Abstract There are several definitions of energy density in quantum mechanics. These yield expressions that differ locally, but all satisfy a continuity equation and integrate to the value of the expected energy of the system under consideration. Thus, the question of whether there are physical grounds to choose one definition over another arises naturally. In this work, we propose a way to probe a system by varying the size of a well containing a quantum particle. We show that the mean work done by moving the wall is closely related to one of the definitions for energy density. Specifically, the appropriate energy density, evaluated at the wall corresponds to the force exerted by the particle locally, against which the work is done. We show that this identification extends to two and three dimensional systems.
Abstract There are several definitions of energy density in quantum mechanics. These yield expressions that differ locally, but all satisfy a continuity equation and integrate to the value of the expected energy of the system under consideration. Thus, the question of whether there are physical grounds to choose one definition over another arises naturally. In this work, we propose a way to probe a system by varying the size of a well containing a quantum particle. We show that the mean work done by moving the wall is closely related to one of the definitions for energy density. Specifically, the appropriate energy density, evaluated at the wall corresponds to the force exerted by the particle locally, against which the work is done. We show that this identification extends to two and three dimensional systems.
There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, e(r) and ${\ensuremath{\sigma}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\mathbf{r})$, in quantum systems. A key issue is the kinetic energy since the well-known forms, $\frac{1}{2}{|\mathbf{\ensuremath{\nabla}}\mathrm{\ensuremath{\Psi}}|}^{2}$ and $\ensuremath{-}\frac{1}{2}\mathrm{\ensuremath{\Psi}}{\ensuremath{\nabla}}^{2}\mathrm{\ensuremath{\Psi}}$, lead to different densities, and analogous issues arise for interaction energy terms. This paper considers the ground state of a system of many interacting particles in an external potential, and presents a resolution to the problems in steps. (1) For the kinetic energy all effects of exchange and correlation are shown to be unique functions defined at each point r; all issues of nonuniqueness are relegated to terms that involve only the density n(r) and are equivalent to an effective single-particle problem with ground state wave function s(r)=n(r)/N. (2) Interactions can be considered in two ways: in terms of potentials acting on particles or in terms of the interaction fields, e.g., the Maxwell form in terms of electric fields. In each case, the problem reduces to a mean field part that is a function of the density and a part due to correlation that is uniquely defined; however, it is different for the two cases. (3) The final results follow from the nature of energy and stress. Because the energy determines the ground state itself through the variational principle, the energy density approach leads directly to kinetic energy in the form $\ensuremath{-}\frac{1}{2}s{\ensuremath{\nabla}}^{2}s$ and interactions in terms of potentials acting on the particles. This leads naturally to density functional theory and provides an interpretation in which the energy density e(r) is equilibrated to minimize fluctuations with the same chemical potential at all points r. On the other hand, stress is related to forces, and the only physically acceptable expressions for the stress field involve the combination $\frac{1}{2}[s{\ensuremath{\nabla}}^{2}s\ensuremath{-}{|\mathbf{\ensuremath{\nabla}}s|}^{2}]$, as derived by Schr\"odinger, Pauli, and others, and Coulomb interactions in terms of electric fields, not potentials. Together these results lead to well-defined formulations of energy density and stress fields that are physically motivated and based on a clear set of arguments.
Abstract We derive expressions for the expectation values of the local energy and the local power for a many-particle system of (scalar) charged particles interacting with an external electrical field. In analogy with the definition of the (local) current probability density, we construct a local energy operator such that the time-rate of change of its expectation value provides information on the spatial distribution of power. Results are presented as functions of an arbitrarily small volume Ω , and physical insights are discussed by means of the quantum hydrodynamical representation of the wavefunction, which is proven to allow for a clear-cut separation into contributions with and without classical correspondence. Quantum features of the local power are mainly manifested through the presence of non-local sources/sinks of power and through the action of forces with no classical counterpart. Many-particle classical-like effects arise in the form of current-force correlations and through the inflow/outflow of energy across the boundaries of the volume Ω . Interestingly, all these intriguing features are only reflected in the expression of the local power when the volume Ω is finite. Otherwise, for closed systems with Ω → ∞ , we recover a classical-like single-particle expression.
Projection operators may be used instead of vectors to represent the states of a quantum-mechanical system and the appropriate equation of motion is obtained. When a basis is used the basic vectors may be oblique or orthogonal. A formulation appropriate to an oblique basis is developed with the help of tensor methods and the summation convention. This is generalized to be applicable to systems containing N electrons subject to Pauli's exclusion principle. The projection operator corresponding to an arbitrary state is the density operator and is represented by an Nth-order tensor. The well-known tensor operation of contraction of indices applied to the density operator furnishes similar, reduced operators giving one-, two-, or more-particle properties of the system. The one-particle properties are of special interest in chemical systems, the corresponding reduced density matrix being called the charge and bond order matrix by L\"owdin. The full formulas for the reduction of the general density matrix are given, showing the origin of the Coulomb and exchange operators.When the symbolic equation of motion is expressed by means of components and the process of contraction (reduction) is applied, the equation of motion of the one-particle reduction is obtained as well as an expression for the energy of the system. In the special case of the Hartree-Fock approximation the time-dependent wave equations for the one-particle orbitals are shown to satisfy the general equation of motion, suggesting that the energies of the individual orbitals need not be stationary. This approximation is characterized by the idempotent property of the one-particle reduced density matrix. This condition alone leads to a close restriction on the possible values of the charge and bond orders of a molecule. Specific formulas for the molecule butadiene show that these can be expressed in terms of two parameters only, in this case.
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.