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.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
Schr\"odinger's method for determining the energy levels of an atomic system is applied to the case of the symmetrical top (moment of inertia about axis of symmetry
C, the other one
A). The energy values are found to be ${W}_{\mathrm{jn}}=\frac{{h}^{2}}{8{\ensuremath{\pi}}^{2}}\left[\frac{1}{A}j(j+1)+\left(\frac{1}{C}\ensuremath{-}\frac{1}{A}\right){n}^{2}\right]$ in agreement with the result obtained by Dennison from the matrix mechanics. The quantum numbers
j and
n must be integers restricted by $0\ensuremath{\leqq}j$, $|n|\ensuremath{\leqq}j$, while half-integral values are not permissible. The intensities of transitions are also calculated.
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.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
This chapter shows that the laws of conventional optics are a consequence of the linearity of Maxwell's equations, which permit the superposition of different waves. Experiments on electron diffraction show that the interference phenomena associated with these "matter waves" obey the familiar laws of physical optics. The chapter provides the linearly independent solutions of the Schrodinger equation form a complete set of functions. The form of the Schrodinger equation is intimately connected with the requirement of gauge invariance . In classical transformation theory, the Hamiltonian H is the generator of displacements in time. In classical mechanics the total linear and angular momenta are constants of the motion if the Hamiltonian is invariant under translations and rotations, respectively. The chapter focuses on eigenfunctions, and in particular, on stationary states. It presents several illustrations of the fact that there frequently are more than one linearly independent energy eigenfunction belonging to an energy eigenvalue.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.