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

  1. Dec 29, 2017 #1
    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(nxπx/a)* sin(nyπy/b)* sin(nzπz/c).

    2. Relevant equations
    Condition for the normalization:
    0adx ∫0bdy ∫0cdz 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:
    0adx ∫0bdy ∫0cdz Ψ*(x,y,z)Ψ(x,y,z) = 1 = (AxAyAz)20asin2(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.)
     
  2. jcsd
  3. Dec 29, 2017 #2

    Dick

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    In multiple integrals like this they sometime write all of the integration variables and limits on the left and the integrand on the right. This is not supposed to mean that the integrand is 'outside' of the integral or anything. It's just supposed to make it a little easier to read. I think you know exactly what to do. Just ignore that writing convention and put the integrand where you know it should be.
     
  4. Dec 29, 2017 #3

    PeroK

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    @Valeria There are two ways to write an integral, whether is has one variable or several

    ##\int f(x)dx##

    Might be considered the normal mathematician's convention.

    Or:

    ##\int dx f(x)##

    which physicists may prefer.

    They both mean the same. When it comes to multivariable integrals, I prefer the former. For example:

    ##\int \int \int f(x,y,z) dx dy dz##
     
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