Classical and quantum spherical pendulum
Date
2022
Authors
Cushman, Richard
Śniatycki, Jędrzej
Journal Title
Journal ISSN
Volume Title
Publisher
Symmetry
Abstract
The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a
revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about
the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–
Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which
is compatible with geometric quantization. In the general case, we use geometric quantization to
prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf .
If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly
quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies
the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum
system of the type we are looking for requires a new treatment of the classical action functions
and their properties. For the sake of completeness we have provided an extensive presentation
of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call
Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions
between different quantum states. Moreover, we relate these shifting operators to quantization of
functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study
the properties of the quantum spherical pendulum, in particular, the boundary conditions for the
shifting operators and quantum monodromy.
Description
Keywords
Bohr-Sommerfeld-Heisenberg quantization, geometric quantization, shifting operators
Citation
Cushman, R. & Śniatycki, J. (2022). “Classical and quantum spherical pendulum.” Symmetry, 14(3), 496. https://doi.org/10.3390/sym14030496