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Eigenfunction vs wave function

  1. Jun 30, 2012 #1
    What is the difference between eigenfunction and wave function?

    I'm always get confused when i am asked to write wave function and eigenfunction..
     
  2. jcsd
  3. Jun 30, 2012 #2

    Simon Bridge

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    An eigenfunction is a type of wavefunction that has an eigenvalue when operated on. It is said to be "an eigenfunction of the operator".
     
  4. Jun 30, 2012 #3
    is this right?

    ψ(x,t) = ψ1(x) + ψ2(x)
    wavefunction = eigenfunction1 + eigenfunction2
     
  5. Jul 1, 2012 #4

    Nabeshin

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    Assuming ψ1,ψ2 are eigenfunctions, then yes... Just because the wavefunction is written as the sum of something doesn't mean those somethings are eigenfunctions!

    Here's the full description of the situation: You have a quantum system which has only two possible states, 1 and 2. That means there are two quantum states, ψ1 and ψ2 describing the system in either state 1 or 2 (1 and 2 can be spin up or spin down, for example). In general, then, any arbitrary wavefunction can be written as a linear superposition of these two states, [itex] \Psi(x,t) = \alpha \Psi_1(x,t) + \beta \Psi_2(x,t) [/itex]. This is possible because the set of eigenstates (ψ1,ψ2) are complete and form a basis.
     
  6. Jul 1, 2012 #5

    Simon Bridge

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    Since the LHS is a function of time as well as position while the LHS is position only, not really. However ... lets say we have a set of wavefunctions [itex]\{ \psi_n \}[/itex] which has been selected so that [tex]\mathbf{H}\psi_n = E_n\psi_n[/tex] .... then each [itex]\psi_n[/itex] is said to be an eigenfunction of the Hamiltonian with eigenvalue [itex]E_n[/itex].

    A system prepared in a superpostion state may have wavefunction [tex]\psi = \frac{1}{\sqrt{2}}\left ( \psi_1 + \psi_2\right )[/tex] (assuming each [itex]\psi_n[/itex] are already normalized.) In this case [itex]\psi[/itex] is not an eigenfunction of the Hamiltonian.

    In general, the set of eigenfunctions of an operator can be used as a basis set. Any wavefunction can. Therefore, be represented in terms of a superposition of eigenfunctions ... including eigenfunctions of another operator. (Just in case someone infers that superpositions of eigenfunctions cannot be eigenfunctions.) It is also possible for a wavefunction to, simultaniously, be an eigenfunction of more than one operator.

    Notice how careful I was in the way I phrased things above?
    In QM it is very important to be careful about what exactly is being said about a system ... when you are starting out it is as well to get really pedantic about this.
     
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