How to operate on product state?

  • Thread starter KFC
  • Start date
  • #1
KFC
488
4

Main Question or Discussion Point

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:

Answers and Replies

  • #2
diazona
Homework Helper
2,175
6
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.
 
  • #3
KFC
488
4
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]
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]

????
 
Top