Probability Distributions in Quantum Mechanics

Most recent answer: 02/28/2014

Q:
Probability in Quantum Mechanics. I'm wondering if there are characteristic probability distributions in quantum mechanics. I'm thinking that the distributions are over integers, but do the probabilities of the outcome of particle interactions range from an impulse of 100% for macroscopic objects all the way to uniform probability distributions for high-energy particle interactions? Or are high-energy outcomes limited to bell-shaped probability distributions? Probably none of the above, but I'm willing to learn. I've read that bosons(like photons), obey Bose-Einstein statistics and that Fermions(like protons), obey Fermi–Dirac statistics. Does that mean what kind of bell-curves they have? I've learned a good deal of probability and statistics and I wonder if QM probability distributions have means and standard deviations and all the other machinery?
- Bill Eshleman (age 68)
Gainesville, Florida, Alachua
A:

Yes, the probability distributions we usually deal with in quantum mechanics have means and standard deviations. There are some common cases in which continuous variables have the bell-shaped Gaussian density functions, but there are many cases with other distributions. Uniform density functions are pretty unusual- I can't think of a single realistic case offhand. The Bose-Einstein and Fermi-Dirac statistics become very different for dense collections of particles. For individual particles, the same sorts of density functions are available for either type.

I'm not sure what you mean by distributions over integers. There are some operators, the ones that we say represent particle numbers, which only take on integer values. Their probabilities are thus confined to those integer values. That's quite a bit different from the distributions for the position or momentum operators, for examples.

Mike W.


(published on 02/28/2014)

Follow-Up #1: more quantum probability functions

Q:
Thanks Mike. I thought the distributions were distributed over integers and I thought the integers were the quantum numbers. Guess I missed the point. Now I'm thinking the distributions in QM are distributed over space?? What are these probability density distributions distributed over?
- Bill Eshleman (age 68)
Gainesville, Florida, Alachua
A:

It all depends on what the variable is, and the situation. Energy provides a good example. For an electron-proton system you can get the electron bound to the proton, forming a hydrogen atom. The possible energy levels of this system form a set of discrete values. (There are some slight complications involving interactions with quantized electromagnetism, but let's ignore those for now.) So you need a list of probabilities for those discrete values.  If the system has more  energy then the particles don't bind. Then a continuous range of energies is possible. So for that part of the distribution you need a probability density function, not a list of probabilities.

In another example, for momentum, a particle confined in a box has a list of discrete possible momentum states. A free particle has a continuum of states.

Mike W.


(published on 03/02/2014)