1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Wave function problem

  1. Mar 1, 2010 #1
    1. The problem statement, all variables and given/known data
    The function

    [itex] \Psi(r) = A(2-{Zr\over a})e^-{Zr\over 2a} [/itex]

    gives the form of the quantum mechanical wavefunction representing the electron
    in a hydrogen-like atom of atomic number Z when the electron is in its first
    allowed spherically symmetric excited state. Here r is the usual spherical polar
    coordinate, but, because of the spherical symmetry, the coordinates θ and φ do
    not appear explicitly in Ψ. Determine the value that A (assumed real) must have
    if the wavefunction is to be correctly normalised, i.e. the volume integral of |Ψ|^2
    over all space is equal to unity.


    3. The attempt at a solution

    [itex] {\int \int \int}_R |\Psi|^2 dV = 1[/itex]

    [itex] \int _0^{\infty }\int _0^{2\pi }\int _0^{\pi }A^2e^{-\frac{Zr}{a}} \left(2-\frac{Zr}{a}\right)^2d\phi d\theta dr = 1[/itex]

    Which implies

    [itex] \int _0^{\infty }A^2e^{-\frac{\text{Zr}}{a}} \left(2-\frac{\text{Zr}}{a}\right)^2dr = {1\over 2\pi^2}[/itex]

    This turns out to be

    [itex] \frac{2aA^2}{Z} = \frac{1}{2\pi^2} [/itex]

    [itex] A = \pm \frac{\sqrt{\frac{z}{a}}}{2\pi} [/itex]

    This is wrong though?
    Is the problem the fact that Psi(r) isnt a function of theta or phi?
     
  2. jcsd
  3. Mar 1, 2010 #2

    Päällikkö

    User Avatar
    Homework Helper

    Looks to me you forgot the r2 sin term from the differential volume element (note that the way you did it, the volume of a unit ball would come out as 2pi2). See http://en.wikipedia.org/wiki/Spherical_coordinate_system
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Similar Discussions: Wave function problem
  1. Wave function (Replies: 2)

  2. Wave functions (Replies: 15)

Loading...