Quantum mechanical operator equivalents used in the theory of magnetism

D. C. Cabra, Pierre Pujol

In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.

Shoibal Chakravarty

Viraht Sahni

ABSTRACT According to the Bohr correspondence principle, the external temporal scalar potential in the classical equation of motion is replicated in quantum theory as a multiplicative operator. An equivalent quantum‐mechanical definition of the scalar potential in Schrödinger‐Pauli/Schrödinger theory is provided. The potential is a known universal functional of the wave function. At each instant of time, it is the work done in a conservative “classical” field representative of internal properties of the system: Pauli and Coulomb correlations, kinetic effects, the density, the Lorentz force, an internal magnetic component, and the current density response. The Hamiltonians are thus rewritten in a new physical manner indicative of these properties. An application to the triplet state of a 2‐electron semiconductor quantum dot in a magnetic field is provided. The significance of the definition to the Hohenberg‐Kohn theorem, and its relationship to Quantum and Kohn‐Sham density functional theory, is discussed.

Thomas Gramespacher, Stefan Weigert

Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because another formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators that may cause divergence of the classical perturbation series. The results that are established here link the concept of quantum-mechanical integrability to a technical question, namely, the behavior of specific perturbation series. \textcopyright{} 1996 The American Physical Society.