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## Homework Statement

Below is a past paper question from a paper i am working through in preperation for my exams. I am having particular trouble with this question, it doesn't seem as straight forward and formulated as other questions. I am unsure if anything i have done is right so any help would be great.

## Homework Equations

(a)

Time independent equation

(b)

Time dependent Schrodinger

(c)

Probabilty Density [tex]=\left|\psi(x,t)\right|^{2}[/tex]

## The Attempt at a Solution

a)

[tex]H_{osc}\phi_{0}(x)=E\phi_{0}(x)[/tex]

[tex]\\\\H_{osc}\phi_{1}(x)=E\phi_{1}(x)[/tex]

b)

I tried simply putting the wavefunction into the Time dependent Schrodinger equation and differentiating. This mustnt be the way since there is an m on the LHS.

c)

Probabilty Density

[tex]\left|\psi(x,t)\right|^{2}=1/2[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)][/tex]

d)

[tex]\int^{\infty}_{0}\left|\psi(x,t)\right|^{2}dx=1/2\int^{\infty}_{0}[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)]dx[/tex]

^ This doesn't work out too well