It is well known that in classical mechanics algebraic integrals of the top equations only exist in the cases of Euler (asymmetric top, no electric moment), of Lagrange (symmetric top, electric moment parallel to the axis of figure) and of Kowalewski. The latter case is that of a symmetric top whose two equal moments of inertia are twice as large as the third (
A=B=2C) with an electric moment perpendicular to the axis of figure. The quantum mechanical analogue to Euler's and Lagrange's cases being well known, Kowalewski's case was tried. If $\ensuremath{\vartheta}$, $\ensuremath{\psi}$, $\ensuremath{\phi}$ are the Euler angles,
QI=Q1+iQ2, ${Q}_{\mathrm{II}}={Q}_{1}\ensuremath{-}i{Q}_{2}$ linear combinations of the momenta around the principal axes and $U={{Q}_{\mathrm{I}}}^{2}+4C\ensuremath{\mu}sin\ensuremath{\vartheta}{e}^{\ensuremath{-}i\ensuremath{\psi}},$ then Kowalewski's integral becomes:
UU∗+U∗U+4h2(QIQII+QIIQI)=Diag.Matrix, which differs from the classical result by the symmetrization and by the last term proportional to
h2.
The theory of a symmetric rotator which in addition to the usual three degrees of rotational freedom has about the axis of symmetry also a degree of torsional freedom between two of its principal parts, is treated quantum mechanically. The potential energy is taken to be expressible in the form: $V=L(1\ensuremath{-}cosm{\ensuremath{\varphi}}^{\ensuremath{'}})$ where
L is proportional to the restoring torque, ${\ensuremath{\varphi}}^{\ensuremath{'}}$ is the angle of displacement, and
m the number of minima of the potential energy curve as ${\ensuremath{\varphi}}^{\ensuremath{'}}$ increases from zero to $2\ensuremath{\pi}$. The Schr\"odinger equation is found to be separable into two differential equations, one being the quantum mechanical equation for an ordinary symmetric rotator, and the other being of the form of Mathieu's equation:
dx2d2M+(a+16qcos2x) M=0 in which
a is proportional to the energy parameter of the oscillator,
q proportional to the restoring torque, and
x to ${\ensuremath{\varphi}}^{\ensuremath{'}}$. It is found that the solutions to this equation must satisfy the condition: $M(x\ensuremath{-}m\ensuremath{\pi})=(\mathrm{exp}\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iK}{{A}_{z}}^{\ensuremath{'}}}{{A}_{z}}) M(x)$ where
K is the quantum number of angular momentum about the axis of symmetry, ${{A}_{z}}^{\ensuremath{'}}$ and
Az the moments of inertia of the lower part of the top and of the whole of the top respectively about the axis of symmetry. This ordinarily demands a general non-periodic solution to Mathieu's equation which, however, degenerates like the ordinary Mathieu functions to an exponential function as $q\ensuremath{\rightarrow}0$. A qualitative discussion is given about the manner in which the energy states in the limiting case where
q=0 go over into the other limiting case where $q=\ensuremath{\infty}$, and the calculation of the intensities and the selection rules for the rotator are finally determined where
q=0 and where $q=\ensuremath{\infty}$. These it is believed will also be valid at least in first approximation in the neighborhoods of these limiting cases where
q no longer is quite zero and not quite equal to infinity.
Applying the systematic method discussed in previous papers, we derive the invariants and the groups of the time-dependent Schrödinger equations for the rigid rotator and the symmetric top. The groups for these systems are found to be SO(3,2) (rigid rotator) and SU(2,2) (symmetric top). For the case of the symmetric top, it is found that under the symmetry breaking I1 = I2 = I3 → I1 = I2 ≠ I3, where I1, I2, and I3 are the moments of inertia of the top, two of the time-independent constants of the motion become time-dependent constants of the motion.
In view of the recent experimental determination of the dispersion by atomic hydrogen [1] it seems interesting to apply the theory of dispersion developed by Schrödinger [2] to this case. In this paper we restrict ourselves to an approximation in which terms of the order of relativistic correction are neglected. For this purpose it is simpler to obtain our wave equation by the operational method of Schrödinger [3] and Eckart [4], as extended by Epstein [5], for in this way we immediately obtain an equation free of relativistic terms.
By means of geometric quantization, we discuss the quantum-deformed-symmetric-top model. We show that the quantum-deformed symmetric top and the quantum-standard symmetric top obey the same Heisenberg equations of motion. The quantum-deformed-symmetric-top model possesses quantum group symmetry; thus the quantum group can be dynamical symmetries of the symmetric-top system. The energy levels and wave functions of stationary states of the quantum-deformed-symmetric-top model are obtained exactly by a quantum-group-theoretic approach. Using the exactly solved model, we investigate the rotational spectra of symmetric-top molecules and diatomic molecules when the moment of inertia about the line joining the nuclei is not exactly zero.
We study exact quantum recurrences in the double kicked top (DKT), a driven spin model that extends the quantum kicked top (QKT) by introducing an additional time-reversal symmetry-breaking kick. Reformulating its dynamics in terms of effective parameters
kr and
kθ, we analytically show exact periodicity of the Floquet operator for
kr=jπ/2 and
kr=jπ/4 with distinct periods for integer and half-odd integer
j. These exact recurrences were found to be independent of
kθ. The long-time-averaged entanglement and fidelity rate function show dynamical quantum phase transition (DQPT) for
kr=jπ/2 at time-reversal symmetric cases
kθ=±kr. In the other time-reversal symmetric case
kθ=0, the DQPT exists only for a half-odd integer
j. Using level statistics, a smooth transition is observed from integrable to non-integrable nature as
kr is changed away from
jπ/2. Our work demonstrates that regular and chaotic regimes can be controlled for any system size by tuning
kr and
kθ, making the DKT a useful platform for quantum control and information processing applications.