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:

• For the kernel function
  

  $$f^{\rm W}(\xi,\eta) \; := \; 1$$ (A.10)

  
  
  one obtains from equation (A.13), integrating over $\xi$ and $x'$ and substituting $x'':=\hbar\eta/2$, the WIGNER distribution function [Wig32]:
  

  $$F^{\rm W}(x,p,t) \; = \; \frac{1}{\pi\hbar} \int\!\! {\mbox{... ...t. % e^{-2ipx''/\hbar}. e^{\textstyle -\frac{2ipx''}{\hbar}}.$$ (A.11)

  
  
  It corresponds to the WEYL ordering of operators [Wey31,DGS72],
  

  $$f^{\rm W}(\xi,\eta)e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \; = ... ...}}+\eta{\hat{p}})} \;\; \widehat{=} \;\; e^{i(\xi x+\eta p)},$$ (A.12)

  
  
  and by equations (A.7, A.8) gives rise to the WEYL transform of the operator ${\hat{A}}$:
  

  $$A^{\rm W}(x,p) \; = \; \hbar \int\!\! {\mbox{d}}\eta\; \big... ...rt x-{\textstyle \frac{1}{2}}\hbar\eta \big> \: e^{-i\eta p}.$$ (A.13)

  
  
  The last two terms of expression (A.16) indicate the association of operators and scalars that is defined by $f^{\rm W}$, as discussed in section A.1.

  Using the WIGNER distribution function, the overlap of two states $\left\vert \psi_1 \right>$ and $\left\vert \psi_2 \right>$ described by the WIGNER functions $F_1^{\rm W}$ and $F_2^{\rm W}$ can be computed as the overlap of the corresponding WIGNER functions in phase space:
  

  $$\big\vert \left< \psi_1 \left\vert \psi_2 \right> \right. \... ...!\! {\mbox{d}}p\; ä F_1^{\rm W}(x,p,t) \, F_2^{\rm W}(x,p,t).$$ (A.14)

  
  

  Besides the HUSIMI distribution function (which is introduced in section A.3 below), the WIGNER function is the most commonly used quantum phase space distribution function.

• The kernel function
  

  $$f^{\rm S}(\xi,\eta) \; := \; e^{-\frac{1}{2}i\hbar\xi\eta}$$ (A.15)

  
  
  defines the standard ordering of operators; it is characterized by all ${\hat{x}}$-dependent terms preceding the ${\hat{p}}$-dependent terms,
  

  $$f^{\rm S}(\xi,\eta)e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \; = ... ...e^{i\eta{\hat{p}}} \;\; \widehat{=} \;\; e^{i(\xi x+\eta p)},$$ (A.16)

  
  
  which leads to the standard-ordered distribution function^A.4:
  

  $$F^{\rm S}(x,p,t) \; = \; \frac{1}{\sqrt{2\pi\hbar}} \left< ... ...t> \right. % e^{ipx/\hbar}. e^{\textstyle \frac{ipx}{\hbar}}.$$ (A.17)

  
  
  Essentially, $F^{\rm S}(x,p,t)$ is the product of the position and momentum representations of the state $\left\vert \psi(t) \right>$.

• Setting
  

  $$f^{\rm AS}(\xi,\eta) \; := \; \frac{1}{f^{\rm S}(\xi,\eta)}$$ (A.18)

  
  
  and thus having all ${\hat{p}}$-dependent terms precede the ${\hat{x}}$-dependent terms, one gets the antistandard-ordered distribution function (also known as the KIRKWOOD distribution function [Kir33] or RIHACZEK distribution function [Rih68]):
  

  $$F^{\rm AS}(x,p,t) \; = \; \frac{1}{\sqrt{2\pi\hbar}} \left<... ... \right. % e^{-ipx/\hbar}. e^{\textstyle -\frac{ipx}{\hbar}}.$$ (A.19)

  
  
  It is obtained from its standard-ordered counterpart $F^{\rm S}(x,p,t)$ by complex conjugation.

