How to operate on product state?

[KFC]

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

$$|\phi\rangle = |n\rangle|x\rangle$$

If an operator defined as

$$A = \left(<br /> \begin{matrix}<br /> \alpha\hat{a}\hat{a}^\dagger & \beta\hat{a}^\dagger\hat{a} \\<br /> \gamma\hat{a}^\dagger\hat{a} & \kappa\hat{a}\hat{a}^\dagger<br /> \end{matrix}<br /> \right)$$
where $$\hat{a}$$ and $$\hat{a}^\dagger$$ is creation and annilation operator will only opeate on Fock state. So how A operate on $$|n\rangle|x\rangle$$?

 

[diazona]

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 $$|\chi\rangle$$ (what you've written $$|x\rangle$$) is actually a spinor, basically a two-component vector:

$$|\chi\rangle = \left(\begin{matrix} a \\ b\end{matrix}\right)$$

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

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

If $$|x\rangle$$ 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" $$|n\rangle$$ over all the elements of the matrix A and wind up with something like

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

I guess the point is, if the creation and annihilation operators only act on $$|n\rangle$$, you just leave $$|x\rangle$$ alone. It's fairly simple.

 

[KFC]

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 $$|\chi\rangle$$ (what you've written $$|x\rangle$$) is actually a spinor, basically a two-component vector:

$$|\chi\rangle = \left(\begin{matrix} a \\ b\end{matrix}\right)$$

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

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

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

$$|x\rangle = \left(\begin{matrix}a \\ b \end{matrix}\right)$$
and
$$|n\rangle|x\rangle = \left(\begin{matrix}a|n\rangle \\ b|n\rangle \end{matrix}\right)$$

so

$$\left(<br /> \begin{matrix}<br /> \alpha\hat{a}\hat{a}^\dagger & \beta\hat{a}^\dagger\hat{a} \\<br /> \gamma\hat{a}^\dagger\hat{a} & \kappa\hat{a}\hat{a}^\dagger<br /> \end{matrix}<br /> \right)|n\rangle\left(\begin{matrix} a \\ b\end{matrix}\right) = \left(<br /> \begin{matrix}<br /> (\alpha a\hat{a}\hat{a}^\dagger + \beta b\hat{a}^\dagger\hat{a})|n\rangle \\<br /> (\gamma a\hat{a}^\dagger\hat{a} + \kappa b\hat{a}\hat{a}^\dagger)|n\rangle<br /> \end{matrix}\right)$$

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

I wonder if
$$\left(\begin{matrix}\alpha|x\rangle \\ \beta|x\rangle \end{matrix}\right) \equiv<br /> \left(\begin{matrix}a|n\rangle \\ b|n\rangle \end{matrix}\right)$$

?