schrodinger equation help?

For what kind of system?

I'll assume you mean a particle in a one-dimensional, square, infinite potential well with zero potential between 0 and L because that happens to give the answer you seek.

For the region with zero potential, the time-independent Schrodinger equation becomes

ψ" = -[2mE/ħ²]ψ

This linear second-order differential equation has the general solution

ψ(x) = Asin(kx) + Bcos(kx), where k = sqrt(2mE)/ħ

Using the boundary conditions ψ(0) = ψ(L) = 0 yields

ψ(x) = Asin(kx) and k = nπ/L

To find A, we assume that once the state collapses (n = 1 for ground state), the probability of finding the particle between 0 and L is 1.

∫ ψ*(x)ψ(x) dx = 1

∫ A²sin²(πx/L) dx = 1

∫ A²[1/2 - (1/2)cos(2πx/L)] dx = 1

A²[x/2 - (L/(4π))sin(2πx/L)] = 1

A²[L/2 - (L/(4π))sin(2πL/L)] = 1

A = sqrt(2/L)

ψ = sqrt(2/L) sin(nπ/L)

Now calculate the probability you need.

∫ (2/L)sin²(πx/L) dx =

(2/L)[x/2 - (L/(4π))sin(2πx/L)] =

(1/L)[x - (L/(2π))sin(2πx/L)] (evaluate from L/2 to 2L/3) =

(1/L)[(2L/3 - L/2) - (L/(2π))(sin(4π/3) - sin(π))] =

1/6 + sqrt(3)/(4π) ≈ 0.3044