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Eigenequation and eigenvalue

  1. Jan 27, 2005 #1
    what is an eigenequation? what is the purpose of the eigenvalue? how does this fit into the schrodinger equation (particle in a box problem) ?
  2. jcsd
  3. Jan 27, 2005 #2
    An eigenequation is for example the following:

    M x = b x

    where M is a Matrix (for example a 3x3), x is a vector (3 components)
    and b is a real number (could also be complex number).
    You see that the Matrix doesn't change the direction of x, only it's length (right hand side of the equation).
    x is called eigenvector and b eigenvalue of M.


    Now in Quantum mechanics you have operators (instead of matrices)
    and so called state vectors,

    for example:

    H |Psi> = E |Psi>

    ( M x = b x )

    H is the Hamilton-Operator, |Psi> is your eigenvector and E the eigenvalue.

    Whats the meaning of the equation above?
    It just says that you got a system represented by the vector |Psi>
    (for example electron in the Hydrogen atom).
    And then you want to measure the energy. This is done by
    'throwing' the operator H on your vector |Psi>. What comes out
    is your eigenvalue E which is the energy.


    Now whats the Schrödinger equation?
    Suppose you want to examine the energy of the electron in the hydrogen atom. So you just apply H on |Psi> and get the energy E on the right hand side of the eigenequation.
    The PROBLEM is, you dont know how your |Psi> looks like.

    So here's where the SCHRÖDINGER equation comes into the play.
    The Schrödinger equation is a differential equation,
    which you have to solve in order to get your |Psi>. (solving the differential equation means you get a solution |Psi>)

    You put your potential (square well potential for particle in a box, or Coloumb potential for hydrogen atom) into the Schrödinger equation and solve it. You get your |Psi> from it.


    I hope I could help you.

  4. Jan 28, 2005 #3
    thanks alot!
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