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Thread: Question about wave functions?

  1. #1 Question about wave functions? 
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    If I have solved a give Schrodinger equation and obtained the wave function, how to I obtain the corresponding probability density of that wave function? And, also could you give the probability density of the wave function Ψ(x,t)=e^(7it+5ix), just as an example.
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  2. #2  
    mvb
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    Quote Originally Posted by muon321 View Post
    If I have solved a give Schrodinger equation and obtained the wave function, how to I obtain the corresponding probability density of that wave function? And, also could you give the probability density of the wave function Ψ(x,t)=e^(7it+5ix), just as an example.
    The probability is |N|2Ψ*(x, t) Ψ(x,t), with N the normalization of the wave function. In principle, N is gotten by requiring that the integral of the probability density over all space is 1. That makes the probability density for your wave function P(x,t) = |N|2 , where you don't yet know N. In this case, the determination of N is problematic if your function is defined over all space. In that event, the normalization is chosen to be a different form that is motivated by Fourier transform arguments.

    If the normalization is what is bothering you about this function, say so, and I will try to say something useful about it.
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    Quote Originally Posted by muon321 View Post
    If I have solved a give Schrodinger equation and obtained the wave function, how to I obtain the corresponding probability density of that wave function? And, also could you give the probability density of the wave function Ψ(x,t)=e^(7it+5ix), just as an example.
    Your function is an element of a Hilbert space. The norm of such a function is obtained by taking the integral of the square of the absolute value of the function and then taking the square root. One normalizes such a wave function by dividing by the norm, which results in a function of norm 1.

    The particular wave function that you specified, being the exponential of a purely imaginary quantity has complex magnitude 1 (Google the Euler formula is this is not clear to you). In order that it be square integrable the domain over which you are integrating must be compact (of finite volume). In this case the norm will be the volume of that domain.

    Once you have normalized the wave function the probability density for position will be just the magnitude of the function squared, in this case just the constant 1. So your position distribution is uniform.
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