View source for 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<code>-infty,x</code>infty) 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 = sum''n a''n cos( k_n * x + theta)

it will stays as

p = sum''n a''n cos( k''n ''' x + theta + k''n ''' 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
----
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]]
----