# Free electron conservation of probability

1. Feb 24, 2012

### dikmikkel

1. The problem statement, all variables and given/known data
Say you have the equation:
$i \dfrac{\partial u}{\partial t} = -\beta \frac{\partial^2 u}{\partial x^2}$
How does one show that the probability change pr. unit time equals zero for ANY wave function which obeys the equation.

2. Relevant equations
dP/dt = 0 and the above equation.

3. The attempt at a solution
I Fourier transformed the equation from x-> k and got a solution:
$u(k) = c(k)\exp(-\beta k^2 i t)$
And here its easy to show that dp/dt = 0, but is it sufficient. Say i wanted to transform like t-> k, then i would have 2 linear independent solutions and that can also be shown true.

My question is: Is my argument strong enough to make sure that all wave functions obeying the equation satisfies dp/dt = 0

Last edited: Feb 24, 2012