Classical notion

Let us consider a random variable $x$ and a classical probability density function $\rho(x)$,

\[ \rho(x) \leq 0,\quad \int_\Omega \rho(x) dx = 1. \]

The $n$-th moment of $x$ is defined as

\[ \langle x^n\rangle = \int dx\ x^n \rho(x) \]

Claim. If all moment $n$ of $x$ is known then $\rho(x)$ is completely known.

This claim can be justified by the introduction of the characteristic function $C(k)$,

\[ C(k) = \langle e^{ikx} \rangle = \int dx\ e^{ikx} \rho(x) = \sum_{n=0}^{\infty} \dfrac{(ik)^n}{n!} \langle x^n \rangle. \]

Then, the probability density function is the Fourier transform of $C(k)$,

\[ \rho(x) = \int dk\ e^{-ikx} C(k) \]

Reversibly, if we are given the characteristic function $C(k)$, we can also calculate the moments,

\[ \langle x^n \rangle = \dfrac{1}{i^n} \dfrac{d^n C(k)}{dk^n}\bigg\vert_{k=0} \]

Quantum mechanical version

Let us now introduce the characteristic function in the quantum case, $C(\lambda, p)$. Other than $\lambda$, we introduce one more parameter $p$ for convenience. The function $C(\lambda, p)$ is called the $p$-ordered characteristic function,

\[ C(\lambda, p) = \text{tr}\left[\rho\exp\left(\lambda a^\dagger - \lambda^* a\right)\right]\exp\left(\dfrac{p}{2}|\lambda|^2\right) \]

Each value of the parameter $p$ corresponds to a specific ordering of the creation and annihilation operators. In particular, the values $p=+1, 0, -1$ corresponds to normal, symmetric, and antinormal ordered characteristic functions. The $p=1$ case is also known as the Wigner $C(\lambda, 1)$.

The number operator

The number operator $n$ is the product of two non-commuting operators. This definition brings about the prescription for ordering these two operators $a, a^\dagger$ so that $n=a^\dagger a \neq a a^\dagger$.

All functions of non-commuting operators require a specification of the operator ordering. For functions of $a$ and $a^\dagger$ there are three special orderings commonly encountered (guess what?): normal, symmetric, and antinormal ordering.

1. Normal ordering: the operators are positioned so that no annihilation operator $a$ ever appears to the left of any creation operator $a^\dagger$.
2. Anti-normal ordering: reverse the above definition.
3. Symmetric ordering, denoted $S(\cdot)$, is the average of all possible orderings of $a$ and $a^\dagger$.

An example for the first two symmetric ordered powers of the number operator,

\[\begin{align} S(n) &= \dfrac{1}{2}\left(a^\dagger a + a a^\dagger\right)\newline S(n) &= \dfrac{1}{6}\left(a^{\dagger 2}a^2 + a^\dagger a^2 a^\dagger + a^\dagger a a^\dagger a+ a^{2}a^{\dagger 2}+ aa^\dagger a a^\dagger + a a^\dagger a a\right). \end{align}\]