I want a clarification in the idea of superposition principle.Perhaps,I should rather say that the distinction of superposition principle in QM and in classical physics. Few weeks back,I was spending time with Gottfried's book. He explains the novelty of superposition principle with the help of a two state system.He does this as for one particle systems, the superposition principle has some classical analogue,he comments.You can take a look at the experiment described in his text; it is available at google book review (Kurt Gottfried,p14).It took some time for me to digest the experiment,and I have written something in my blog:http://www.gradqm.blogspot.com/ But my present question is how superposition principle for one particle system [QM] (say, double slit experiment with electrons) different from the superposition principle in classical optics.In classical optics,the superposition occurs between two waves (generated from the different parts of the wavefront of the same primary wave) and interference effect is produced by the variation in the cross term(real of course). In double slit experiment in QM(with photon or electron or whatever),we can reach a situation where a single state is exhibiting interference.It is interpreted as the actual state is a linear combination of two base states: \(\psi\ =\ a\ |1>\ +\ b\ |2>\) And interference occurs between the two base states beyond the slit.Here, the probability amplitude \(\ <1|2>\) is complex.It looks conceptually similar to the classical optics to me.Am I missing something?Or is it that the wavefunction is complex and that is making all the difference? While going through the same,I also found that the idea of coherence is modified in QM.In particular,Gottfried comments that coherence has a richer meaning in multi-particle system,which he did not explain.I found some rigorous treatment in Ballentine's book (many body theory).But what I want is some simpler way to look at it.How to conceive the meaning of coherence in double slit experiment and how it is different from its meaning in classical optics.And what possible modification is needed when we consider multi-particle system.Can anyone shed some light?
I think its a little misleading for you to have a blog intended to help others with QMs. I've seen you ask a single question across half a dozen forums and often ones which you really should grasp if you're doing a physics degree. The notion of a 2 state superposition is basic 2nd or 3rd year stuff yet you're putting yourself forward as someone who explains graduate textbooks? 'Blind leading the blind' springs to mind. The overlapping pattern in quantum mechanics is equivalent to the non-interfering pattern seen in classical wave mechanics. Pure superposition is used, where the two states are ultimately from the same source else you don't get a static formation on the wall or detector. Interference is something different, where the two waves do more than just pass through one another in superposition. The real/complex stuff is largely irrelevant other than the fact the wavefunction isn't anything physical, its the amplitude of a probability.
Alphaneumeric,I did not opened that blog to help you,one who knows everything and discourage elementary physics discussions.And I do not feel ashamed to go through a good literature even on Newton's laws in any point of my career.And if I find some point unclear,I will try to discuss with people to get the point clarified. If they were 2nd/3rd yr stuff (never to be referred to any further),standard graduate texts would not have included them.Better you go through the experiment (Gottfried's book p-14,in google books or whatever---I am pretty sure you have NOT done that,in that case you would have seen it was "two particle system" I wanted to refer to;by mistake,I wrote two state system) and then put forward whatever you have understood. From your words,it seems as though I am asking you to do it for me.It is NOT that.The idea of posting the same thread in 2 (at most 3, and NEVER half a dozen) forums is to connect to as much people as possible.You also do the same for the same reason.The difference is that I consider myself as a student and you consider yourself "a true scientist" who knows everything. -------------------------------------------------------------------------- The point of asking the question was,it is often said the superposition principle in QM is different from in classical physics.However,as far as single particle interferometry is concerned,except from some difference in real/complex calculation,they look the same to me.I wanted people to comment on this point. And as usual,you repeated whatever I told in your language and avoiding the point.This is the problem with you;you do physics to command others,not to learn. If I am blind,at least I know that...you even do not know that you are also the same.
The original quantum theory mathematics were based on light and radiation, at least in Schrodinger's original approach. So it's good that you observe a correspondence between classical optics and quantum mechanics, because theorists were looking for a description of matter in a way that it could self-interfere just like ordinary light.
Precisely.The descriptions are quite similar.In the case of classical optics, the primary wave is present at the two slits,simultaneously.In QM, a single photon also does the same.In QM,the photon interferes with itself; and in optics,the primary wave interferes with itself (we can think like this).Apart from the calculations(vector sum of real vectors in optics and scalar product of complex wave functions in QM),superposition treatment is fairly similar.This is expected because of the linear character of EM theory and QM. The difference is that in QM,the interference pattern destroys when there is an evidence which path the photon follows.In this case,the photon is NOT present at both the slits and single photon interference does not occur. Whereas in optics,interference destroys if the two components of the prmary wave lose their coherence,even if it is present at the two slits at the same time.
Hi neelakash, You asked a lot of questions, so maybe we can just start small and work our way up. One very basic question is do we need complex numbers? It seems the answer is no because we can always take a complex vector space of dimension n and interpret it as a real vector space of dimension 2n. Recall that we can regard complex multiplication as a particular kind of multiplication in the plane involving only real numbers. So we don't need complex numbers, but we do need negative numbers. Another basic thing we should understand is that the mathematics is more or less identical in both cases. The superposition principle, for example, appears in both theories, although it has a quite different physical meaning in each. Do you agree so far?
I think you definitely do need to use complex numbers. If you replace complex numbers with 2-component vectors, you still have to define a product on those vectors, which will end up being entirely equivalent to complex numbers in everything but notation.
Physicsmonkey,complex numbers are needed when we are thinking about something imaginary.\(\vec{E}\) and \(\vec{B}\) field in EM theory are real "concrete"objects,so we need not use complex numbers.A three component vector serves our purpose. However,a wave function is a mathematical description corresponding to a physical state.You get it when you take the <x| of the ray (resing in the Hilbert space) describing the state.The wave function is no way less abstract than the ray.Unlike classical mechanics,the abstract concept of a wavefunction is not idealization of "concrete" things.Therfore, we need to use the complex numbers.In this point,I will agree with CptBork: However,we must take care that we must use a linear vector space.So,we must have access also to -ve numbers. Regarding the 2nd point: I agree that the structure of mathematics is the same.Because both are "linear superposition". And this gives me the impression that the physical meaning of superposition is the same.I am not understanding where the superposition principle is getting different meaning.
Hi CptBork, Of course I agree with you. If you keep the same algebraic structure then you are using complex numbers even if you don't say it aloud. However, I think it is potentially still useful to make the statement I made because it draws our attention to the algebraic structure. We now have a better question than why complex numbers. Instead, we can try to directly abstract the algebraic structure necessary to describe quantum mechanical experiments. For example, is the complex multiplication essential, why don't we think about SO(2n) instead of SU(n), etc. I believe this is the sort of thing the OP was interested in.
My eyes! Ze goggles, they do nothing! Please don't take labels such as "complex", "imaginary" and "real" to imply anything about their usage.
This and the idea of coherence were my concern...real and complex numbers come along the calculation,but that is not what I am really worried about. If your point is to show people there is nothing in this question,then let me tell people that you always confuse people by saying big big things and giving hand-waving arguments.When you are presented with a simple and correct solution,you never write back as you know your ability. The follwoing was when I just started learning tensors: http://www.sciforums.com/showthread.php?t=94493 The following was at a time,when I did not take a course on STR or tensors: http://www.physforum.com/index.php?showtopic=15685&hl= If you do not understand the question,please leave it.
