# Why the wave function

1. Nov 27, 2012

### aaaa202

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. Nov 27, 2012

### cattlecattle

For one example, were it not for the wave function, you can't explain the double slit interference pattern which is caused by the addition of wave function of different phase as opposed to the addition of the probabilities. It's true that probability carries physical meaning, but it's false to claim wave functions don't. They carry physical meaning in an implicit way. It's like nobody can ever isolate individual quarks, but so far only by using the quarks model can we explain the results of scattering experiments.

3. Nov 28, 2012

### nanosiborg

Try it. Does it work?

4. Nov 28, 2012

### DiracPool

Strictly speaking, the wave function isn't squared, it is multiplied by its complex conjugate. Doing this produces two benefits, 1) it gets rid of the imaginary form of the wave function, which can't be plotted, i.e., the two Euler exponential parts of the wave function reduce to unity when multiplied. 2) it get's rid of any negative values of the wave function so that there is a clean, easy to understand graphical presentation of the probablility distribution.

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. Nov 29, 2012

### akhmeteli

I don't know about aaaa202, but someone Schrödinger tried:-). And it does work. Please see references and details in the following thread: https://www.physicsforums.com/showthread.php?p=3008318#post3008318 (for example, my posts 11 and 73 there). Briefly: a scalar wave function can be made real by a gauge transform (the relevant unitary gauge may seem inconvenient though). After that you may rewrite its time evolution in terms of its square, but it won't be linear.

6. Nov 30, 2012

### remnax

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. Nov 30, 2012

8. Dec 1, 2012

### TheBlackNinja

how can probability waves interfere destructively?

9. Dec 1, 2012

### RUTA

Exactly. In the Euclidean path integral approach to lattice gauge theory for example, the transition amplitude works like a partition function for computing expectation values of observables and one can obtain information (e.g., particle masses) without Wick rotating back to real time. But, when used to compute correlation functions, you have an amplitude and must Wick rotate back to real time, add amplitudes and square to produce a probability. The reason is precisely what BlackNinja points out -- different configurations can interfere, unlike classical stat mech. So, I'm also interested in the answer to this question.

10. Dec 2, 2012

### akhmeteli

Dear RUTA,

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 Klein-Gordon equation in electromagnetic field, the solution is generally complex, but it can always be made real by a gauge transform (at least locally). The four-potential 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/wp-content/uploads/2011/08/JMAPAQ528082303_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 fourth-order 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 four-potential 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 Klein-Gordon 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 non-relativistic 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 non-relativistic 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 Klein-Gordon equation and the Dirac equation, the answer is not clear: we should remember that, for example, for the Klein-Gordon 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 non-relativistic 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 four-potentials 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 four-potentials of electromagnetic fields, or both, but I have not considered this issue in any detail.

11. Dec 2, 2012

### HomogenousCow

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. Dec 2, 2012

### TheBlackNinja

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. Dec 2, 2012

### RUTA

The Klein-Gordon eqn reduces to the Schrodinger eqn in the non-relativistic 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 Klein-Gordon solutions by squaring.

14. Dec 2, 2012

### RUTA

Exactly, the twin-slit experiment “has in it the heart of quantum mechanics. In reality, it contains the only mystery.” R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics III, Quantum Mechanics (Addison-Wesley, Reading,1965), p. 1-1.

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 Wick-rotated 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. Dec 2, 2012

### akhmeteli

Approximately - maybe, but still the expression for the relevant component of the conserved current is quite different for the Klein-Gordon: see, e.g., http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation , equation 9.2.10, - it is not even positive definite in a one-particle theory.

16. Dec 2, 2012

### RUTA

But the current is still obtained using the square of an amplitude.

17. Dec 2, 2012

### akhmeteli

I'm not sure I understand that - I gave the reference to the expression for the conserved current for the Klein-Gordon equation - its zeroth (temporal) component (which is supposed to correspond to probability density) cannot be a squared absolute value of \phi (other than approximately), again, this component is not even positive definite. Or, maybe you have something else in mind when you mention amplitude?

18. Dec 2, 2012

### RUTA

You solve the KG eqn for psi, use psi*(operator)psi to obtain the current, and psi is the amplitude whose non-relativistic limit is found via the SE. So, this doesn't strike me as a promising method for producing a theory of probabilities a la classical physics.

