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Special Distribution Functions

In the same way in which the kernel $f(\xi,\eta)$ determines the association between operators and scalars, it also defines an operator ordering [Meh77,AM77]. In this context an expansion (A.7) of an operator is called ordered if it is written as a superposition of terms that are of the form $f(\xi,\eta)e^{i(\xi{\hat{x}}+\eta{\hat{p}})}$. The essential point of this definition is that the multiplication of the exponential $e^{i(\xi{\hat{x}}+\eta{\hat{p}})}$ with the kernel function yields a characteristic composition of exponential operators, the explicit form of which depends on the particular choice of $f$.

The implications of this definition become clearer when some specific examples are considered:

Since there are infinitely many different functions $f(\xi,\eta)$ that can be chosen as kernel functions, there exist equally many different distribution functions $F^f(x,p,t)$. The important point is that all these different $F^f$ are equivalent: each of them contains the same information about the state $\left\vert \psi(t) \right>$, and each $F^f$ can be used to compute the expectation values (A.9a) of any operator ${\hat{A}}({\hat{x}},{\hat{p}})$ that is expanded as in equation (A.7).

The connection between the WIGNER distribution function and the antinormal-ordered distribution function is an important example:

F^{\rm AN}(x,p,t) \; = \;
\frac{1}{\pi\hbar} \int\!\! {\mbo...
...le \frac{1}{\hbar m_0\omega_0}}(p-p')^2 }
F^{\rm W}(x',p',t).
\end{displaymath} (A.21)

This unveils the antinormal-ordered distribution function as the convolution of the WIGNER function with a Gaussian in phase space; I discuss this fact in some more detail in section A.5.

Formula (A.31) may be proved in the following way: using similar arguments as those leading to equation (A.13), with the definition (A.6) I first rewrite $F^{\rm AN}(x,p,t)$ as

F^{\rm AN}(x,p,t) \; = \;
\frac{1}{4\pi^2} \int\!\! {\mbox...
...xi{\hat{x}}+\eta{\hat{p}})} \right>_t \;
e^{-i(\xi x+\eta p)}
\end{displaymath} (A.22)

and obtain for the expectation value in the integrand:
$\displaystyle \left< f^{\rm AN}(\xi,\eta) \, e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \right>_t$ $\textstyle =$ $\displaystyle f^{\rm AN}(\xi,\eta)
\left< f^{\rm W} (\xi,\eta) \, e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \right>_t$  
  $\textstyle =$ $\displaystyle f^{\rm AN}(\xi,\eta)
\int\!\! {\mbox{d}}x'\! \int\!\! {\mbox{d}}p'\; e^{i(\xi x'+\eta p')}
F^{\rm W}(x',p',t);$ (A.23)

equation (A.31) then follows by integration over $\xi$ and $\eta$.

Other conversion formulae between arbitrary different distribution functions $F^{f_1}(x,p,t)$ and $F^{f_2}(x,p,t)$ with $f_1\neq f_2$ can be obtained by similar computations. Some formulae of this type are listed in [Lee95].


... functionA.4
This slightly inaccurate naming convention is common practice. In order to avoid misunderstandings I want to stress here that in the narrow sense it is only operators that are (or are not) standard-ordered. For distribution functions (standard) ordering is not defined at all. Therefore, the standard-ordered distribution is not standard-ordered. Similar statements hold for all the other orderings of operators and their associated distribution functions.
... useful.A.5
Typical applications may be found in quantum optics; see [Vou94,KH95b] for some examples. Another field where $F^{\rm N}$ and $F^{\rm AN}$ are utilized frequently is the modelling of a heat bath by an ensemble of harmonic oscillators [Coh94,HB95,HB96]. Cf. also the monographs by Louisell [Lou64,Lou73] and Dineykhan et al. [DEGN95].
... computation:A.6
For more information on (anti-) normal ordering of operators see [DEGN95].
... space,A.7
A Gaussian wave packet like $\left\vert q',p' \right>$ is often referred to as a coherent state. Such a state is characterized by $\left< {\hat{x}} \right>=x'$, $\left< {\hat{p}} \right>=p'$, $\Delta{\hat{x}}=\sqrt{\hbar/(2m_0\omega_0)}$ and $\Delta{\hat{p}}=\sqrt{\hbar m_0\omega_0/2}$, such that $\left\vert q',p' \right>$ is a state with minimum uncertainty product: $(\Delta{\hat{x}})(\Delta{\hat{p}})=\hbar/2$. In section A.3 I discuss coherent states from a more general point of view and an important generalization of this concept.

next up previous contents
Next: Minimum Uncertainty States and Up: Quantum Phase Space Distribution Previous: Definition of Quantum Phase   Contents
Martin Engel 2004-01-01