The first point to note when it comes to the practical application of
phase space distribution functions is that their numerical computation for
a given quantum state
does not cost as much effort as it
might seem at first sight of equations like (A.13).
For example, the WIGNER function can be written as
and for the HUSIMI distribution one has
These expressions show that both and can be computed efficiently from by FOURIER transformation, and the same is true for other as well. Typically the method of fast FOURIER transformation (FFT) [CT65,EMR93] is employed here which drastically reduces the numerical effort, as compared with conventional methods of computation.A.14
In the literature, initially mainly the WIGNER function has been used. But in the last years the situation has changed in favour of the HUSIMI function which is in widespread use by now. In the following I want to motivate this transition with two arguments, the first being analytical while the second is a numerical example of a typical application.
In the context of HUSIMI functions, equation (A.31) can be
The numerical example below shows that the rapid oscillations of the WIGNER function have no obvious counterpart in the classical phase portrait, whereas the phase space patterns of the HUSIMI function follow the classical structures much more closely. In this sense the smoother HUSIMI function can be interpreted as a (quasi-) probability density in a much more intuitive way than the WIGNER function. This property makes the first choice distribution function for the purpose of visualization of quantum nonlinear dynamical systems, especially when the classical dynamics is chaotic.
argument explains why the normal-ordered distribution function
is utilized rarely and why the anti-HUSIMI distribution function is
not used at all for
creating an intuitive picture of a quantum state.
(A.89) one has
Finally there exists a strong argument in favour of the HUSIMI distribution from the experimenters' point of view: TAKAHASHI and
SAITÔ argue that is much better suited to any
experimental setup than since the averaging in equation
(A.89) acts in a similar way as the coarse graining effect that
is inherent to all experimental measurement processes
In a particular experimental setup, on the other hand, the WIGNER function still has the striking advantage that it can be measured
at each point of phase space,
without the need for any further complex
computation: in [MCK93] it is shown that, after some
algebra, can be written as
At this point one is now in the position to further interpret the additional ``squeezing'' parameter of the (anti-) HUSIMI distribution. Equations (A.89, A.91) show that controls the way in which the averaging discussed above is performed, namely by specifying the width of the Gaussian (A.90) in - and -direction. The effective area over which is averaged is an ellipse with widths and in - and -direction, respectively -- compare these values with the uncertainties (A.64) of general squeezed states. Therefore, larger values of yield an averaging area that is squeezed in the direction of and broadened in the direction of . The particular choice , which is used in many applications and corresponds to the coherent state representation (A.51), accordingly specifies a circular averaging area.
In order to exemplify the above statements, in figures A.1-A.3 I compare the most important ways to graphically represent quantum states. The states used for this demonstration are the familiar ground state of the harmonic oscillator and its tenth eigenstate .
Figure A.1a shows a classical phase space trajectory of the harmonic oscillator at the energy . The corresponding quantum mechanical wave function in the position representation (figure A.1b) bears no obvious resemblance to the classical trajectory and is therefore of little use for the purpose of comparing the classical and quantum states. For this purpose, figure A.1c is much better suited: the contour plot of the WIGNER function of that is shown here is essentially a bell-shaped function that is peaked at the origin of phase space -- near the classical trajectory. The contour plots of the HUSIMI distributions for different values of in figures A.1d-A.1f are not much more informative; all of them show bell-shaped curves centered about the origin. Note that the HUSIMI function for is the one that is closest to the classical trajectory, since for one obtains distributions that do not reproduce the classical rotational symmetry of the trajectory.
While the advantages of the HUSIMI distribution are not obvious with respect to the simple ground state they become clearly visible when more complex states are considered. Figure A.2 shows the same kind of plots as figure A.1 but now for the state . The highly oscillatory character of the WIGNER function is illustrated in the contour plot A.2c and in its three-dimensional counterpart in figure A.3a. These plots do not show much similarity to the corresponding classical path in figure A.2a.
Only when turning to the HUSIMI distribution in figures A.2d-A.2f -- and in figures A.3b, A.3c with the 3D versions -- (some aspects of) the classical trajectory can be identified in quantum phase space. This is best achieved for : essentially is a circular ridge which takes on its maximum value at the points of the classical trajectory. For and rotational invariance is lost and two characteristic peaks emerge with no obvious counterpart in the classical picture. But even in these cases the location of the classical path can clearly be identified.