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#1
Nov2712, 05:05 PM

P: 1,005

As I understand it the wave function itself does not carry any physical interpration. Rather it is the square of it's absolute value. But that forces the question: Why construct a theory with the basic equation being about the time evolution of the wave function, when you could (I guess just as well) set up an equation for the time evolution of the absolute value of it squared. It just seems weird to me that we make this middle step, where we calculate something which actually carries no importance for the physical system.



#2
Nov2712, 05:15 PM

P: 32




#3
Nov2812, 01:30 AM

P: 79




#4
Nov2812, 01:57 AM

P: 579

Why the wave function
Squaring that real part of the wave function doesn't change the probability distribution, as the normalized squared result still retains the relative amplitude relations across the distribution. I think you could even raise the modulus to the 4th power and it wouldn't change anything. 


#5
Nov2912, 11:31 PM

P: 597




#6
Nov3012, 03:58 PM

P: 2

I don't think it is possible to construct a useful theory with the absolut square of psi (or its
square) as variable: psi as a variable allows for gauge freedom  and the gauge mechanism describes the way external fields act on the objects described by psi. But it is a good question 


#7
Nov3012, 06:36 PM

P: 79




#8
Dec112, 03:15 PM

P: 21

how can probability waves interfere destructively?



#9
Dec112, 10:26 PM

P: 660




#10
Dec212, 02:05 AM

P: 597

I intended to avoid replying to TheBlackNinja's (TBN) post, partially because his question may contain several different questions, so it may require a long answer, but your post was "the last straw", so I'll try to answer. 1) So one question that may be implicit in TBN's question is: irrespective of quantum theory, can a real, rather than a complex function, describe destructive interference? I guess we can answer this question affirmatively, as, in general, wave equations can be written for real functions. 2) Another possible implicit question in TBN's question: can quantum theory be reformulated in terms of a real, rather than complex, wavefunction (not pairs of real functions)? I gave an affirmative answer in post 5 in this thread. Let me explain in a slightly more explicit form here. As Schrödinger noted (Nature, v.169, p.538(1952)), if we have a solution of the KleinGordon equation in electromagnetic field, the solution is generally complex, but it can always be made real by a gauge transform (at least locally). The fourpotential of electromagnetic field changes in the process as well, but electromagnetic field does not. Schrödinger intended to extend his results to the Dirac equation, but it seems there was no sequel to his 1952 work. However, such extension is indeed possible (J. Math. Phys., v. 52, p. 082303 (2011), http://akhmeteli.org/wpcontent/uplo...28082303_1.pdf ). It turns out that, in a general case, three out of four complex components of any Dirac spinor solution of the Dirac equation in arbitrary electromagnetic field can be algebraically eliminated, yielding a fourthorder partial differential equation (PDE) for just one complex component. This equation is generally equivalent to the Dirac equation. As there is just one complex component left, Schrödinger’s trick can be used to make this component real by a gauge transform. Again, the fourpotential of electromagnetic field changes in the process as well, but electromagnetic field does not. So the Dirac equation is generally equivalent to an equation for one real wave function. 3) Yet another possible implicit question in TBN's question: can quantum theory be reformulated in terms of the squared absolute value of wave function? At least sometimes (meaning: for some equations of quantum theory), it is possible. For example, as the KleinGordon equation in arbitrary electromagnetic field can be rewritten as an equation for a real, rather than complex, wave function, obviously, it can be rewritten in terms of the square of the wave function (see, e.g., equations 29, 30 in http://arxiv.org/abs/1111.4630). However, the resulting equations are not linear. Probably, the same can be done with the nonrelativistic Schrödinger equation, but I did not try that. As for the Dirac equation, it can be rewritten in terms of just one real component, so it can be rewritten in terms of the square of this component. However, the resulting equation will not be linear. It is not clear if the Dirac equation can be rewritten in terms of the sum of squares of absolute values of four components of the wave function. 4) Yet another possible implicit question in TBN's question: can quantum theory be reformulated in terms of probability? Probably, yes,  for the nonrelativistic Schrödinger equation. It can be rewritten in terms of the squared real wave function, which square equals probability. But the equation for probability will not be linear. As for the KleinGordon equation and the Dirac equation, the answer is not clear: we should remember that, for example, for the KleinGordon equation in electromagnetic field, probability does not equal \psi*\psi. 5) Finally, the explicit TBN's question: how can probability waves interfere destructively? Based on the above, it looks like they can interfere for the nonrelativistic Schrödinger equation. However, the relevant wave equation for probability is not linear, so there is no linear superposition (however, it is my understanding that interference is possible in some sense for nonlinear equations). Furthermore, there is another complication. When we consider \psi_3, which is a linear superposition of two other solutions of the Schrödinger equation, \psi_1 and \psi_2, then the relevant real wave functions \phi_1, \phi_2, and \phi_3 may correspond to different fourpotentials of electromagnetic fields (but to the same electromagnetic field). As for your arguments based on the path integral approach… I guess they can be circumvented in this case due to either nonlinearity or ambiguity of fourpotentials of electromagnetic fields, or both, but I have not considered this issue in any detail. 


