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Since $F^f(x,p,t)$, for each $f$, contains exactly the same information as the HILBERT space vector $\left\vert \psi(t) \right>$ it is possible to give a complete formulation of quantum mechanics by means of phase space distribution functions only, that is without referring to quantum state vectors or wave functions. In such a framework $F^f(x,p,t)$ describes the state of the system at time $t$, whereas the time evolution of $F^f(x,p,t)$ is determined by the equation of motion

$\displaystyle \frac{\partial F^f(x,p,t)}{\partial t}$ $\textstyle =$ $\displaystyle \displaystyle
\frac{2}{\hbar} \;
\frac{ \displaystyle f \left( -i...
...rac{\partial}{\partial x_1},
-i\frac{\partial}{\partial p_1} \right) } \;\times$  
    $\displaystyle \times\;
\sin\left\{ \frac{\hbar}{2}
\left( \frac{\partial}{\part...
...\partial}{\partial x_2}
\frac{\partial}{\partial p_1}
\right) \right\} \;\times$  
    $\displaystyle \times\;
\tilde{H}^f(x_1,p_1) \; F^f(x_2,p_2,t)
\, \bigg\vert _{\, x_1=x_2=x,\; p_1=p_2=p \;\;\displaystyle ,}$ (A.50)

which takes the role normally occupied by the SCHRÖDINGER equation in the conventional formulation of quantum mechanics. Here, $\tilde{H}^f(x,p)$ is the $f$ transform of the Hamiltonian; it is obtained by first ordering the Hamiltonian $\H({\hat{x}},{\hat{p}})$ according to the kernel function $1/f(-\xi,-\eta)$, as described in section A.1, and then replacing the operators ${\hat{x}}$, ${\hat{p}}$ with the scalars $x$, $p$. COHEN was the first to formulate the equation of motion (A.80) [Coh66]; a derivation starting from the VON NEUMANN equation can be found in [Lee95].

As an example, for the special case of the WIGNER distribution function equation (A.80) becomes

$\displaystyle \frac{\partial F^{\rm W}(x,p,t)}{\partial t}$ $\textstyle =$ $\displaystyle \displaystyle \frac{2}{\hbar}
\sin \left\{ \frac{\hbar}{2}
...\partial}{\partial x_2}
\frac{\partial}{\partial p_1}
\right) \right\} \;\times$  
    $\displaystyle \displaystyle \times\; \tilde{H}^{\rm W}(x_1,p_1) \; F^{\rm W}(x_2,p_2,t)
\, \bigg\vert _{\, x_1=x_2=x, \; p_1=p_2=p \;\;\displaystyle .}$ (A.51)

Obviously, the equations of motion (A.80) and (A.81) are quite awkward to work with. This is the most important reason why in practical applications quantum mechanics hardly ever is discussed with distribution functions completely replacing quantum states and wave functions. Rather, even when the final task is to obtain distribution functions, typically the well-established version of quantum mechanics is used, i.e. one starts by solving -- numerically, if necessary -- the SCHRÖDINGER equation and then inserts the solution $\left\vert \psi(t) \right>$ into equation (A.13) or (A.74), for instance, in order to compute the desired $F^f(x,p,t)$.

But in addition to the computation of distribution functions, the equation of motion for WIGNER's function is of importance for the investigation of the semiclassical approximation with $\hbar\approx 0$. For Hamiltonians of the type $H(x,p)=p^2/2m_0+V(x)$ equation (A.81) can be ordered with respect to powers of $\hbar$:

\frac{\partial F^{\rm W}}{\partial t} \; = \;
\frac{\partial^{2n+1}F^{\rm W}}{\partial p^{2n+1}}.
\end{displaymath} (A.52)

This equation was first derived by WIGNER in 1932 [Wig32]. Using the POISSON bracket $\{\cdot,\cdot\}$, it can be rewritten in the form
\frac{\partial F^{\rm W}}{\partial t}
\; = \; \left\{ H,F^{\rm W} \right\} + {\mathcal O}(\hbar^2),
\end{displaymath} (A.53)

with ${\mathcal O}(\hbar^2)$ as usual denoting terms that are at least quadratic in $\hbar$ and that give rise to the quantum mechanical corrections to the classical phase space distribution function $F(x,p,t)$ as obtained from the classical LIOUVILLE equation
\frac{\partial F}{\partial t} \; = \; \left\{ H,F \right\}.
\end{displaymath} (A.54)

Therefore in the classical limit $\hbar=0$ the equation of motion (A.83) for $F^{\rm W}$ formally becomes the LIOUVILLE equation (A.84); this makes WIGNER's function a useful tool for the investigation of the correspondence issue. Furthermore, for potentials $V(x)$ that do not contain powers of $x$ exceeding 2, the dynamics of $F^{\rm W}$ is completely classical, since in this case the equations (A.83) and (A.84) coincide. But note that $F^{\rm W}$ itself depends on $\hbar$, such that the similarity of these quantum mechanical and classical equations of motion is formal in the first place, and the details of obtaining equation (A.84) from (A.83) in the classical limit are nontrivial in general.

I return to equations (A.82) and (A.83) in section A.5 when I discuss the interpretation of $F^f(x,p,t)$ as a (quasi-) probability distribution function in quantum phase space.

next up previous contents
Next: On the Interpretation as Up: Quantum Phase Space Distribution Previous: The HUSIMI Distribution Function   Contents
Martin Engel 2004-01-01