Show that the normalized wave function for a particle in a three-dimensional box with sides of length a, b, and c is:
Ψ(x,y,z) = √(8/abc) * sin(nxπx/a)* sin(nyπy/b)* sin(nzπz/c).
Condition for the normalization:
∫0adx ∫0bdy ∫0cdz Ψ*(x,y,z)Ψ(x,y,z) = 1.
The Attempt at a Solution
From the 1-D case I know that I should arrive to this:
∫0adx ∫0bdy ∫0cdz Ψ*(x,y,z)Ψ(x,y,z) = 1 = (AxAyAz)2 ∫0asin2(nxπx/a)dx ∫0bsin2(nyπy/b)dy ∫0csin2(nzπz/c)dz
However, I do not understand why Ψ*(x,y,z)Ψ(x,y,z) (unlike a 1-D case) is outside of the integrals. How exactly Ψ*(x,y,z)Ψ(x,y,z) is related to ∫0adx ∫0bdy ∫0cdz then? I do not understand how it is split between 3 integrals to give the formula above.
(I know how to proceed with the solution once I can arrive to this formula, so actually finding the normalization constant for me is not important.)