ABSTRACT
The function G, with σ σq p ≥ /2, is thus physically measurable through simultaneous measurements of position and momentum.
14.3 Rule of ordering It should be emphasized that, at least in principle, the function G is as good a quantum phase-space distribution function as the Husimi function or the Wigner function. The expectation value of any arbitrary operator can be calculated using the function G as well as the Husimi function or the Wigner function. Only the rule of ordering of noncommuting operators is different. In order to nd the rule of ordering associated with the function G, we begin with the equation (Lee 1995)
Tr{ ( , )} ( , )ρe f dp dpe G q pi q i p i q i pξ η ξ ηξ η+ += ∫ ∫ (14.5) It can be shown that the function f(ξ, η) that determines the rule of ordering for the
function G is given by
f e q p( , ) /ξ η σ ξ σ η= − −2 2 2 22 2/ (14.6)
At this point, we nd it convenient to introduce two parameters κ and s dened as
κ
σ
σ =
q (14.7)
and
s q p q= − = −
σ σ κσ
/ /2 2
(14.8)
The parameter к has a dimension of mω [mass/time]. The parameter s is real and negative, and its absolute value measures the product of the widths σq and σp associated with the function G being considered with respect to that of the minimum uncertainty Gaussian wave packet. Once σq and σp are given, к and s are determined, and vice versa. Equation 14.6 can be rewritten, in terms of κ and s, as
We further introduce dimensionless parameters v and β and an operator bˆ as
v i= −ξ
κ η κ
2 2 (14.10)
β
κ
κ = +
2 2 q
i p
(14.11)
ˆ ˆ ˆb q
i p= +
κ
κ2 2 (14.12)
Equation 14.5 with f (ξ, η) given by Equation 14.6 can then be rewritten as
The rule of ordering for the function G can now be determined from Equation 14.13 using the same method that Cahill and Glauber (1969) adopted for their s-parameterized distribution function. The nal result is
{ } ! ,
( )b k n
k
m
k s
b bbn m
=
− −
=
0 2
1 2
(14.14)
where { }b bn m † represents the rule of ordering for the function G, the symbol (n, m) denotes
the smaller of the two integers n and m, and n
k
is a binomial coefficient. Equation 14.14
yields, for example, { }b b† †= , { }b b= , { } ( )( )b b bb s † †= − +1 2 1/ , { ( )}b b b b s b † †2 2 1= − + ,
{ } ( )b b bb s b † † †2 2 1= − + , and { ( ) ( )( )}b b b b s bb s † † †2 2 2 2 22 1 1 2 1= − + + +/ . As an illustration, let us nd the expectation value of ˆ ˆqp2 with the function G(q, p).
