Show that this Equation Satisfies the Schrodinger Equation

[Zinggy]

Homework Statement

Show directly by using the wave function in the Schrodinger equation, that this equation satisfies the Schrödinger equation

Relevant Equations

V(x)=1/2kx^2
E=(5ħ/2)√(k/m)
α=(mk/ħ)^1/4
Ψ(x,t) =ψ (x)φ (t) = √(α)/[2√(2π^1/4)]*e^((-1/2)α^2x^2)[-2+α^2 x^2]e^(-iEt/ħ)

I apologize for the bad formatting:

To start off, I'm trying to use the Schrödinger Equation in the form: (ħ/2m) d^2Ψ(x,t)/dx^2+V(x,t)Ψ(x,t)=EΨ(x,t)

I couldn't remember if I need to also take the partial derivative with respect to T as well, but I started off with just X.

I plugged in my known values into the equation which gives me the very messy:
(ħ/2m)d^2[√(α)/[2√(2π^1/4)]*e^((-1/2)α^2x^2)[-2+α^2 x^2]e^(-iEt/ħ)]/dx^2 +(1/2kx^2)(√(α)/[2√(2π^1/4)]*e^((-1/2)α^2x^2)[-2+α^2 x^2]e^(-iEt/ħ)) = (5ħ/2)√(k/m)* √(α)/[2√(2π^1/4)]*e^((-1/2)α^2x^2)[-2+α^2 x^2]e^(-iEt/ħ)

This already seemed wrong to me, but I carried on and tried to take the second derivative with respect to X and got the following:

√(α)/[2√(2π^1/4)]*α^2 e^[-1/2(α^2 x^2)] (α^2 x^2 -1) [-2+8α^2)e^[-iEt/ħ] = d^2Ψ(x,t)/dx^2

From here I extracted common terms, giving me:

√(α)/[2√(2π^1/4)] * e^[-iEt/ħ]*e^[-1/2(α^2 x^2)] * {(ħ/2m)α^2(α^2 x^2 -1)[-2+8α^2)+1/2kx^2[-2+4α^2x^2]} = EΨ(x,t)

From this point I divided both sides by √(α)/[2√(2π^1/4)] and e^[-1/2(α^2 x^2)] giving me

e^[-iEt/ħ]* {(ħ/2m)α^2(α^2 x^2 -1)[-2+8α^2)+1/2kx^2[-2+4α^2x^2]} = E([-2+α^2 x^2]e^(-iEt/ħ)]

After this point I don't really know where to go. I don't understand how to isolate and solve for either ψ(x) or φ (t), I also don't know if I should've also taken the second derivative with respect to T as well.
I appreciate any time and Input. Thanks.

 

[vela]

Start by looking up the Schrödinger equation. The version you're using isn't correct (or you made the same typo multiple times). Also, note that the function you're working with is a function of $x$ and $t$.

A comment on terminology... You're not trying to show that an equation satisfies the Schrödinger equation; you're trying to show that a wave function satisfies the Schrödinger equation.

 

[Zinggy]

I may have not gotten my formatting correct, this is what I'm using as my Schrödinger equation

I found this equation in my Harmonic Oscillator section in my textbook, is this wrong?

 

[vela]

Here's what you said in your original post:

(ħ/2m) d^2Ψ(x,t)/dx^2+V(x,t)Ψ(x,t)=EΨ(x,t)

You were missing the minus sign in front and you didn't square $\hbar$. The minus sign makes a big difference.

Note that in the equation you cited in post #3, the function $u$ is a function of only $x$ whereas you're given a function $\Psi$ of $x$ and $t$, so that's not the equation you want to use.

 

[Zinggy]

Ah yes, those were just typos, sorry I didn't do a great job of representing my equations.. Would the question use an equation like this?

 

[vela]

Essentially, yes, though that's an intermediate step.

You shouldn't have to be guessing here. Your book should tell you what the time-dependent and time-independent forms of the Schrödinger equation are.