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## Homework Statement

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(n

_{x}πx/a)* sin(n

_{y}πy/b)* sin(n

_{z}πz/c).

## Homework Equations

Condition for the normalization:

∫

_{0}

^{a}dx ∫

_{0}

^{b}dy ∫

_{0}

^{c}dz 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:

∫

_{0}

^{a}dx ∫

_{0}

^{b}dy ∫

_{0}

^{c}dz Ψ*(x,y,z)Ψ(x,y,z) = 1 = (A

_{x}A

_{y}A

_{z})

^{2}∫

_{0}

^{a}sin

^{2}(n

_{x}πx/a)dx ∫

_{0}

^{b}sin

^{2}(n

_{y}πy/b)dy ∫

_{0}

^{c}sin

^{2}(n

_{z}πz/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 ∫

_{0}

^{a}dx ∫

_{0}

^{b}dy ∫

_{0}

^{c}dz 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.)