#11
Dec212, 02:27 AM

P: 362

The whole deal of adding the amplitudes and not the probabilities creates the weirdness in QM, take the double slit experiment as an example, once you add the amplitudes for both slits and square the modulus you get an "inteference term" and the usual classical probabilities, this inteference term is what caused that whole "the electron is going through both slits" thing



#12
Dec212, 12:59 PM

P: 21

I do not have a lot of knowledge in this, but OP's question seems pretty direct.
What I was thinking the initial quesiton was like "could exist an equivalent of schroedingers equation with the born rule 'already applied'?" he mentioned "absolute of it squared", and thats a probability density function. And what I though was that things which can cancel are things that are allowed to have different signals. They may be vectors in opposite directions, scalars with opposite signals etc. But what you get form the born rule are probability density functions, which maps to positive scalars. So I don't see how these can interfere. Not real scalars, positive real scalars Sure that 'to do the math' you can invent anything, like negative probability(if you are famous enough), or maybe if the underlying phenomenon already accounts for destructive interference in some way. But thats not his quesiton. , As homogenousCow said, its like the 'news' quantum physics brought are something like "existence itself is 'vectorial'", it can sum up and cancel. If you get rid of it in Schroedingers equation you will end putting it somewhere else. So akhmeteli, this is my quesiton and my answer. would you give a word on it? 


#13
Dec212, 01:02 PM

P: 660

The KleinGordon eqn reduces to the Schrodinger eqn in the nonrelativistic limit and its solutions are related to SE amplitudes by a simple phase:
http://users.etown.edu/s/stuckeym/SchrodingerEqn.pdf Therefore, one would expect to obtain probabilities from the KleinGordon solutions by squaring. 


#14
Dec212, 01:21 PM

P: 660

I've attended many foundations conferences in the past 18 years and I've seen many researchers attempting to do QM with classical probability theory in the manner alluded to by akhmeteli. I have not seen anyone succeed for this very reason, i.e., they can't explain quantum interference without introducing negative probabilities, which doesn't make sense in physics (experimentally at least). Like I said supra, it's very tempting to interpret QFT a la stat mech since the Wickrotated or Euclidean path integral works like a partition function for expectation values of observables. However, its correlation functions must be rendered amplitudes to produce probabilities, so every QFT text contains a caveat warning against taking the stat mech analogy too far. QFT is still a quantum formalism in that configurations can cancel in the computation of probability and it is this "destructive interference" that, as Feynman says, makes quantum physics "mysterious," i.e., it contradicts intuition per classical physics. 


#15
Dec212, 02:09 PM

P: 597




#16
Dec212, 02:27 PM

P: 660




#17
Dec212, 03:10 PM

P: 597




#18
Dec212, 03:17 PM

P: 660




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