Number of different ways, particle with E and delta_E.

[Spinnor]

If we prepare a particle with energy E and some small spread in energy delta E then its wave function can be expanded with some large countable set of momentum states (exclude states above some very high energy)?

If we have a very large countable set of momentum states it seems we could exclude some states and still make a very good approximation for the wave function for a particle of energy E and energy spread delta E?

Thanks for any help!

 

[xts]

Countable? Why? Continuous!

And of course, the mix of continuous states excluding some values should not cause any problems.

 

[Spinnor]

Countable? Why? Continuous!

And of course, the mix of continuous states excluding some values should not cause any problems.

In a large box if we exclude very high energy don't we just need a countable set?

 

[xts]

In a large box if we exclude very high energy don't we just need a countable set?

OK - in a large box, you have countable set. Don't dispute about order of infinity...
But I still can't see the problem? Yes, you may exclude a subset of measure 0 (or close to 0) and still get the same results of all statistical predictions.

 

[Spinnor]

OK - in a large box, you have countable set. Don't dispute about order of infinity...
But I still can't see the problem? Yes, you may exclude a subset of measure 0 (or close to 0) and still get the same results of all statistical predictions.

I'm trying understand so that I might answer the question in the title, "Number of different ways, particle with E and delta_E". If I read you correctly it seems we can have many states that are for practical purposes the same state?

 

[xts]

If I read you correctly it seems we can have many states that are for practical purposes the same state?

Sure!
If the number of possible states (number of degrees of freedom) counts - then if you prohibit some of the large number of still possible - it won't be noticeable.
In any case - prohibiting measure-0 subset won't cause any visible effects (like black lines on the spectrum)

 

[Spinnor]

Sure!
If the number of possible states (number of degrees of freedom) counts - then if you prohibit some of the large number of still possible - it won't be noticeable.
In any case - prohibiting measure-0 subset won't cause any visible effects (like black lines on the spectrum)

Say we scatter a spin-less charged particle of energy E and energy spread delta E off a potential. Now say we have the wave function made of different subsets of the full set (minus the highest energy) of momentum states for this spin-less charged particle of energy E and energy spread delta E. Will those wave-functions of different subsets scatter off the potential the same? Can we manipulate things so that small changes in the subsets give rise to large variations in scattering?

 

[xts]

As long as your potential is smooth and 'nice' small gaps in a set constituting your wavefunction have no effect. You may create artificial potentials, e.g. forming diffraction grid, differentiating small (but still finite) variances in your wavefunction. But even then, if the eliminated subset has measure of 0, they won't affect final outcome.