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Finding the probability density function given the eigenfunction

  1. Mar 5, 2014 #1
    1. The problem statement, all variables and given/known data

    I need to find the probability density function given the eigenfunction

    2. Relevant equations

    [itex]\psi=C\exp^({\frac{ipx}{\hbar}-\frac{x^2}{2a^2}})[/itex]

    3. The attempt at a solution

    I tried to square the function but that gave me a nasty integral that I could not solve. I also tried to factorise the index so I could've used change of variables for the Gaussian integral. I haven't come across such a question before so I am not quite sure what to do. Please help me.
    Cheers
     
  2. jcsd
  3. Mar 5, 2014 #2

    vanhees71

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    2016 Award

    The probability density is given by
    [tex]P(x)=|\psi(x)|^2.[/tex]
    Here [itex]\psi[/itex] is the wave function of the particle. Calculate this square, and you'll find a pretty well-known probability distribution!
     
  4. Mar 5, 2014 #3
    I've realised my mistake...face palm -_- Thanks! But I can't make progress at this point.
    Edit: I need the integral to find the normalisation constant, not for the actual density function. The thing that worries me is the x^2 and x term being together at the index.
     
    Last edited: Mar 5, 2014
  5. Mar 5, 2014 #4

    strangerep

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    Show details of your attempt to perform the integral.
     
  6. Mar 6, 2014 #5
    Hey Guys!
    I've made some progress and everything looks neat. It turns out the imaginary part vanishes when the function is squared(due to its conjugate). If someone could verify my work I'd appreciate it. Thanks!
    [itex]\psi=C\exp^({\frac{ipx}{\hbar}-\frac{x^2}{2a^2}})[/itex]
    [itex] |\psi(x)|^2=C^2 exp^{\frac{ipx}{\hbar}-\frac{x^2}{2a^2}} . exp^{\frac{-ipx}{\hbar}-\frac{x^2}{2a^2}}[/itex]
    [itex] |\psi(x)|^2=C^2 exp^{-\frac{x^2}{a^2}}[/itex]
    Let [itex]\beta=\frac{1}{a^2}[/itex], [itex] |\psi(x)|^2=C^2 exp^{-\beta x^2}[/itex]

    Now I need to integrate this from [itex]-\infty[/itex] to [itex]\infty[/itex] and equate to 1 in order to find the normalisation constant.
    [itex] C^{2}\int_{-\infty}^{\infty} exp^{-\beta x^2}=1[/itex]
    [itex] C^{2}\sqrt{\frac{\pi}{\beta}}=1[/itex]
    [itex] C^{2}=\sqrt{\frac{\beta}{\pi}}→ C=\frac{1}{\sqrt{a}\pi^{\frac{1}{4}}}[/itex]

    I am new to Latex...so I'm trying my best :D
     
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