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Homework Help: Quantum mechanics HW problem on infinite square well.

  1. Oct 6, 2016 #1
    1.
    ##<x>= \int_{0}^{a}x\left | \psi \right |^{2}dx##
    ##\psi (x)=\sqrt{\frac{2}{a}}\sin\frac{n\pi x}{a}##
    then ##<x>= \frac{2}{a} \int_{0}^{a}x \sin\frac{n\pi x}{a}dx##


    2. Relevant equations

    1) ##y=\frac{n\pi x}{a}## then ##dy=\frac{n\pi}{a}dx##
    and
    2)
    ##y=\frac{n\pi x}{a}## then ##dx=\frac{a}{n\pi}dy##

    then
    ##\psi (x)=\sqrt{\frac{2}{a}} \sin(y)##
    ##<x>= \frac{2}{a}\int_{0}^{a=n\pi}y \sin^{2}ydy \times \frac{a}{n\pi} \times \frac{a}{n\pi}##


    3. The attempt at a solution

    I don't need help solving the general problem for the expectation value of x...I have the solution manual. The question I have is about how/why they chose to solve the integral this way by substituting y for (n*pi*x)/(a)? I understand how 1) works but I need help clarifying how 2) works.

    I need a general walkthrough of why they are doing this integral this way.

    Thank you

    <Moderator's note: formatting tidied up. OP, please make sure your posts are readable and use the proper LaTeX tags>
     
  2. jcsd
  3. Oct 6, 2016 #2

    BvU

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    2017 Award

    Hi quell,
    It is pretty customary to work around integrands with ##\sin ax## to get integrands with ##\sin y## : it makes it easier to get this factor a outside the integral, especially when higher powers occur and/or partial integrations are involved.
     
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