The Mathieu-Schrödinger equation, which describes the behavior of a quantum pendulum, depending on the value of the parameter l (pendulum filament length), can have the symmetry of the Klein's four-group or its invariant subgroups. The paper shows that the mean values of z-components of the angular momentum of nondegenerate quantum states (the symmetry region of the four-group) tend to zero and their root mean square fluctuations are non-zero. Consequently, in this region of parameter values, the fluctuations overlap the mean values of the angular momentum and they become indistinguishable. Therefore, it can be argued that if, with an increase in the parameter, the system goes into a non-degenerate state, then after the inversion of the parameter change and the transition to the region of degenerate states, the initial states will not be restored. This behavior of the average values of angular momenta is caused by the combined actions of two factors: discontinuous change in the system at the points of change of its symmetry and the presence of quantum fluctuations in nondegenerate states.
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.
Complex and spinorial techniques of general relativity are used to determine all the states of SU(2) invariant quantum mechanical systems in which the equality holds in uncertainty relations for the components of the angular momentum vector operator in two given directions. The expectation values depend on a discrete quantum number and two parameters, one of them is the angle between two angular momentum components and the other is the quotient of two standard deviations. Allowing the angle between two angular momentum components to be arbitrary, a new genuine quantum mechanical phenomenon emerges: it is shown that although standard deviations change continuously, one of the expectation values changes discontinuously on this parameter space. Since physically neither of the angular momentum components is distinguished over the other, this discontinuity suggests that the genuine parameter space must be a double cover of this classical one: it must be diffeomorphic to a Riemann surface known in connection with the complex function z. Moreover, the angle between angular momentum components plays the role of the parameter of an interpolation between the continuous range of expectation values in the special case of orthogonal angular momentum components and the discrete point spectrum of one angular momentum component. The consequences in the simultaneous measurements of these angular momentum components are also discussed briefly.
The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.
Calculates expectation values and uncertainties of the radial coordinate and momentum, and the azimuthal angle and angular momentum components for a three-dimensional harmonic oscillator in a coherent state. The operators corresponding to powers of the angular variable are defined by their Fourier expansions. For large distances of the particle from the origin, where the oscillator becomes classical, radial and angular uncertainty products approach the value h(cross)/ square root 2, except for the point of discontinuity, where the angular uncertainty becomes infinite. For small distances, where the oscillator tends to its quantum ground state, the radial uncertainty stays finite and the angular uncertainty product tends to zero. It is also shown that the concept of a polar momentum operator is meaningless.
Abstract Zitterbewegung is the exotic phenomenon associated either with relativistic electron–positron rapid oscillation or to electron–hole transitions in narrow gap semiconductors. In the present work, we enlarge the concept of Zitterbewegung and show that trembling motion may occur due to dramatic changes in the symmetry of the system. In particular, we exploit a paradigmatic model of quantum chaos, the quantum mathematical pendulum (universal Hamiltonian). The symmetry group of this system is Klein’s four-group that possesses three invariant subgroups. The energy spectrum of the system parametrically depends on the height of the potential barrier, and contains degenerate and non-degenerate areas, corresponding to the different symmetry subgroups. Change in the height of the potential barrier switches the symmetry subgroup and leads to trembling motion. We analyzed mean square fluctuations of the velocity operator and observed that trembling is enhanced in highly excited states. We observed a link between the phenomena of trembling motion and the uncertainty relations of noncommutative operators of the system.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.