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The HUSIMI Distribution Function

With the results of the previous subsections it is now but a small step to the definition of the HUSIMI distribution function. Replacing in equation (A.27) the frequency $\omega_0$ of the harmonic oscillator with an arbitrary $\kappa\in\mathbb{R}$ (the interrelation of which with squeezed states is illustrated by equation (A.69)), a complex number $w$ is defined:

% w := \sqrt{\frac{\hbar}{2m\kappa}} \left( -m\kappa\eta+i\...
...{\frac{\hbar}{2m_0\kappa}} \left( -m_0\kappa\eta+i\xi \right).
\end{displaymath} (A.44)

At this point the discussion has moved quite a distance away from the original harmonic oscillator: direct contact with it was lost when the generalized ladder operators $\b$, $\b^\dagger$ were introduced in subsection A.3.2 in order to replace the original ${\hat{a}}$, ${\hat{a}}^\dagger$ in an ad hoc fashion; the oscillator frequency $\omega_0$ has been substituted by the abstract quantity $\kappa$, and the only remaining original parameter of the oscillator is its mass $m_0$. Therefore it is reasonable to leave the original oscillator terminology behind altogether and define the new real parameter

\zeta \; := \; m_0\kappa
\end{displaymath} (A.45)

for the sake of simplicity, as $m_0$ and $\kappa$ enter the equations (A.68, A.70) for $\b$ and $w$ via this product only. For reasons which are explained in section A.6 and motivated by equations (A.64), $\zeta$ is called the squeezing parameter. One obtains:
\b\,(\zeta) \hspace*{0.8cm}
& = & \frac{1}...
...\hbar}{2\zeta}} \left( -\zeta\eta+i\xi \right).

With these $\b$ and $w$ replacing ${\hat{a}}$ and $v$ in equation (A.28b), one finally arrives at the ``celebrated'' HUSIMI distribution function [Hus40]:

F^{\rm H}(x,p,t;\zeta) \; = \;
\int\!\! ...
...b}e^{w\b^\dagger} \big\vert x' \big> \;
e^{-i(\xi x+\eta p)}.


$F^{\rm H}(x,p,t;\zeta)$ depends on the single parameter $\zeta$ only, and it is formally -- after the above exchange of operators and parameters -- identical with the antinormal-ordered distribution function (A.28b) which has two parameters, namely $m_0$ and $\omega_0$. As in the case of $F^{\rm AN}$, usually not the ``complicated'' formula (A.73) is used but its more concise version
F^{\rm H}(x,p,t;\zeta) \; = \; \frac{1}{2\pi\hbar} \;
...\!\left< \beta \left\vert \psi(t) \right> \right.
\end{displaymath} (A.46)

\beta \; = \; \beta(x,p;\zeta)
\; =: \; \frac{1}{\sqrt{2\hbar\zeta}} \left( \zeta x+ip \right),
\end{displaymath} (A.47)

which is completely analogous to equations (A.51, A.38). For the proof of equation (A.74) one uses the representation (A.62) of the squeezed states $\left\vert \beta \right>_{\rm s}$ and follows the same steps as in the proof of equation (A.51).

For this calculation, which is an application of the formalism outlined in section A.1, the kernel function $f^{\rm H}(\xi,\eta;\zeta)$ defining $F^{\rm H}(x,p,t;\zeta)$ is needed. It is easy to see that this kernel function is

f^{\rm H}(\xi,\eta;\zeta) \; := \; \exp
\fr...\xi^2}{4\zeta}+\frac{\hbar \zeta\eta^2}{4}
\end{displaymath} (A.48)

since with this $f^{\rm H}$ the general definition (A.13) of distribution functions in fact yields equation (A.73).

For the practical evaluation of the formula (A.74) for the HUSIMI function normally the position representation of the squeezed state $\left\vert \beta \right>_{\rm s}$,

\left< x' \left\vert \beta \right> \right._{\rm s}
\; = \;...
e^{-i{\textstyle \frac{xp}{2\hbar}}} \; ,
\end{displaymath} (A.49)

is used which is analogous to equation (A.50).

In line with the formalism of section A.1, as a byproduct the definition (A.76) in a natural way leads to the introduction of yet another distribution function which is the counterpart of the HUSIMI function in the same way in which the antinormal-ordered distribution function is the counterpart of the normal-ordered distribution: with the definition

f^{\rm AH}(\xi,\eta;\zeta) \; := \; \frac{1}{f^{\rm H}(\xi,\eta;\zeta)}
\end{displaymath} (A.50)

an ``anti-HUSIMI distribution function'' $F^{\rm AH}(x,p,t;\zeta)$ is obtained that -- to the best of my knowledge -- has not yet been mentioned in the literature (although LEE comes close to $F^{\rm AH}$ when he defines the ``anti-HUSIMI transform'' of an operator [Lee95]).

Finally, it remains to specify the operator orderings themselves that are associated with these distribution functions. Not surprisingly it turns out that the anti-HUSIMI and the HUSIMI distribution functions correspond to standard ordering and anti-standard ordering with respect to the generalized ladder operators $\b^\dagger$, $\b$, respectively:
f^{\rm AH}(\xi,\eta;\zeta) \, e^{i(\xi{\h...
& = & e^{-w^*\b}e^{ w\b^\dagger}.

In section A.5 I discuss in some more detail the significance of the parameter $\zeta$ and how this parameter can be used in the course of the analysis of a physical state $\left\vert \psi \right>$.

next up previous contents
Next: Dynamics Up: Minimum Uncertainty States and Previous: Squeezed States   Contents
Martin Engel 2004-01-01