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Discussion of the Parameter Dependence

In this section the parameter dependence of the quantum map is discussed qualitatively, with some emphasis on the semiclassical limit. Because of the observation stated in the last paragraph of the previous section, the equations of that section are not well suited for this purpose. Therefore, in the present section mainly the FLOQUET operator in the form of equations (2.37) and (2.45, 2.51) is investigated.

Generally speaking, $U_{mm'}$ depends on the three parameters $T$, $V_0$ and $\hbar$. As mentioned above, only ${\hat{U}}_{\mbox{\scriptsize free}}$ (as opposed to ${\hat{U}}_{\mbox{\scriptsize kick}}$) is $T$-dependent; similarly, only ${\hat{U}}_{\mbox{\scriptsize kick}}$ (and therefore $K_{mm'}$) depends on the kick strength $V_0$. From inspection of (2.51) it is clear that $V_0$ enters the formulae via the quotient $V_0/\hbar$ only, an obvious consequence of the form (2.35a) of the kick propagator. In this sense, the quantum map in essence depends on the three parameters $T$, $V_0/\hbar$ and $\hbar$, rather than $T$, $V_0$ and $\hbar$.

The intricacy with respect to the parameter $\hbar$ makes the analysis of the semiclassical limit $\hbar\to 0$ particularly difficult: all nontrivial terms in equation (2.51) depend on $\hbar$ in a nontrivial way. This is not an artificially created consequence of the scaling (1.15, 2.4) used here, but a general phenomenon that occurs within all the different scalings that up to now have been used for the analysis of the kicked harmonic oscillator (see for example [BRZ91,SS92]).

The first thing to note about the quantum map (2.37) is that for $V_0/\hbar\to 0$ the kick propagator becomes the identity operator and the quantum map is essentially determined by just the rotation given by the free harmonic oscillator propagator. This becomes more clear by considering equations (2.45) and (2.51); they show that in that limit $K_{mm'}$, and thus $U_{mm'}$, become diagonal, such that the contributing harmonic oscillator eigenstates each perform their respective rotation dynamics in quantum phase space (cf. section A.6 of the appendix) without mutual perturbation.

There are several ways to obtain the case of $V_0/\hbar\to 0$; here I discuss two of them. First, one can keep $\hbar$ fixed and let $V_0$ tend to zero. This scenario agrees with the discussion of the dynamics of the classical POINCARÉ map (1.21) near the origin of phase space when the kick strength is decreased and approaching zero.

Second, when keeping $V_0$ fixed and increasing $\hbar$, i.e. moving away from the semiclassical towards the full quantum regime, another scenario with $V_0/\hbar\to 0$ is obtained which is quite different from the first: clearly the evaluation of the series in equation (2.51) becomes increasingly difficult with growing $\hbar$, as large degree LAGUERRE polynomials and powers of large arguments have to be evaluated and summed up. The numerical intricacy of the evaluation of the matrix elements for large values of $\hbar$ makes it quite impossible to follow this approach by direct calculation on a computer.

The reverse direction, the semiclassical limit with $V_0$ fixed and $\hbar$ tending to zero, which leads to $V_0/\hbar\to \infty$, is even more difficult to analyze, because for semiclassically small values of $\hbar$ many more terms contribute to the series in equation (2.51), due to the exponential $\exp(-\hbar l^2/4)$. This mathematical observation can also be understood from a physical point of view: for expanding a quantum state in the same part of phase space a much larger basis of (e.g. harmonic oscillator) eigenstates is needed for smaller values of $\hbar$, because the structures to be resolved become smaller, too. The only clear observation is that in this case there is no argument by which the kick propagator attains a simple form or even becomes trivial (in a similar way as for $V_0/\hbar\to 0$). So in the semiclassical limit there seems to be no way to make the desired contact with the classical POINCARÉ map analytically.

Summarizing, in order to study the correspondence between classical and quantum dynamics -- especially in the semiclassical approximation -- one has to revert to numerical means. After some preparations in chapter 3, this approach is followed in chapters 4 and 5 in the cases of resonance and nonresonance, respectively.

next up previous contents
Next: Numerical Methods Up: Quantum Mechanics of the Previous: The Classical Limit   Contents
Martin Engel 2004-01-01