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.

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

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

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:

$$C=(a_1, a_2)A\begin{pmatrix}b_1\\ b_2\end{pmatrix}$$

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

$$c=\begin{pmatrix} a_1 \\ a_2 \end{pmatrix}, \quad c^\dag=(a_1^\dag, a_2^\dag)$$

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.

 

[Entropia]

I can provide some insights into the concept of a root of number operator and the matrix representation shown in the content.

Firstly, the root of number operator is a mathematical concept used in quantum mechanics to represent the square root of a number operator. In quantum mechanics, operators are used to represent physical observables such as position, momentum, and energy. The number operator, denoted by \hat{N}, is an operator that represents the number of particles in a given quantum state. The square root of this operator, denoted by \sqrt{\hat{N}}, is known as the root of number operator.

In the matrix representation provided, the elements of the matrix contain the root of number operator, which is represented as \sqrt{\hat{a}\hat{a}^\dagger} and \sqrt{\hat{a}^\dagger\hat{a}}. These operators act on quantum states to yield the square root of the number of particles in that state. The real numbers \alpha, \beta, and \gamma are coefficients that determine the strength of the operators in the matrix.

The symbol A^\dagger represents the Hermitian conjugate of the matrix A. In other words, it is the transpose of the complex conjugate of A. This operation is commonly used in quantum mechanics to represent the adjoint of an operator. In the given matrix representation, A^\dagger can be written as the second matrix shown in the content, where the elements are the complex conjugates of the elements in A.

Finally, if the matrix A is operated on a Fock state, which represents a quantum state with a specific number of particles, the operators in the matrix will act on that state to yield the square root of the number of particles. This can be seen by applying the matrix operation on a Fock state and observing the resulting state.

In conclusion, the concept of a root of number operator and its matrix representation are important tools in quantum mechanics for understanding the behavior of quantum systems. The matrix representation provided in the content showcases how these operators can be used to manipulate quantum states and extract information about the number of particles in a given state.