Question

how do i show that the wavefunction: psi= sqrt(2/L)sin(3pix/L) satisfies the schrodinger equation

Answer 1

Find the Schrodinger equation [probably without a potential]
and substitute psi into both sides and see if they are equal.

Answer 2

So, you're wave function is a, uhm, function of x. The time independent Schrodinger equation is:

\[iħ\frac{ ∂ }{ ∂t }\psi \left( x \right)=\frac{ -ħ^{2} }{ 2m }∇^{2}\psi \left( x \right)+V \left( x \right)\psi \left( x \right)\]If you're not given a potential, then the last term on the right side of the equation is zero:\[V \left( x \right)\psi \left( x \right)=0\]

It's just a matter of "plugging and chugging" as they say, with a bit of differentiation thrown in.