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Homework Help: Probability of finding a particle in a region

  1. Dec 16, 2017 #1
    1. The problem statement, all variables and given/known data
    A particle is restrained to move in 1D between two rigid walls localized in ##x=0## and ##x=a##. For ##t=0##, it’s described by:

    $$\psi(x,0) = \left[\cos^{2}\left(\frac{\pi}{a}x\right)-\cos\left(\frac{\pi}{a}x\right)\right]\sin\left(\frac{\pi}{a}x\right)+B $$

    , determine the probability of finding the particle between 0 and ##\frac{a}{4}##.

    2. Relevant equations
    (1) ##\phi_{n}(x)=\sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)##
    (2) ##B=\sum_{n} C_{n}\phi_{n}(x)##

    3. The attempt at a solution

    Using some trigonometry and the orthonormal base (equation 1), I can write the wave function as:

    $$\psi(x,0)=\sqrt{\frac{a}{32}}\phi_{1}(x)+\sqrt{\frac{a}{8}}\phi_{2}(x)+\sqrt{\frac{a}{32}}\phi_{3}(x)+B$$

    I still can’t use the evolution operator. I must find an expression to ##B##, so I can put it in terms of the base.

    I used the equation 2 and after finding the value of ##C_{n}##, and noticing that only odd values of n contributes to the wave function:

    $$B\rightarrow -\frac{B}{\pi}\sqrt{8a}\sum_{0}^{\infty}\frac{1}{2n+1}\phi_{2n+1}(x)$$

    I can write the wave function like this:
    $$\psi(x,0)=\bigg( \sqrt{\frac{a}{32}}-\frac{B}{\pi}\sqrt{8a}\bigg)\phi_{1}(x) + \sqrt{\frac{a}{8}}\phi_{2}(x) + \bigg(\sqrt{\frac{a}{32}}-\frac{B}{3\pi}\sqrt{8a}\bigg)\phi_{3}(x) - \frac{B}{\pi}\sqrt{8a}\sum_{2}^{\infty}\frac{1}{2n+1}\phi_{2n+1}(x)$$


    My problem is how should I continue. Am I sure this is normalized? Probably not, but how to find a constant so I can be sure it's normalized? It’s just finding the value ##B## using ##\langle\psi|\psi\rangle##? Or there is other way? Because using ##\langle\psi|\psi\rangle## I get a quadratic equation, and I’m not so sure that is this way.
     
  2. jcsd
  3. Dec 16, 2017 #2

    PeroK

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    What are you trying to find? Do energy eigenfunctions help you in this?
     
  4. Dec 16, 2017 #3

    jtbell

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    Staff: Mentor

    To clarify, do you want this probability only at time 0 or do you want it as a function of time?
     
  5. Dec 16, 2017 #4
    You are right, it's for t>0, I will edit it. Thank you

    Edit: Well, I noticed I can't edit the original post.
     
    Last edited: Dec 16, 2017
  6. Dec 16, 2017 #5
    I'm trying to determine the probability of finding the particle between 0 and a/4 for a t>0
     
  7. Dec 16, 2017 #6

    jtbell

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    Staff: Mentor

    Hmmm, I'm confused. You have the following initial wave function:

    When I first saw this, I thought B is a constant, maybe for normalization although it's in an odd place for a normalization constant. Then you have this:

    OK, (1) are the energy eigenfunctions for the "particle in a box." But where does (2) come from? Is it saying that B isn't a constant after all, but instead, a function? A linear combination of all the energy eigenfunctions, with constant coefficients ##C_n## that are as yet undetermined?
     
  8. Dec 16, 2017 #7

    I understand, sorry I wasn't very clear. Yes, B it is a constant. But even if it is a known constant, when I make it act the evolution operator, I can't just make it evolve a constant. I only con aply the evolution operator to a state given by the base (1). So, what did I do so I can aply the evolution operator to the constant? I expanded in a fourier serie, so now it's in term of the orthonormal base, basically that's the equation (2). Now it's expanded I can aply the evolution operator, and try to determine the probability asked.
    I hope it is clear now.

    Now I want to make sure it is normalized, and there is where I have my doubts of how to do it.
     
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