Proving solutions to the schrodinger equation

[Bacat]

Homework Statement

Verify that the following are not solutions to the Schrodinger equation for a free particle:

(a) $$\Psi(x,t) = A*Cos(kx-\omega t)$$

(b) $$\Psi(x,t) = A*Sin(kx-\omega t)$$

Homework Equations

Schrodinger equation: $$\frac{-hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} = \frac{i*hbar*\partial \Psi}{\partial t}$$

The Attempt at a Solution

For part a:

Let $$y=kx-\omega t$$

Calculating the derivatives gives...

$$\frac{\partial^2 \Psi}{\partial x^2}=-Ak^2*Cos y$$
$$\frac{\partial \Psi}{\partial t} = A*\omega*Sin y$$

Substituting and rearranging, we see that:

$$Sin y = \frac{hbar * k^2}{2mi\omega}Cos y$$

Let $$f=\frac{hbar * k^2}{2mi\omega}$$

Then $$Sin y = f Cos y$$

The equality holds when $$y=\pm ArcCos(\pm\frac{1}{\sqrt{1+f^2}})$$

If we substitute back in...

$$kx-\omega t = \pm ArcCos(\pm\frac{1}{\sqrt{1+\frac{hbar^2k^4}{4m^2\omega^2}}})$$

If I assign some values to omega, m, hbar, and k, I can graph x as a function of t for one of the cases. It shows up as a straight line with a positive slope.

This does not seem to prove that (a) is not a solution to Schrodinger's equation. I think I may be making this more complicated than it is. How should I approach this problem differently?

 

[gabbagabbahey]

[
Substituting and rearranging, we see that:

$$Sin y = \frac{hbar * k^2}{2mi\omega}Cos y$$

Let $$f=\frac{hbar * k^2}{2mi\omega}$$

Then $$Sin y = f Cos y$$

This right here should tell you that $\Psi(x,t) = A*Cos(kx-\omega t)$ is not a solution to the schrodinger equation. If it were, you would expect it to satisfy the schodinger equation everywhere (i.e. for all values of y)--- which it clearly doesn't.

 

[Bacat]

Ah, good point. So it IS a lot simpler than I was making it. Haha.

Any ideas on this one:

https://www.physicsforums.com/showthread.php?t=292957

?

 

[vaibhav1803]

a faster way would be to express cos(kx-$$\omega$$ t) in terms of e^{i(kx-$$\omega$$ t)}
and substitute in the time dependant equation both will never satisfy it(coz the wavefunc. is of the form Ae^{i(kx-$$\omega$$ t} )
any ideas on this one mate..?
https://www.physicsforums.com/showthread.php?p=2218109#post2218109