(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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).

2. Relevant equations

Condition for the normalization:

∫_{0}^{a}dx ∫_{0}^{b}dy ∫_{0}^{c}dz Ψ*(x,y,z)Ψ(x,y,z) = 1.

3. 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.)

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# Homework Help: Normalization constant for a 3-D wave function

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