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Probability with a normalized wave function

  1. May 22, 2013 #1
    1. The problem statement, all variables and given/known data
    So I have this normalized wave function

    psi=sqrt(2/a) * sin^2(pi*x/a)

    with limits of 0 and a.

    I'm supposed to find the probability for a bunch of points if this form:

    p(x=0.00a,x=0.002a)


    2. Relevant equations

    P(a,b)=int(psi^2)dx

    3. The attempt at a solution

    So, partially solving the integral, I'm stuck here:

    (2/a) * [ (x/2) - (a/4pi) * sin(2*pi*x/a) ] between 0 and a

    My first question. If I continue to solve the definate integral P=1. This is appearently not the correct answer. I know my calculus is right and I've verified it in wolfram alpha and octave.

    Once I figure out how to proceed I need to plug in these values which have a in them, not really sure how that is going to work out either.

    Any suggestions?
     
  2. jcsd
  3. May 22, 2013 #2

    Simon Bridge

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    $$P(\alpha<x<\beta)=\int_\alpha ^\beta \psi^\star(x)\psi(x)\; dx$$

    You have been given ##\psi(x)= \sqrt{\frac{2}{a}}\sin^2(\pi x/a)##

    - there has to be more to the function than that because this is not normalized - there must be part where it says ##\psi = 0## for some values of x.

    You need to be able to find probabilities of form:
    $$P(0<x<ra)=\frac{2}{a}\int_0 ^{ra} \sin^4(\pi x/a)\; dx\; :\; 0<r\leq 1$$

    (eg. in your example, post #1, r=0.002)

    this will be easier with a substitution like: ##\theta=\pi x/a##

    $$P(0<x<ra)=P(0<\theta <\pi r)=\cdots$$

    ... and you can look up the indefinite integral for sin4(x)
     
  4. May 23, 2013 #3
    Ok, I'm pretty new to this stuff but if by normalized you mean is equal to one than it is normalized.
    If thats not what you mean then maybe I'm misunderstanding the meaning of the word normalized.

    I've already taken it to my professor once, because I thought there was a mistake. He insists there is not.

    Here is a link to the actual assignment
    https://dl.dropboxusercontent.com/u/22548767/CHEC%20352%20Exam%20II%20Assignment.pdf [Broken]


    BTW: is that Latex you are using? can you just type that in to a message or does it have to be encapsulated by something?
     
    Last edited by a moderator: May 6, 2017
  5. May 24, 2013 #4

    Simon Bridge

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    By "normalized" I mean that:
    $$\int_{-\infty}^\infty \psi^\star\psi\; dx = 1$$

    ... the wavefunction given in post #1 is:
    ... that wavefunction is periodic and positive everywhere - so the integral is going to diverge. If we restrict the range to a single wavelength, [0,a], assuming zero elsewhere, then the integral does not come to 1 (check). Therefore, it is not normalized. Therefore there is missing information =)

    Excellent <reads>

    The wavefunction given in your assignment sheet has some important differences from the one in post #1 - it's much simpler for a start. Please go over it more carefully.

    Anyway - even with the differences - the same advise will hold as of post #2 but with the actual wavefunction this time.

    Well spotted - yes it is ##\LaTeX##
    ... below my post, to the right, there is a button marked "quote" - click on it and you will see how I did it ;)
     
  6. May 24, 2013 #5
    got it.
    thanks for your help.
     
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