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American Journal of Physics

Tight-binding-like expressions for the continuous-space electromagnetic coupling Hamiltonian

Timothy B. Boykin

Published Jul 1, 2001• 22 citations
Publisher

Abstract

In quantum mechanics texts one sometimes encounters the unqualified (and generally untrue) assertion that the solution of the Schrödinger equation for a charged particle in the presence of an externally applied vector potential can be found from that in the absence of the vector potential simply via multiplication with an r-dependent phase factor. The confusion caused by this assertion is only increased when one examines the expression for the matrix elements of the Hamiltonian including electromagnetic interactions (via the vector potential) appropriate for a tight-binding model, for this expression indeed takes a different form from that of the usual minimal coupling Hamiltonian in continuous space. Motivated by the tight-binding (i.e., discretized) result, we derive its continuous-space analog. We show that the aforementioned perplexing assertion actually arises from a confusion between the wavefunction and the matrix elements of the continuous-space Hamiltonian in the position basis and discuss both the tight-binding and continuous-space expressions.

DOI
10.1119/1.1344169

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In quantum mechanics texts one sometimes encounters the unqualified (and generally untrue) assertion that the solution of the Schrödinger equation for a charged particle in the presence of an externally applied vector potential can be found from that in the absence of the vector potential simply via multiplication with an r-dependent phase factor. The confusion caused by this assertion is only increased when one examines the expression for the matrix elements of the Hamiltonian including electromagnetic interactions (via the vector potential) appropriate for a tight-binding model, for this expression indeed takes a different form from that of the usual minimal coupling Hamiltonian in continuous space. Motivated by the tight-binding (i.e., discretized) result, we derive its continuous-space analog. We show that the aforementioned perplexing assertion actually arises from a confusion between the wavefunction and the matrix elements of the continuous-space Hamiltonian in the position basis and discuss both the tight-binding and continuous-space expressions.
We consider the state vector of a system containing a particle, in the approximation treating the particle (mass = m) as if it followed a classically prescribed trajectory R(t). Since a real particle has position and momentum operators r and p of its own, one must find a way effectively to demote these from dynamical variables to mere time-dependent parameters. The traditional method is to subject the Hamiltonian system to a formal constraint r — R(t) = 0. Here we consider instead the particle trapped in a sufficiently tight-binding potential V(r — R(t)); the conclusions remain the same but the reasoning becomes much more accessible. The state vector acquires an overall phase factor exp{—i ft dt' (-1/2mR2(t'))} surprising in that the integrand features not the enforced kinetic energy, but its negative. We elucidate the physics of this phase factor, sketch a scenario for verifying it experimentally and show that it persists essentially unchanged when the trapped particle is coupled to the Maxwell field. The standard adiabatic approximation routinely applied misses this factor altogether, but a more careful modification recovers the right result.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian.In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with.As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. 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Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.