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Results in Physics

Deriving the Kramers’ dispersion formula from the canonical transformations of Heisenberg’s equation of motion

Yong-Yi Huang

Published Mar 15, 2024• 0 citations
Publisher

Abstract

We obtain the canonical transformations for the time-dependent Hamiltonian from Heisenberg’s equation of motion, based on which we derive the Kramers’ dispersion formula via the perturbation theory of matrix mechanics to the second order approximation.
DOI
10.1016/j.rinp.2024.107578

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A generalization of Dirac's transformation of the off-resonance single-photon scattering amplitude that leads to the Kramers-Heisenberg dispersion formula is described. The amplitude is first calculated to order e2{e}^{2}e2 in perturbation theory with the minimal-coupling interaction Hamiltonian as perturbing energy, and is then transformed so that the effects of multipole transition moments of all orders and types (electric, magnetic, and diamagnetic) readily become manifest. No appeal is made to any multipolar expansions, however, as exact integral expressions are used for the summed series. It is shown that the same dispersion formula is obtained by using the multipolar Hamiltonian partitioned in the usual way, and this despite the fact that the unperturbed Hamiltonians are different in the two calculations.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Abstract The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory.