Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Getting Psi(x,t) from Psi(x,0)

  1. Mar 1, 2007 #1
    Hey guys, first time poster.

    I am doing some quantum physics homework, and I came across the following problem:

    A particle in an infinite square well has the initial wave function
    Psi(x,0) = Ax 0?x?a/2
    A(a-x) a/2 ?x?a

    Find Psi (x,t)

    Now after normalizing it, I tried plugging it in Schrödinger's equation, however I'm still having problems.

    Thanks in advance, Rob.
  2. jcsd
  3. Mar 1, 2007 #2

    Meir Achuz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Psi(x,t)=Sum exp{-iE_n hbar t}phi_n(x)<phi_n|Psi(x,0)>,
    where phi_n are the estates and E_n the eignevalues of the square well.
  4. Mar 1, 2007 #3
    Why did you do the summation of the function instead of integrating it??
  5. Mar 1, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The energies are discrete, hence the summation.

    You must write
    [tex] \Psi(x,t=0) = \sum_n c_n \psi_n(x) [/tex]
    where the [itex] \psi_n(x) [/itex] are the energy eigenstates, [itex] \sqrt{2/a} \, sin(n \pi x/a) [/itex] (for a well located between x=0 and x=a). What you have to do is to find the coefficients c_n using the orthonormality of the sine wavefunctions. Once you have that, the wavefunction at any time is

    [tex]\Psi(x,t) = \sum_n c_n e^{-i E_n t / \hbar} \psi_n(x) [/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook