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Are state vectors and wavefunctions the same?

  1. Oct 1, 2005 #1
    Hi,
    State vectors ("kets") live in Hilbert space. Do wavefunctions also live in Hilbert space? I've read that they both do, but how can functions and vectors reside in the same space? Or do wave functions simply map from coordinate space to Hilbert space?
    Thanks
     
  2. jcsd
  3. Oct 1, 2005 #2

    Hurkyl

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    Any collection of things can form a vector space, as long as the axioms of a vector space are satisfied.
     
  4. Oct 1, 2005 #3
    Hello emob2p,

    I just found in wikipedia
    http://en.wikipedia.org/wiki/Wave_mechanics#The_wavefunction

    "We define the wavefunction as the projection of the state vector |ψ(t)> onto the position basis:
    [tex]\Psi(r,t) = \langle r|t \rangle[/tex]"

    I think the difference, as mentioned above, is that a state vector (in Dirac representation) looks like this: |Psi> , |a> , |b> ......, it has so to say no representation, only this strange "|blabla>" one. This |Psi> contains all the information you can have about a physical system.

    But when you talk of a wavefunction, you have a certain representation, for example: space, then you have [tex]\Psi (x) = \langle x|\Psi (x) \rangle[/tex], or in momentum representation, you have [tex]\Psi (p) = \langle p|\Psi (x) \rangle[/tex].

    Mathematicians don't like the Dirac notation ([tex]|\Psi \rangle[/tex]), if I remember correctly. They only have the wavefunctions ([tex]\Psi (x)[/tex], but not the state vectors alone.
    For them, the state vector alone makes no sense (correct me if I'm wrong), so they don't use this term "state vector".

    A second interpretation could be that mathematicians do use the term "state vectors" for the wavefunctions. The reason would be that wavefunctions are elements of a vector space, and mathematicians call all the elements of V vectors.

    Maybe a mathematician could tell you more about this and the rigorous definitions.
     
    Last edited: Oct 1, 2005
  5. Oct 1, 2005 #4

    Hurkyl

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    Any collection of things can form a vector space, as long as the axioms of a vector space are satisfied. :biggrin:

    You can form a Hilbert space from the set of all square-integrable functions on the real line. When working with that Hilbert space, you would call a square-integrable function of the real line a vector. If you take that Hilbert space to be your state space, then you would call a square-integrable function of the real line a state vector.
     
  6. Oct 1, 2005 #5

    wrt vectors and functions both reside in the same space its important to know that functions and vectors are isomorphic, that is, can be treated the same mathematically. The set of polynomials all form an orthogonal basis, that is x is orthogonal to [tex]x^2[/tex] which is orthogonal to [tex]x^3[/tex] etc... (there is some normalization coeficients I've missed off but you get the picture). You can look this up more - Gram Schmidt Orthogonalization and I think Legendre Polynomials.
     
  7. Oct 2, 2005 #6
    The key benefit of Dirac notation is that it is basis independent, which makes it more general in a sense. By writing down a wave function you have implicitly chosen a basis (the position representation, in most textbooks).
     
  8. Oct 3, 2005 #7

    dextercioby

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    There's something simple here. Or at least it seems simple. The "wavefunction" is a term one uses to name the mathematical objects (either functions,functionals or vectors, "objects" in general) which fully describe pure states in the Schrödinger representation (so-called "wave mechanics") of the Born-Jordan CCR. While "state vector" is the vector in the Hilbert space (or the linear functional continuous on the nuclear subspace associated to an unbounded selfadjoint densly defined linear operator) which "pinpoints" the (unit) ray which describes the pure quantum state in the fundamental formulation on (nonrelativistic) quantum mechanics.

    Daniel.
     
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