The Bohr—Sommerfeld Theory of Atomic Structure

Daniel H. Deutsch

Richard H. Cushman, Jędrzej Śniatycki

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.

Kamal Barley, Andreas L. Ruffing, Sergeĭ K. Suslov

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics.The derivation of quantization rules and energy levels is revisited using semiclassical methods.Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schr odinger and Dirac equations are established.Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

Fanica Cimpoesu, Mihai V. Putz, Marilena Ferbinţeanu