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wrongusername

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## Homework Statement

Find the ground state wave function for the 1-D particle in a box if V = 0 between x = -a/2 and x = a/2 and V = [tex]\infty[/tex]

## Homework Equations

I would guess -- Schrodinger's time-independent equation?

[tex]\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V\left(x\right)\psi=E\psi[/tex]

## The Attempt at a Solution

[tex]V\left(x\right)=0[/tex] for [tex]-\frac{a}{2}\leq x\leq\frac{a}{2}[/tex]

So then, we don't need the V(x) term in the equation, and it simplifies to [tex]\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}=E\psi[/tex]

So now I move the nasty stuff over to the right side:

[tex]E\psi+\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}=0[/tex]

[tex]\frac{2mE}{\hbar^{2}}\psi+\frac{d^{2}\psi}{dx^{2}}=0[/tex]

Now I use a trick from my book:

[tex]k^{2}=\frac{2mE}{\hbar^{2}}[/tex]

[tex]\frac{d^{2}\psi}{dx^{2}}+k^{2}\psi=0[/tex]

And now there's 2 solutions to this differential equation: [tex]\psi=A\cos\left(kx\right)[/tex] and [tex]\psi=A\sin\left(kx\right)[/tex]. Where do i go from here? I am quite lost unfortunately