What Is the Root of Number Operator in Quantum Mechanics?

[KFC]

I seen in some paper, there is an matrix whose element has a square root of number operator, e.g.

<br /> A = \left(<br /> \begin{matrix}<br /> \alpha &amp; \gamma \sqrt{\hat{a}\hat{a}^\dagger} \\<br /> -\gamma \sqrt{\hat{a}^\dagger\hat{a} &amp; \beta<br /> \end{matrix}<br /> \right)<br />
where \alpha, \beta, \gamma are real number.

What is A^\dagger? Can I write it as the following?
<br /> A^\dagger = \left(<br /> \begin{matrix}<br /> \alpha &amp; -\gamma \sqrt{\hat{a}^\dagger\hat{a}} \\<br /> \gamma \sqrt{\hat{a}\hat{a}^\dagger &amp; \beta<br /> \end{matrix}<br /> \right)<br />

By the way, if I have it operate on any Fock state, how could the operators in the matrix operating those states?

 

[jensa]

The square root of the number operator is probably defined by it's taylor expansion. This means that letting the the root of the number operator act on a Fock-state gives you \sqrt{N} times the state. Also since the number operator is self-adjoint this implies that any function of it will be self-adjoint, in particular (\sqrt{a^\dag a})^\dag=\sqrt{a^\dag a}.

One can think of the number operator as matrices themselves (and thus also the square root of the number operator) so what you have is a 2x2 block-matrix. The hermitian adjoint of which is given by what you wrote.

I don't think that this matrix can act on the Fock-space generated by the algebra of a^dag and a. I could imagine that the matrix never acts on the Fock-space by itself only in the form of some "scalar product". What I mean with this is basically some new operator C:

<br /> C=(a_1, a_2)A\begin{pmatrix}b_1\\ b_2\end{pmatrix} <br />

where a_1/2 and b_1/2 can be numbers or operators. Then C can act on the Fock-space.

Compare it with how one sometimes uses "vectors" of operators like

<br /> c=\begin{pmatrix}<br /> a_1 \\<br /> a_2<br /> \end{pmatrix}, \quad c^\dag=(a_1^\dag, a_2^\dag)<br />

but they never really act on Fock-space in this vector form but only in a "scalar product" form. For example the Hamiltonian may look something like

\mathcal{H}=c^\dag H c

usually the elements of the matrix H are real numbers but in principle they could also be operators.