Feynman told"the entire mystery of quantum is single photon double slit experiment? what is his exact purpose?

Question

in very ideal condition what kind of answer can satisfy above mentioned problem?
is there any exact quantum mathematical solution that can interpret above experiment in a very exact formulation?
I mean a mathematical statement like fresnel and fraunhofer mathematical model for near field and far field experiment or very classical mathematical model for many photon double slit experiment?
it is obvious that the pattern of single electron (or photon)double slit experiment differes from classical mathematical model, and
then it is necessary to have correct mathematical statement, does such solution exist? I mean a solution that is originated from quantum mathematics and describes the intensity and shape of fringes in single electron (photon) double experiment?

Answer 1

You seem to be interested in two separate questions,

(1) "What was Feynman's exact purpose ...[in promoting double slit experiments as generically illustrating the quantum 'mystery' --- ed.]?" and

(2) "Is there an exact quantum mechanical solution or formulation?"

With the greatest respect, I suggest that these two questions have very little to do with each other, apart from the experimental procedure that provides them a common context. One question is about a human being's motivation for doing something, the other about the possibility of obtaining an exact mathematical solution to a specific physical problem.

So, I'll largely address these two issues separately.

(1) I suggest that there is no better person than Feynman himself to explain exactly what his purpose was in using the single photon double slit experiment as his personal exemplar of "the entire mystery of quantum mechanics." (But by the way, are you SURE of your [apparent] quotation? Can you CITE an EXPLICIT SOURCE where this can be checked? I had the pleasure of sitting in on some of his fabled lectures, and of talking with him occasionally at Caltech. My admittedly fading recollection is that he always emphasized the following about the SINGLE ELECTRON double slit experiment. It showed that an uninterfered with and previously supposed "single rock solid particle" would pass through the two slits "at once," thereby exhibiting the interference phenomena previously thought obtainable only in wave phenomena. In doing this it knocked classical determinism off its long standing pedestal and replaced it by something more probabilistic. What is more, the particle "cared" or "knew" whether both or only one of the slits was open!

He deals with this in the first chapter of one of his fairly short, but still quite challenging books (at least, I found it so!) --- I THINK it's "The Theory of Fundamental Processes (Advanced Book Classics)," although I can't absolutely swear to this, as I don't have it to hand right now.

Of course, he also expounds his point of view on this in one of his most celebrated publications, "The Feynman Lectures On Physics," available in "The Complete And Definitive Issue, Second Edition (2005)." He's a bit more discursive there, since these lectures were delivered to freshmen --- though they were, of course, the rather daunting Physics freshmen at Caltech !

This his main purpose, as I understand it, was to get across the idea that a Q.M. calculation is ultimately probabilistic rather than deterministic, and that while the so-called (complex) "amplitudes" may be calculated in a "linear" fashion, the process of converting those amplitudes into the requisite "probabilities" involves "interference effects" due to the nature of complex quantities.

I don't think that an "exact" description of the resulting functional forms produced for the line shapes concerned him anywhere near as much if at all; it was rather the fundamental difference between looking at the world as basically a "quantum world" with interference and therefore wave-like properties versus the purely deterministic classical world. Thus, while for some purposes the behaviour of light (long thought to be of "just" a wavelike nature following Young) could also be thought of as particulate following Planck and Einstein, the electron "particle" could also be treated as having wavelike properties following de Broglie and Schroedinger.

(2) It isn't quite clear what you mean by an "exact" Q.M. solution. For example, treating the distances from the two slits to a given position on the screen as "exact" assumes an idealization that the slits themselves have zero width! Such treatments that I recall seeing in advanced physics texts also tend to assume a small angle approximation, concentrating on the shapes of the lines produced very close to the "axis" bisecting the distance between the slits themselves.

An "exact" solution that does not make such idealizations or simplifications would involve both finite widths for the slits (including the simply awful truth that their widths would NOT be "exactly" equal, in practice), AND demand that the solution be valid for an arbitrarily wide screen for which the "small angle approximation" would no longer suffice.

The point I'm trying to make is that many important solutions to Physics problems are NOT "exact" if you examine them carefully, and rightly so. The art of being an excellent and insightful Physicist and/or Astrophysicist is knowing how to concentrate on the centralmost issue(s), and ignore needless further detail or sheer irrelevances. In other words, being able to approximate wisely.

Live long and prosper.

P.S. If we are to talk in terms of the Klein-Gordon equation, the following historical point may be of interest. Many people don't know that Schroedinger first found a completely relativistic treatment of his wave equation idea. However, he shelved that theory because it did not give the correct COEFFICIENT for the second order term in the energy levels for the electron in the hydrogen atom. (It gave the correct FUNCTIONAL form of the second order term, but not the right coefficient! Both the form of this term AND its coefficient were known experimentally, and --- fortuitously and rather remarkably --- already predicted by the Bohr-Sommerfeld theory.) He later said that he had not wanted to publish a theory that already seemed to be either incorrect or at the very least deficient at the "exact level" to which it could be taken. (Note the resonance with the topic of "an exact solution" above.)

