Wavefunction Evolution Problem: Solving for \phi_1

[Domnu]

Problem
Describe the evolution in time of $$\phi_1 = A \sin \omega t \cos k(x+ct)$$.

Attempt at Solution
We have that

$$\partial^2 \phi_1 / \partial x^2 = -Ak^2 \sin \omega t \cos k(x+ct)$$
$$\partial \phi_1 / \partial t = A \sin \omega t (-kc \cdot \sin k(x+ct)) + A \omega \cos \omega t \cos k(x+ct)$$

Now, by Schroedinger's Equation,

$$-h^2/2m \cdot \partial^2\phi_1 / \partial x^2 = i \hbar \cdot \partial \phi_1 / \partial$$

So, substituting, we have

$$\hbar^2 / 2m \cdot Ak^2 \sin \omega t \cos k(x+ct) = ihA \sin \omega t \cdot kc \sin k(x+ct) - i\hbar A \omega \cos \omega t \cos k(x+ct)$$

$$\iff \hbar^2/2m \cdot k^2 \cdot \sin \omega t = i \hbar \sin \omega t \tan k(x+ct) - i \hbar \omega \cos \omega t$$

$$\iff \hbar^2/2m \cdot k^2 = i\hbar \tan k(x+ct) - i \hbar \omega \cot \omega t$$

$$\iff \hbar k^2 = i \cdot 2m (\tan \k(x+ct) - \omega \cot \omega t)$$

Is this it? What am I to do now?

 

[maverick280857]

What is $\phi$?

(Sidenote: Looking at the function, the first term is a time dependent term whereas the second term is basically a "traveling wave" (for want of a better term...its actually the wavefunction of a free particle).)

I don't know what $\phi$ is, but if its the total wavefunction (we usually use $\psi(x,t)$ for it..lol ) and if your algebra is correct, the last equation doesn't make sense: the left hand side is a constant, whereas the right hand side is a function of space and time...for it to hold, $x+ct$ and $\omega t$ both should be constants (by a simple argument).

You could play around a bit by writing the whole thing as a bunch of complex exponentials, using Euler's Theorem...

$$\phi(x,t) = A\left(\frac{e^{i\omega t}-e^{-i\omega t}}{2i}\right)\left(\frac{e^{i(kx+kct)}+e^{-i(kx+kct)}}{2}\right)$$

See if that helps...