• For systems that can be described as a harmonic oscillator with mass $m_0$ and frequency $\omega_0$ -- in this appendix no scaling as in section 2.1 is employed, such that the parameters $m_0$ and $\omega_0$ are retained in the formulae -- or as an ensemble of harmonic oscillators, the normal-ordered and the antinormal-ordered distribution functions $F^{\rm N}(x,p,t)$ and $F^{\rm AN}(x,p,t)$ are useful.^A.5They are defined by requiring operators to be (anti-) standard-ordered not with respect to ${\hat{x}}$, ${\hat{p}}$, but with respect to the ladder operators, i.e. with respect to the creation operator ${\hat{a}}^\dagger$ and the annihilation operator ${\hat{a}}$, where ${\hat{a}}$ is given by
  

  $${\hat{a}}\; := \; \frac{1}{\sqrt{2\hbar m_0\omega_0}}\left(m_0\omega_0{\hat{x}}+i{\hat{p}}\right),$$ (A.20)

  
  
  as usual. (Equation (2.30) is obtained from this definition in the case of the scaling (1.15, 2.4), i.e. by formally setting $m_0=\omega_0=1$.)

  This (anti-) standard ordering of operators is achieved by using the kernels
  
  as can easily be confirmed by direct computation:^A.6
  
  where the complex parameter $v$ is defined as
  

  $$v(\xi,\eta) \; := \; \sqrt{\frac{\hbar}{2m_0\omega_0}} \left( -m_0\omega_0\eta+i\xi \right).$$ (A.19)

  
  
  The corresponding distribution functions are
  
  

  The normal-ordered distribution $F^{\rm N}$ is also called GLAUBER-SUDARSHAN distribution function or $P$-function [Gla63b,Gla65,Sud63], while the antinormal-ordered distribution $F^{\rm AN}$ is sometimes referred to as the $Q$-function [Gla65].

  Defining $\left\vert x',p' \right>$ as a Gaussian wave packet centered at $(x',p')^t$ in phase space,^A.7
  

  $$\left< x \left\vert x',p' \right> \right. \; = \; \sqrt[4]{\... ...0\omega_0}{2\hbar}(x-x')^2} e^{\textstyle \frac{ip'x}{\hbar}}$$ (A.19)

  
  
  with $x',p'\in\mathbb{R}$, the antinormal-ordered distribution function can be written in a particularly concise way:
  

  $$F^{\rm AN}(x,p,t) \; = \; \frac{1}{2\pi\hbar} \; \big\vert \left< x,p \left\vert \psi(t) \right> \right. \big\vert^2,$$ (A.20)

  
  
  as is easily verified by substituting formula (A.29) into equation (A.30) and comparing the result with the definition (A.13) for $f(\xi,\eta)=f^{\rm AN}(\xi,\eta)$.

  Therefore, $F^{\rm AN}$ is essentially obtained by computing the convolution of the state $\left\vert \psi(t) \right>$ with a Gaussian in position space. The physical meaning of this convolution becomes clearer in section A.5. From equation (A.30) it is also clear that $F^{\rm AN}$ is non-negative; this is of importance for the interpretation of $F^{\rm AN}$ as a (quasi-) probability distribution function in section A.5. $F^{\rm AN}$ in the form of equation (A.30) finds its main application in the discussion of the HUSIMI distribution function in section A.3.

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:

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

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

and obtain for the expectation value in the integrand:

$$\left< f^{\rm AN}(\xi,\eta) \, e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \right>_t = f^{\rm AN}(\xi,\eta) \left< f^{\rm W} (\xi,\eta) \, e^{i(\xi{\hat{x}}+\eta{\hat{p}})} \right>_t$$

$$= f^{\rm AN}(\xi,\eta) \int\!\! {\mbox{d}}x'\! \int\!\! {\mbox{d}}p'\; e^{i(\xi x'+\eta p')} F^{\rm W}(x',p',t);$$ (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].

Footnotes

... function^A.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.