connected. In fact, according to Bell, physicists must hold a non-local view of the universe and this should be no reason for them to reject BM. Furthermore, there are at least a number (how many I don't know) of respected mathematicians who now support BM. -- I recomend that we move any further discussion of the Bohm interpretation to the talk page for that entry though.

-Also, as to shut up and calculate, this seems to me to be a grievous prognostication for thwarting all further inquiry into the nature of quantum reality. If you want a perfectly analogous example to this, look at the history of Ludwig Boltzmann and atomic theory at the turn of the 20th century. He was told for decades and vehemntly renounced by the scientific community as the last man standing, holding that atomism was a model that could account for the kinetic theory of gases among other things. He was told, 'don't worry about models, that's a bunch of philosophical rubbish, just do the math' and 'you believe in something that is so small it can't be seen or measured, ludicrous'. Well eventually people went on to prove atoms existed and the microscope soon became powerful enough to study them. The point is that many thought him absurd for even trying to draw a model of reality out of the math much less one that was not observable or even measurable at that time. But if not for his stubborn insistence, it is highly likely that atomic theory would not have fully developed till much later than it did. - Mike C

I think that some paragraphs in the section describing the theory are not entirely correct, or at best misleading. I mean the text around "During the process of wavefunction collapse...". These few paragraphs, talk about wavefunction collapse as if it's "really happening". It is well known that there is more than one interpretation of quantum mechanics that tell us what's "really happening". Even though the Copenhagen interpretation is the most commonly cited one, sooner or later it will start a heated debate as to which one should be used in the article.

I suggest rewriting that passage as follows: (a) state that experiments produce measurements that are sometimes discrete in values and the same experiment performed on the same state may result in more than one measured value. (b) explain that these experimental facts are predicted by quantum mechanics (c) offer an interpretation of these experimental facts in terms of wavefunction collapse but stress that this is only an interpretion and acknowledge other ones.

Lastly, I really don't like the sentence "During the process of wavefunction collapse, the wavefunction does not obey the Schrodinger equation." The wavefunction of a system obeys the Schrodinger equation only if it is closed (no external influences). And if a system is closed, its wavefunction always obeys the Scrodinger euqation. To avoid misconceptions, it is better to say that during the process of measurement the system can no longer be considered isolated, and the state of the measurement aparatus must be also taken into account. The analysis of measurement taking the measuring aparatus into account is called the study of decoherence and leads to results that can potentially explain the standard problem of measurement.

I'm going to think about the best way to put this into words and then make changes. I can incorporate comments if any accumulate. --igor

This is a great explanation of quantum mechanics to a lay audience, and the number of good relatively non-technical explanations quantum mechanics is shamefully small. Obviously a lot of work has gone into it, and it shows. I do have a couple of comments.

First, according to the article, "For example, an electron in an unexcited atom is pictured classically as a particle circling the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric probability cloud surrounding the nucleus." This is true only for the electron whose ang. momentum quantum number is 0. I think you meant by this the electron in the Hydrogen atom...

Secondly, it would be nice if you included the photoelectric effect in the list of observations that could not be explained by classical mechanics, since this led to the idea that light is made of packets called "quanta." Also, it is for this that Albert Einstein won the Nobel prize.

Also, igo I find your comments interesting, particularly since "elementary" quantum mechanics is usually taught using the Copenhagen interpretation. It was taught this way to me, so I wasn't aware that the concept of "wave function collapse" is not intrinsic to the theory, but is rather a part of the Copenhagen interpretation. However, something observable happens to the system which is interpreted as wavefunction colapse, so how else do you refer to this observable change to make it free from any particular interpretation of quantum mechanics?

I also never realized that the Schrodinger equation only holds in a closed system-- well, it's more like I never thought about it, but of course, that makes sense-- the system becomes entangled with another, and so the "measuring" system plus the "measured" system comprise a closed system and so can be modelled with the Schrodinger equation (right?)

Assessment comment

The comment(s) below were originally left at Talk:Quantum mechanics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This would benefit from presentational improvements. The lead does not properly define the subject or give an overview the article, and some of the sections are too short, some too long. Lack of referencing remains a barrier to GA or FA status. Geometry guy 18:58, 9 June 2007 (UTC)

Last edited at 18:58, 9 June 2007 (UTC). Substituted at 21:57, 3 May 2016 (UTC)