Abstract A Hamiltonian in two space dimensions whose kinetic-energy contributions have opposite signs is studied in detail. Solutions of the time-independent Schrödinger equation for fixed energy are superpositions of plane waves, with wavevectors on hyperbolas rather than circles. The local velocity (e.g. in the Madelung representation) is proportional to the kinetic momentum, i.e. local particle velocity, not the more familiar canonical momentum (phase gradient). The patterns of the associated streamlines are different, especially near phase singularities and phase saddles where the kinetic and canonical streamline patterns have opposite indices. Contrasting with the superficially analogous circular smooth solutions of kinetically isotropic Hamiltonians are wave modes that are anisotropic in position and also discontinuous. Pictures illustrating these phenomena are included. The occurrence of familiar concepts in unfamiliar guises could be useful for teaching quantum or wave physics at graduate level.
Abstract A Hamiltonian in two space dimensions whose kinetic-energy contributions have opposite signs is studied in detail. Solutions of the time-independent Schrödinger equation for fixed energy are superpositions of plane waves, with wavevectors on hyperbolas rather than circles. The local velocity (e.g. in the Madelung representation) is proportional to the kinetic momentum, i.e. local particle velocity, not the more familiar canonical momentum (phase gradient). The patterns of the associated streamlines are different, especially near phase singularities and phase saddles where the kinetic and canonical streamline patterns have opposite indices. Contrasting with the superficially analogous circular smooth solutions of kinetically isotropic Hamiltonians are wave modes that are anisotropic in position and also discontinuous. Pictures illustrating these phenomena are included. The occurrence of familiar concepts in unfamiliar guises could be useful for teaching quantum or wave physics at graduate level.
Abstract The Madelung trajectories associated with a wavefunction are the integral curves (streamlines) of its phase gradient, interpretable in terms of the local velocity (momentum) vector field. The pattern of trajectories provides an immediately visualisable representation of the wave. The patterns can be completely different when the same wave equation describes different physical contexts. For the time-independent Schrödinger or Helmholtz equation, trajectories circulate around the phase singularities (zeros) of the wavefunction; and in the paraxially approximate wave, streamlines spiral slowly in or out of the zeros as well as circulating. But if the paraxial wave equation is reinterpreted as the time-dependent Schrödinger equation, its Madelung trajectories do not circulate around the zeros in spacetime: they undulate while avoiding them, except for isolated trajectories that encounter each zero in a cusp singularity. The different local trajectory geometries are illustrated with two examples; a local model explains the spacetime cusps.
According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
Wave streamlines are integral curves of the local wavevector (phase gradient). An exact formula is derived, giving the curvature of streamlines as the component transverse to the local wavevector of the gradient of the logarithm of the local wavenumber. The formula is applied to quantum particles moving in a potential and classical light in the presence of a refractive-index gradient. Three limiting regimes are encompassed. The first is geometrical, in which the bending of streamlines arises solely from the classical force or optical index gradient. The second and third limits concern singularities in the pattern of wave streamlines, of two types: optical vortices, near which the streamlines are asymptotically circular, and phase saddles, near which the streamlines are asymptotically hyperbolic.
Abstract In single-particle Madelung mechanics, the single-particle quantum state Ψ ( x → , t ) = R ( x → , t ) e i S ( x → , t ) / ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R ( x → , t ) 2 and local momentum ∇ → S ( x → , t ) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q ( x → , t ) = − ℏ 2 2 m ∇ → R ( x → , t ) R ( x → , t ) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q ( x → , t ) < 0 . For energy superposition states with band-limit
We analyze the trajectories of a massive particle in one space dimension whose motion is guided by a spin-half wave function that evolves according to the free Dirac equation, with its initial wave function being a Gaussian wave packet with a nonzero expected value of momentum k.
We prove that at large times, the wave function is approximately equal to the superposition of two wave packets traveling in opposite directions, which results in trajectories with approximately constant asymptotic momentum k and asymptotic energy ±c2m2+k2, with m the rest mass of the particle and c the speed of light. The sign of the asymptotic energy is determined by the initial position of the particle. Particles with negative energy will have an asymptotic velocity that is in the opposite direction of their momentum.
The proof uses the stationary phase approximation method, for which we establish a rigorous error bound.
According to Schrodinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrodinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.