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How to operate on product state?

  1. May 2, 2009 #1

    KFC

    User Avatar

    Suppose a proudct is form by a Fock state [tex]|n\rangle[/tex] and any other state [tex]|x\rangle[/tex], i.e.

    [tex]|\phi\rangle = |n\rangle|x\rangle[/tex]

    If an operator defined as

    [tex]
    A = \left(
    \begin{matrix}
    \alpha\hat{a}\hat{a}^\dagger & \beta\hat{a}^\dagger\hat{a} \\
    \gamma\hat{a}^\dagger\hat{a} & \kappa\hat{a}\hat{a}^\dagger
    \end{matrix}
    \right)
    [/tex]
    where [tex]\hat{a}[/tex] and [tex]\hat{a}^\dagger[/tex] is creation and annilation operator will only opeate on Fock state. So how A operate on [tex]|n\rangle|x\rangle[/tex]?
     
    Last edited: May 2, 2009
  2. jcsd
  3. May 2, 2009 #2

    diazona

    User Avatar
    Homework Helper

    It sounds like you're asking about the states of a spin-1/2 particle in a Fock space (e.g. a harmonic oscillator). In that case, the state [tex]|\chi\rangle[/tex] (what you've written [tex]|x\rangle[/tex]) is actually a spinor, basically a two-component vector:

    [tex]|\chi\rangle = \left(\begin{matrix} a \\ b\end{matrix}\right)[/tex]

    where [tex]\langle\chi |\chi\rangle = 1[/tex]. Then the operator A, a 2x2 matrix, can be applied to that 2-component vector by the usual rules of matrix multiplication.

    [tex]A|\phi\rangle = \left(
    \begin{matrix}
    \alpha\hat{a}\hat{a}^\dagger & \beta\hat{a}^\dagger\hat{a} \\
    \gamma\hat{a}^\dagger\hat{a} & \kappa\hat{a}\hat{a}^\dagger
    \end{matrix}
    \right)|n\rangle\left(\begin{matrix} a \\ b\end{matrix}\right) = \left(
    \begin{matrix}
    (\alpha a\hat{a}\hat{a}^\dagger + \beta b\hat{a}^\dagger\hat{a})|n\rangle \\
    (\gamma a\hat{a}^\dagger\hat{a} + \kappa b\hat{a}\hat{a}^\dagger)|n\rangle
    \end{matrix}\right) = \left(
    \begin{matrix}
    \alpha a\sqrt{n + 1} + \beta b\sqrt{n} \\
    \gamma a\sqrt{n} + \kappa b\sqrt{n + 1}
    \end{matrix}
    \right)|n\rangle[/tex]

    If [tex]|x\rangle[/tex] is not a 2-component vector, then I don't know that there's much you can do with it. You'd probably just "distribute" [tex]|n\rangle[/tex] over all the elements of the matrix A and wind up with something like

    [tex]A|\phi\rangle = \left(
    \begin{matrix}
    \alpha\hat{a}\hat{a}^\dagger|n\rangle & \beta\hat{a}^\dagger\hat{a}|n\rangle \\
    \gamma\hat{a}^\dagger\hat{a}|n\rangle & \kappa\hat{a}\hat{a}^\dagger|n\rangle
    \end{matrix}
    \right)|x\rangle = \left(
    \begin{matrix}
    \alpha\sqrt{n + 1} & \beta\sqrt{n} \\
    \gamma\sqrt{n} & \kappa\sqrt{n + 1}
    \end{matrix}
    \right)|x\rangle[/tex]

    I guess the point is, if the creation and annihilation operators only act on [tex]|n\rangle[/tex], you just leave [tex]|x\rangle[/tex] alone. It's fairly simple.
     
  4. May 2, 2009 #3

    KFC

    User Avatar

    Thank you so much. However, I found it quite confusing to understand the following operation, in above calculation, it seems that you write

    [tex]|x\rangle = \left(\begin{matrix}a \\ b \end{matrix}\right)[/tex]
    and
    [tex]|n\rangle|x\rangle = \left(\begin{matrix}a|n\rangle \\ b|n\rangle \end{matrix}\right)[/tex]

    so

    [tex]
    \left(
    \begin{matrix}
    \alpha\hat{a}\hat{a}^\dagger & \beta\hat{a}^\dagger\hat{a} \\
    \gamma\hat{a}^\dagger\hat{a} & \kappa\hat{a}\hat{a}^\dagger
    \end{matrix}
    \right)|n\rangle\left(\begin{matrix} a \\ b\end{matrix}\right) = \left(
    \begin{matrix}
    (\alpha a\hat{a}\hat{a}^\dagger + \beta b\hat{a}^\dagger\hat{a})|n\rangle \\
    (\gamma a\hat{a}^\dagger\hat{a} + \kappa b\hat{a}\hat{a}^\dagger)|n\rangle
    \end{matrix}\right)
    [/tex]

    But I readed a text, in which the product state of two states is given by
    [tex]
    |n\rangle|x\rangle = |n\rangle \otimes |x\rangle =
    \left(\begin{matrix}\alpha|x\rangle \\ \beta|x\rangle \end{matrix}\right)
    [/tex]
    where [tex]|n\rangle = \left(\begin{matrix}\alpha \\ \beta\end{matrix}\right)[/tex]

    I wonder if
    [tex]\left(\begin{matrix}\alpha|x\rangle \\ \beta|x\rangle \end{matrix}\right) \equiv
    \left(\begin{matrix}a|n\rangle \\ b|n\rangle \end{matrix}\right)
    [/tex]

    ????
     
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