Gaussian Wave Packets - Fiction?

[LarryS]

Forgetting about spin and polarity for the moment, do individual identical particles of the same type (electrons, photons, etc.) really have their own individual wave functions (Gaussian packets)? The mathematical definition of probability (relative frequency, etc.) has meaning only at the ensemble level. Obviously, there is something that I do not understand, but how can one assign a wave function to an individual particle, a sample space of one! As always, thanks in advance.

 

[Fra]

The mathematical definition of probability (relative frequency, etc.) has meaning only at the ensemble level. Obviously, there is something that I do not understand, but how can one assign a wave function to an individual particle, a sample space of one! As always, thanks in advance.

The frequentist or ensemble view is just one possibility, and not necessarily the most sensible one.

Some ponderings of relevance is an old text from John Baez.
http://math.ucr.edu/home/baez/bayes.html

/Fredrik

 

[alxm]

Well, looking at some of the practical aspects: If two particle don't interact, then the S.E. is separable and the overall wave function is exactly represented by a product of single-particle solutions. So for an interacting system of particles, the aforementioned product makes for an approximation in the non-interacting limit.

Also, if you neglect the interaction, the single-particle solutions form a complete set. So you can use your single-particle solutions to form a basis for your many-particle system. That's a good idea mathematically if the interaction energy is small, and a good idea intuitively since it tends to be a lot easier to think about stuff in terms of single particles.

Of course, that requires an infinite number of functions, in theory. So you've got to approximate by truncating your description somewhere, and that's where the interesting physics comes into it. :)

Another issue here is that if you're talking about fermions, you have to satisfy the antisymmetry requirement/Pauli principle. That's pretty easy if you're working in a single-particle basis (form a Slater determinant), but easily gets quite difficult if you're not. (c.f. the difficulties of developing DFT methods)

Now for indistinguishable particles the 'identities' are of course just arbitrary labels. But that doesn't bother me. If I'm looking at an atom or molecule, I'm not interested in measuring an individual electron to find out which one it is; I know I can't. What does interest me, though, is finding out the respective states of the different electrons and how they contribute to the overall picture. Because that's how we think about and rationalize electronic structure. Apart from the energy I get from it, the total wave function is in fact rather uninteresting to look at.

 

[Truecrimson]

There're also a lot of ensemble interpretations.
http://www.dipankarhome.com/ENSEMBLE%20INTERPRETATIONS.pdf
This paper reviews various meanings of probability and ensemble interpretations proposed since Einstein and up to Ballentine.

 

[LarryS]

There're also a lot of ensemble interpretations.
http://www.dipankarhome.com/ENSEMBLE%20INTERPRETATIONS.pdf
This paper reviews various meanings of probability and ensemble interpretations proposed since Einstein and up to Ballentine.

Interesting paper. Thanks.