Understanding the Complexity of Wavefunctions in Quantum Mechanics

[Fra]

Very generally speaking, I don't have problems with subjective bayesian view. For example, I very much like Jaynes' information theory approach to statistical physics.

I'm glad to hear this.

It is important to understand, however, whether such an approach is necessary at the fundamental level or at some higher level (statistical physics may be an example of such higher level).

I agree.

As for the fundamental level, I have yet to be convinced that the bayesian view simplifies the matter or, although adding complexity, is just necessary.

I understand your skepsis. I am personally now quite confident that this is the way, but it wasn't straightforward. I've spent some efforts analysing this from the POV of philosophy of science as I prefer it, and I've arrived at a pretty confident personal position of the foundations. I judged this as absolutely necessary as I could not justify the high degree of speculation that I ended up with as a student.

I think the bayesian view alone is insufficient to explain everything, but it is a step in the right direction. I even think that our inability to understand how the "quantum world" scales from an information theoretic point, is correlated to our lack of understanding how QM and GR can be unified. I think separating them, is part of the problems also to understand the QM foundations.

This traces down to the philosophy of science.

IMO statistics, inductive inference and even probability theory really touches the essence of science. What is knowledge? How _confident_ are we in the "knowledge" we think we have? How do we _measure_ confidence? All these questions touches foundations of statistics and probability theory. Many of these things have issues, that are non-trivial, but by tradition are treated as philosophy by many physicists. I could never accept such attitude, but science also has social dimensions. And if this behaviour is accepted in the scientific society, because everybody does it, well then there you go. And from group dynamics it sure is going to take more than one opinion to change the group behaviour. And reflecting over this, I see deep connections to physical interactions and information theory.

This leads me to a new idea of simplity. Risk taking and speculation.

Sometimes you have to speculate and take a risk based in incomplete information, beucause that's life. Because choosing to not take any actions at all may actually be more risky, than taking one of a set of possible smaller risks.

I interpret "as simple as possible, but not simpler" in a more specific interpretation as:

One should not take unjustified risks, and should only take the justified risks necessary to optimise your self-preservation. The "no risk" options usually never exists. When you connect this game thinking, to probability theory and it's foundations many interesting things appear. But I think it would take that someone works out something explicit and applies to to make new predictions and accomplish at least part unification before the collective pays any interest to it. For some reason very few people seems attracted to this.

/Fredrik

 

[Xeinstein]

As I mentioned in some of my earlier posts, quantum mechanics does not necessarily needs complex wavefunctions (real wavefunctions may be enough). This is not just my personal opinion. Shroedinger demonstrated this for a Klein-Gordon field interacting with electromagnetic field. The reasons offered in the preceding post do not work there as the probability density does not equal \psi^2 in that case

Here is a related thread about "complex wavefunctions" that you might be interested

https://www.physicsforums.com/showthread.php?t=207417

 

[akhmeteli]

Here is a related thread about "complex wavefunctions" that you might be interested

https://www.physicsforums.com/showthread.php?t=207417

Thank you,

actually, I follow the thread that you started, but have not had time to participate in the discussion.

I still stand by what I have written in this thread. The arguments to the contrary in your thread did not convince me.

 

[reilly]

Hi there, i have been studying a bit about QM, but ther's one fundamental question
about the wavefunction i can't understand: why is the wavef. defined complex? I mean,
couldn't one work from the beginning with a real wave?

Thanks

It all started, in earnest, with Euler's exp(2πi) = 1. (At least I think it was Euler.)The rest is history; practical history. We use complex variables in all manner of sciences, engineering, and, yes, even economics. Why? Complex variables, through contour integration, analytic continuation, conformal mapping, etc. give us enormous mathematical power. (Not much different than the choice of Arabic vs. Roman numerals.)

Why do we let electric current be complex in RLC circuits? Convenience. Nothing more, nothing less. Think about the various phase relationships in circuits -- complex variables allow a particularly simple way of dealing with phases.

It's no big deal. If you don't like it, try QM without i. Why do we use 3 or 99, or x= something?

For the record, when I taught QM I required first that my students knew basic atomic and nuclear physics, particularly the key early experiments -- like Davisson Germer --And I also required students to have at least an undergraduate course in complex variables, and an E&M course that covered basic partial differential equations including special functions, radiation, etc. and advanced mechanics, including contact transformations and the Hamilton -- Jacobi EQ. I thought then, and still do, that without this basic background, it will be agonizingly difficult to understand and cope with QM,
Regards,
Reilly Atkinson

 

[danime]

If the wave-function represent a confined particle, there's allways some way to write it as a real function. See Landau-Lif****z volume 3.

 

[reilly]

Why use Arabic numerals rather than Roman numerals in physics?
Regards,
Reilly Atkinson