Schrodinger Ensemble Theory

[RAGORDON2010]

Introduction to Schrodinger Ensemble Theory

Some years ago, I chose to pursue a different approach to the study of the time-independent Schrodinger equation, particularly as it is commonly applied to the following situations:

1. a particle in an infinite potential well,
2. a particle in a finite potential well,
3. the harmonic oscillator,
4. the hydrogen atom.

The first group of examples I will discuss are all one-dimensional. The work will generalize when I deal with the hydrogen atom.

My concept is simple.  For a given potential V(x), suppose \(\psi(x)\) 

is the solution to the Schrodinger equation in the form:

\( (E)(\psi) = ((h/2(\pi))^2/2m)(d^2(\psi)/dx^2) + (V)(\psi) \).

Suppose further that an ensemble of identical, non-interacting particles is distributed in real space at time t=0 such that the fraction of particles in the region (x, x + dx) is given by

\( \psi\psi*dx \).

Suppose, in addition, that these particles exhibit an initial momentum distribution such that the fraction of particles with momentum in the range (p, p + dp) is given by \( \phi\phi*dp \), where \( \phi(p) \) and

\( \psi(x) \) are Fourier transforms of each other according to the usual rules.

I then require that the fractional density functions be consistent across the two spaces - real space and momentum space.  That is, I insist that the fraction of ensemble particles initially positioned in the region (x, x + dx) equals the fraction of ensemble particles with initial values of momentum in the region (p, p + dp).  That is, I require that my consistency relationship

\( \psi(x)\psi(x)*dx = \phi(p)\phi(p)*dp \) is satisfied.

Finally, I use this consistency relationship to seek a momentum function p(x). On the one hand, it may be possible to find p(x) by inspection or via trial and error. Otherwise, it might be possible to integrate each side of the relationship separately and isolate p(x) from the result.  Even then, there will still be some freedom left to decide on the direction of the momentum vectors.

Please note that for these Schrodinger ensembles, total particle energy is not a "sharp" variable.  The expectation energy averaged across the entire ensemble remains the eigenvalue E, but the energy of any individual particle is always computed from

\( p(x)^2/2m + V(x) \) 

in the usual manner.

Also note that since psi(x) and phi(p) only represent initial conditions placed on the ensemble, the subsequent development of the ensemble over time is determined by applying Liouville's theorem to the ensemble.

I have not found a way to develop Schrodinger Ensemble Theory for the time-dependent Schrodinger Equation.

 

[swansont]

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Moderator Note

You were told to do this kind of blogging elsewhere