Why does the Schrodinger equation work(q)

"patrick" <networkone@eircom.net> wrote in message news:<S4dV9.3059$V6.4841@news.indigo.ie>... > hello, > > Why does the Schrodinger equation work? > The only clue from QM books is that solutions to it are plane waves...these > being de Broglie matter waves. There are other diff equations whose > solutions are plane waves.

Interesting question.

Something interesting to consider.

You probably heard of the continuity equation.

d/dx( v * p ) + d/dt( p ) = 0

where p is probability at each instant of time.

and v is the velocity of probability current.

Suppose, v is contant.

A continuity converses total p . i,e if int_(x-infty,xinfty) p(x) dx = 1, it will remain 1, at each instant of time.

Now, suppose, p randomly happens to be, at instant time t=0,

p(0) = a_0 cos( k * x)

it will stay in this wave shape forever.

consider p = a_0 cos( k*x - wt + theta)

d/dx( v cos( k  x  - wt + theta ) ) + d/dt( cos( k  x - wt  +

theta)) = 0

 v  k -sin (k ''' x - wt +theta)  - w - sin(k *x - wt + theta) 

=0

 v *k = w

So, if a wave magically happened to start with the form

p = a_0 cos( k * x + theta )

it will stay in that shape forever.

[side note: There are problems with this approach, because cos(x) is not normalizable.]

Even more interesting is an interesting property of fourier transforms.

Almost any physically possible function can be decomposed into a bunch of sinusoid.

So, if p starts with

p = sumn an cos( k_n * x + theta)

it will stays as

p = sumn an cos( kn x + theta + kn v * t)

forever.

So, even if the phenomenon did not magically start as a wave, it will evolve like one ,which is all and all a curious property of the continuity equation.

> > One would imagine some simple physical model is the basis of the plane > waves. > It is strange no such model has been found.(The various interterpretations > dont seem to give any simple practical model.) > > The obvious question: the Bohr model is simple. It is too near to being > correct to be > pure coincidence. Why that coincidence.? > > patrick

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It turns out that the Schrodinger equation emerges from random population sampling, like when you sample 100 people out of 1000, find one with a desease, and assume there will be 10 in the total population.

Too bad I can't finds the reference to the proof. The article was discussed somewhere on yahoo i believe last year. This was an important finding. -- JimScarver

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