In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.
In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.
"We study the issues of average forces and the Ehrenfest theorem for a particle restricted to a semi-infinite interval by an impenetrable wall. We consider and discuss two specific cases: (i) a free particle in an infinite step potential, and (ii) a free particle on a half-line. In each situation, we show that the mean values of the position, momentum and force, as functions of time, verify the Ehrenfest theorem (the state of the particle being a general wave packet that is a continuous superposition of the energy eigenstates for the Hamiltonian). However, the involved force is not the same in each case. In fact, we have the usual external classical force in the first case and a type of nonlocal boundary quantum force in the second case. In spite of these different forces, the corresponding mean values of these quantities give the same results. Accordingly, the Ehrenfest equations in the two situations are equivalent. We believe that a careful and clear consideration of how the two cases differ but, in the end, agree, is pertinent, and has not been included in the literature."
We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.
The Schrodinger equation for a bound state includes the classical potential V(r) for which –d/dxj V represents Fj or force in the j direction. It has been argued in a previous note (1), that one may attribute physical meaning to the Fourier series of V(r). In such a case, V(r) and W(r) (the wavefunction) are quantities averaged over momentum (or vectors) as well as being averages over time. One may write the product of V(r)W(r) as one of two Fourier series and then reorganize into a series with basis exp(ikr). In such a case, one examines an interaction with V at each momentum wave level. Then the average is only over time as different k momentum states of in W combine with different p momentum states of the Fourier series of V(r). For a p momentum wave, one may write: p2/2m fp + Sum k Vk f(p-k) = E fp where W= Sum over p fp exp(ipx) in one dimension ((1)) The objective of this note is to observe differences in the interaction of a single plane wave with a potential ((1)) and the interaction of W with V in the Schrodinger equation. In particular, for a particle in a box with infinite walls, V(x)=0 inside the one dimensional box, yet the Fourier series of W includes all p values. From ((1)), even though V(x)=0 as an average, there seems to still be an interaction with various Vk values (Vk= the kth Fourier transform) in the interior of the box as ((1)) holds at all x points.
Abstract We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V 0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V 0 − mc 2, for a given V 0. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, V 0 → E + mc 2, for a given relativistic energy E( < V 0), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when V 0 → E + mc 2 does not seem to be compatible with the existence of the impenetrable barrier.
We develop a systematic approach to determine the large |p| behavior of the momentum-space wave function, ϕ(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to 1/pk+3. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.
Abstract We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a . We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a , not for vanishing a . The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.