Why does the Schrodinger equation work?
"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,x=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