Schroedinger's next tack had then been to produce a non-relativistic or Newtonian "approximation" to his original relativistic treatment.

[Why did he originally even treat the problem relativistically? Because --- and again, modern physics texts do a simply DREADFUL job of describing the preceding developments --- de Broglie's discovery of his famous (lamda, p) relationship was NOT just a useful "idea" that he was lucky enough to stumble upon. (Many books wrongly describe it as his "idea.") Instead, it was a theoretical consequence of a much deeper and beautiful approach : an ABSOLUTE BELIEF in the certainty and power of complete Special Relativistic invariance. He ONLY used Einstein's ideas as of the end of 1905 (!) in deriving his noted relationship, ideas that ANYONE else could have used in the intervening period of almost two decades; but only he had the insight to do so.

I have seen written statements that de Broglie was guided all the time by Bohr's model of the atom. That is simply NOT TRUE, and badly misrepresents the historical record. de Broglie USED the Bohr model, after the fact, to confirm and illustrate the CONSEQUENCES of his own deeper ideas.

Whatever de Broglie did subsequently (and Pauli almost destroyed him with a scathing attack at a later Solvay conference), we should nevertheless honour him for a truly deep and original use of Special Relativistic invariance. The basis of Quantum Mechanics IS Special Relativistic invariance, even if teachers and authors don't now find that pedagogically convenient to acknowledge!

It is simply disgraceful that modern Physics texts, and even so-called and much quoted "histories of the development of quantum mechanics" slight de Broglie's method of discovery, if they even mention it at all.]

O.K., so we now understand why it was that Schroedinger first developed a relativistically invariant wave equation. However, although he actually mentioned it and its consequences briefly at the end of his celebrated paper on what is now known as THE "Schroedinger Equation," Klein and Gordon seem to have been unaware of it. Upon learning of his "non-relativistic treatment," they independenty set about making it relativistic! This was because they, too, appreciated that de Broglie's work involved S.R. invariance absolutely centrally; indeed, they may have wondered "how Schroedinger didn't realize that." (!!) Thus, they unknowingly recovered the very solution he had in fact shelved.

Why did Schroedinger fully publish his second theory, but not his first treatment? Because HIS SECOND THEORY only went to FIRST order, but not SECOND, in calculating the energy levels in the hydrogen atom! Therefore, as far as it went, it WAS "accurate and exact." (It would in fact take the later discovery or postulate of electron spin, and the contribution of spin energy to the electron's energy levels, to supplement the non-relativistic treatment and bring the second order terms into agreement with the experimentally determined values.)

This is a remarkable story of judgments made involving "exact" or "approximate" treatments of a most significant problem in the development of modern physics. To honour what Schroedinger himself did, I sometime refer to either the FIRST Schroedinger equation (in other words, what everyone now calls the Klein-Gordon equation), and the SECOND Schroedinger equation (which other people call THE Schroedinger equation).

Answer 2

Yes, such a statement exists. For a photon, which is a boson (integer spin) this statement is the Klein-Gordon equation, the solution of which is the wavefunction of the photon. In the case of the electron (half-integer spin) the statement is the Schrodinger equation.

Feynman's point was that the correct mathematical solution describes the photon entirely as a wave, while the experimental set up (single photon, double slit) insists that the photon is a particle, and that ***both of these are true simultaneously***

Answer 3

Even for calculation of intensity, expectation or probabiility functions at the image plane, there is no "exact model" for the single electron double-slit intereference because even the Schrodinger wave equation is still only an aprpoximation, as relativistic effects are not taken into consideration. Furthermore, the agreement between quantum interference and classical electromagnetic interference isn't exact either, because phase in EM theory has a spatial direction, while phase in quantum theory does not, leading to very slight differences in the way things come out after computation. The most precise current model of electron behavior is quantum electrodynamics, which is a field theory that has yielded excellent precision in prediction. It is doubtful if any models in physics are "mathematically exact" representations of reality. At best, they are all approximations, even if some are highly accurate ones.

Answer 4

last poster was right on. I actually just listened to the Feynman Lecture you're referring to the other day, and the whole point of that lecture is explaining the fundamental wierdness in QM. Feynman said he refuses to use analogies to explain quantum mechanics to his students, so he explains the most fundamental example of the wierdness of QM, ie the fact that objects are both particles and waves and behave as such at the same time. Basically i guess to see who gets it and who doesnt in the class!