19. Dec 2, 2012

### akhmeteli

I did not say anything about "promising" or "not-promising" methods. TBN asked: "how can probability waves interfere destructively?", so I only discussed possibility or impossibility. Specifically, I said about the non-relativistic Schrödinger equation in electromagnetic field that it can be rewritten in terms of probabilities only as follows: for each solution \psi of the Schrödinger equation in electromagnetic four-potential A^\mu you can build (using a gauge transform) a real solution \phi of the Schrödinger equation in electromagnetic potential B^\mu, where A^\mu and B^\mu produce the same electromagnetic field. Therefore, the non-relativistic Schrödinger equation in electromagnetic field is generally equivalent to the non-relativistic Schrödinger equation in electromagnetic field for a real wave function (to prove it, you just need to use the unitary gauge where the wave function is real). In the resulting equation, you can express the real wave function via its square, which is probability density (I emphasized that the equation for probability density is nonlinear). That will work, at least locally. Whether this is "promising" or not, I don't know and I don't care for now :-) Let me note that this approach does not use the Klein-Gordon equation in any way. Again, I just offered some information about little-known mathematical results. If you think the mathematics is flawed, please advise. As for interpretation of these results... That is a different story.

20. Dec 2, 2012

### akhmeteli

I'm not sure the double slit experiment creates the weirdness in QM; however, if it does, this weirdness can be reproduced in classical mechanics. You may wish to look at this article: Yves Couder, Emmanuel Fort, Single-Particle Diffraction and Interference at a Macroscopic Scale, Phys. Rev. Lett. 97, 154101 (2006), where, in particular, two-slit diffraction is successfully modeled by classical objects. I guess the following recent article of the same authors is publicly available: http://iopscience.iop.org/1742-6596/361/1/012001/pdf/1742-6596_361_1_012001.pdf (I don't want to copy their abstract here or rephrase what they did - it would be better if you look at the original articles, if you're not aware of them yet). Nobody doubts that Feynman was a greatest physicist, but your quote is 50 years old. My understanding is nowadays people tend to see quantum weirdness in experiments with more than one particle, such as experiments on the Bell inequalities, rather than in experiments with one particle, such as double-slit experiments.

21. Dec 2, 2012

### akhmeteli

I told RUTA how the non-relativistic Schrödinger could be rewritten in terms of probability density only (my posts 10 and 19). Does that contradict what you're saying? I don't know. First, the resulting equation would be nonlinear, so there are no superpositions, second, there may be some difficulties with the square root, third, there may be some subtleties with the gauge choice. But it seems the equation can be written. Again, I just mentioned some mathematical results. You choose how to interpret them (I tend to think they allow different interpretations).

22. Dec 3, 2012

### RUTA

My response was to your reference to the KG equation and the possibility of it providing probability interference per TBN's question. My statement stands.

23. Dec 3, 2012

### akhmeteli

I may be somewhat confused about which statement you have in mind, but if it is "one would expect to obtain probabilities from the Klein-Gordon solutions by squaring", then again, this statement does stand, but only as an approximation.

24. Dec 4, 2012

### RUTA

The twin-slit experiment does not "create the weirdness of QM," it serves as an example of it, i.e., the interference of probability amplitudes. This is what distinguishes QM probability from CM probability, a fact many in the foundations community would like explained 50 years after Feynman's quote.

The experiment you allude to will not explain the interference of amplitudes in QM or QFT, nor does it map onto the SE, since the particle can only receive updates for changing boundary conditions at the finite wave speed of the vibrating liquid. In that sense it has less explanatory power than DBB (admittedly a relatively popular interpretation of QM).

You are correct, there are other things that foundationalists want explained, e.g., entanglement.

25. Dec 4, 2012

### akhmeteli

I am not trying to draw more profound conclusions from this experiment than warranted. I am just saying that it emulates some unusual features of the quantum double-slit experiment. It is not an explanation of everything under the sun. However, the Feynman's quote is about the double slit experiment, as far as I understand, not about "interference of probability amplitudes" in general, and Couder's experiment does tell us something new about the double slit experiment. Furthermore, as far as I understand, there is no positive experimental evidence of particles receiving updates for changing boundary conditions at infinite speed so far, whereas theoretical proof of the violations of the Bell inequalities has its share of problems as well, as we discussed elsewhere.