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Eigenfunctions versus wavefunctions

  1. Jun 29, 2006 #1
    Hi - hope that someone can help me with this.

    I am new to quantum mechanics - trying to answer a question about eigenfunctions and don't have a decent text book at the moment.

    Can someone tell me please, what is the difference between a wavefunction and an eigenfunction for a particle in an infinite square well?

    Cheers

    Peter
     
  2. jcsd
  3. Jun 29, 2006 #2
    An eigen function is the [tex]\psi[/tex] in the relationship A[tex]\psi[/tex] = a[tex]\psi[/tex]. the [tex]\psi[/tex] can be either a wave function or can represent the state of a system, i.e. [tex]\psi[/tex] can be a "ket", which is a quantity (a generalized vector) which represents the system. A wavefunction, in general, can be either an eigenfunction/eigenket or a linear sum of eigenfunctions/eigenkets.

    Pete
     
  4. Jun 29, 2006 #3
    There is a particular subset of wavefunctions known as "eigenfunctions" of an observable A, they are defined as those satisfying, for some scalar a,

    [tex]\hat{A}\psi(x)=\alpha \psi(x)[/tex]

    (This is linear algebra actually.)
     
  5. Jun 29, 2006 #4
    As you may know, the eigenvalues which one customarily computes in connection with the infinite square-well are energy eigenvalues: E0, E1, E2, ...

    Corresponding to each of these energy eigenvalues is a wavefunction φ0(x),
    φ1(x), φ2(x), ...

    These particular wavefunctions, φn(x), are said to be the energy eigenfunctions associated with the infinite square-well. Thus, for example, a particle in the state φn(x) will have energy En.

    But the state of a particle in the well doesn't have to be just a particular one of these φn(x). The state could be any normalized complex-valued function ψ(x) whose value is zero for x outside of the well. Such a ψ(x) is said to be a wavefunction for a particle in an infinite square-well.

    Thus, every energy eigenfunction is a wavefunction, but not every wavefunction is an energy eigenfunction.

    Nevertheless, it turns out that any such wavefunction ψ(x) can be written as a superposition of the eigenfunctions φn(x). That is, we can write

    ψ(x) = ∑nanφn(x) ,

    for some complex coefficients an, where

    n|an|2 = 1 .
     
  6. Jun 30, 2006 #5
    That's a good, thorough post, Eye_in_the_Sky. However, I would prefer to start my energy eigenvalues from E1 and energy eigenfunctions from φ1(x), because we are dealing with a particle in an infinite square well.
     
  7. Jun 30, 2006 #6
    Yes, masudr, I see what you mean. Starting with n=1 (instead of n=0) makes the actual expressions for En and φn(x) somewhat easier on the eye, since in that case only an "n" (instead of an "n+1") will appear.
     
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