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Describe the evolution in time of [tex]\phi_1 = A \sin \omega t \cos k(x+ct)[/tex].

Attempt at Solution

We have that

[tex]\partial^2 \phi_1 / \partial x^2 = -Ak^2 \sin \omega t \cos k(x+ct)[/tex]

[tex]\partial \phi_1 / \partial t = A \sin \omega t (-kc \cdot \sin k(x+ct)) + A \omega \cos \omega t \cos k(x+ct)[/tex]

Now, by Schroedinger's Equation,

[tex]-h^2/2m \cdot \partial^2\phi_1 / \partial x^2 = i \hbar \cdot \partial \phi_1 / \partial [/tex]

So, substituting, we have

[tex]\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)[/tex]

[tex]\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[/tex]

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

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

Is this it? What am I to do now?

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