Plane Rotator in a Stark Field

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

“…(20), we first exhibit the zeroth order energy eigenvalue predictions (compared to the 'numerically exact' results) in The effect of a constant external field (Stark effect) on the spectrum of the rotor is a well-known example of second-order degenerate perturbation theory [9] - [17] and the general result for the second-order energy shift is…”

“…Note that the first-order, and all odd-order shifts vanish for symmetry reasons. The special case of |m| = 1 must be treated separately with the result that the +1, −1 states are split to this order [9] - [10], [16], [17]. More generally, it can be argued [6] that the degeneracy in a r , b r is split in r-th order in perturbation theory, as in Eqn.…”

“…The corresponding quantum version was first considered early in the history of quantum mechanics by Condon [1] and is sometimes discussed in the more physical context of hindered internal rotations [2] - [5] in chemistry. The problem is also a staple of the pedagogical literature [6] - [8] and is often seen as an example of perturbation theory methods [9] - [11], especially in collections of problems in quantum mechanics [12] - [17], where is often described as the plane rotator in an electric field. Most treatments focus attention on aspects of the structure of the energy eigenvalues spectrum, with only a handful (see, e.g., [7]) examining the question of time-dependence of wave packet solutions.…”

Hypervirial analysis of enclosed quantum mechanical systems. II. von Neumann boundary conditions and periodic potentials

“…Formulas (71) and (75) are not valid for n = 1, because the degeneration is "broken" in the second order, as it is well known [8,9]. However, even so it is possible to apply the previous procedure taking into account the following facts: the solutions for the Mathieu equation can be divided into two disjoint sets (even and odd functions) [lo].…”

Atom optics quantum pendulum

“…Pendulum in quantum mechanics [1] is a subject of great interest when the question comes to explain hindered internal rotations in chemistry [2], quantum features of scattering atoms in quantum optics [3,4], perturbation theory methods to study weak field effects in quantum mechanics [5][6][7], dynamics of Bose-Einstien condensates in optical lattices for small nonlinearity [8] and many other physical systems. Comprehensive study of quantum pendulum has brought to light its various aspects which include the structure of energy spectrum, the time evolution focusing on the quantum revivals [9][10][11][12] and asymptotic Mathieu solutions using algebraic methods [13].…”