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!

Is the wave function normalized?

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data

    The ground state wave function for the electron in a hydrogen atom is:

    [tex]\psi(r) = \frac{1}{\sqrt (\pi a_o^3)} e^\frac{-r}{a_o}[/tex]

    where r is the radial coordinate of the electron and a_o is the Bohr radius.

    Show that the wave function as given is normalized.

    2. Relevant equations

    Any wave function satisfying the following equation is said to be normalized:

    [tex]\int_{-\infty}^{+\infty} |\psi|^2\dx = 1[/tex]

    3. The attempt at a solution

    Because the sum of all probabilities over all values of r must be 1,

    [tex]\int_{-\infty}^{+\infty} (\frac{1}{\sqrt (\pi a_o^3)} e^\frac{-r}{a_o})^2 dr = \frac{1}{\pi a_o^3} \int_{-\infty}^{+\infty} e^\frac{-2r}{a_o^2} dr = 1[/tex]

    Since the integral can be expressed as the sum of two integrals, we have,

    [tex]\frac{1}{\pi a_o^3} \int_{-\infty}^{+\infty} e^\frac{-2r}{a_o^2} dr = \frac{2}{\pi a_o^3} \int_0^{+\infty} e^\frac{-2r}{a_o^2} dr = 1 [/tex]

    After integrating, I obtain,

    [tex]\frac{1}{\pi a_o^3} = 1[/tex]

    which is definitely incorrect. However, I do not see any other way to proceed. could someone give some assitance.

    Thanks for your kind assistance

  2. jcsd
  3. Jan 27, 2009 #2
    you integrated wrong -
    [itex]\int_0^{\infty} e^{-\frac{2r}{a_0^2}} dr = [-\frac{a_0^2}{2} e^{-\frac{2r}{a_0^2}]_0^{\infty}=\frac{a_0^2}{2} [/itex]

    which gives [itex]\frac{1}{2 \pi a_0}=1[/itex]
  4. Jan 27, 2009 #3
    the a_0 in your exponent shouldn't be squared, also the normalization condition says you should integrate over the bounds of your function. if r is the radial coordinate then how can it have a value of negative infinity? similarly your no longer normalizing in 1-d but 3D I recommend using spherical coordinates in which case your integral inherits a r^2 sin theta
  5. Jan 28, 2009 #4
    Tks. I wil try again using spherical coordinates
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Is the wave function normalized?