Some exact equations are derived which clarify some potentials that are hidden in the classical theorems such as the Hellmann–Feynman (H–F) and the integral Hellmann–Feynman (I-H–F) theorems. The differential form of the density equation given previously includes not only the classical force operator but also the force operator associated with the quantum-mechanical potential introduced by Bohm. The latter arises essentially only from the noncommuting property of coordinates and momenta in quantum mechanics. However, after integration only the classical force term survives and results in the H–F theorem. The important role of the quantum force term is completely hidden in the H–F theorem. This fact would be closely related to the nondeterminicity and the classical interpretation of the H–F theorem. A similar role of the quantum potential is also shown for the I-H–F theorem. We have also investigated the origin of force density (the integrand of the H–F theorem) and isolated the roles of the generalized exchange and correlation effects which are also hidden in the H–F theorem.
Some exact equations are derived which clarify some potentials that are hidden in the classical theorems such as the Hellmann–Feynman (H–F) and the integral Hellmann–Feynman (I-H–F) theorems. The differential form of the density equation given previously includes not only the classical force operator but also the force operator associated with the quantum-mechanical potential introduced by Bohm. The latter arises essentially only from the noncommuting property of coordinates and momenta in quantum mechanics. However, after integration only the classical force term survives and results in the H–F theorem. The important role of the quantum force term is completely hidden in the H–F theorem. This fact would be closely related to the nondeterminicity and the classical interpretation of the H–F theorem. A similar role of the quantum potential is also shown for the I-H–F theorem. We have also investigated the origin of force density (the integrand of the H–F theorem) and isolated the roles of the generalized exchange and correlation effects which are also hidden in the H–F theorem.
Theorems giving exact expressions for the derivative of the energy with respect to a parameter in the Hamiltonian for cyclic classical mechanical problems in both the relativistic and nonrelativistic cases are derived. Comparison is made with similar well-known theorems in the quantum mechanical case. Applications of both classical and quantum derivative relations are made to deduction of parametric dependence of the energy in simple cases, deduction of a meaning to the expectation value of the Dirac operator β, and construction of a test of consistency for a well posed problem in the theory of a Dirac particle in a bound state.
Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any signs of a special role of these potentials in the appropriate experiments (Feinberg, 1963) is thereby explained, and c) discovery of classical analogues of the Aharonov-Bohm effect (Berry et al., 1980) is also explained by a classical nature of this effect. Elucidation of the "unlocal" interaction problem was made.
We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).
Since the inception of quantum mechanics, the classical limit has been an area of unease. Explanations proposed on this matter include the usage of Plank's constant, the decoherence parameter of the master's equation, or the quantum potential of Bohm's theory. An equally obscure related topic is the mixing of classical and quantum quantities in the dynamical equations. In this thesis, the subject of classical-quantum mixing will be studied. A method will be explored which will demonstrate one of several candidate methods whereby mixing may be removed and replaced by fully quantum non-relativistic equations. Through use of the quantum potential concept and the Feynman path integral, state-dependent forms of the classical scalar and vector potentials will be derived by assuming the existence of separate component wave functions for the particle and potential. The specific wave component | e attributed to the potential, will hold the effects of an environment. The resulting equations are devoid of classical potentials and can therefore be considered as purely quantum--no mixing takes place. The standard classical potentials emerge from the state-dependent equations by a condition which will be referred to as state-dependence reduction (SDR). Through SDR, the semi-quantum and purely quantum equations are qualitatively and quantitatively equivalent. The new purely quantum equations will be used both to interpret gauge symmetry and to clarify the reasoning behind the Aharonov-Bohm effect. It will be argued that the freedom of gauge is related to the freedom to choose from many different and distinct environments which reproduce the same experimental outcome. Likewise, the AB-effect is understood through the environment from which the topological structure of the electromagnetic field is represented. Several aspects of the environment responsible for the effective potentials will be calculated numerically for well-known physical situations including one-dimensional scattering, two-dimensional double-slit setup, two-dimensional Aharonov-Bohm effect, and the two-dimensional Dirac equation. The proposal of using state-dependent potentials to replace the semi-quantum equation represents a consistent generalization and unification of the quantum and classical potentials, and may offer some experimental results yet untapped.
We clarify certain confusions in the literature of the density operator in quantum mechanics, and demonstrate that the quantum Liouville theorem has the same form in both the Schrodinger and the Heisenberg pictures.Our starting point is to treat the density operator as an observable which has its specific time dependence in each of the two pictures.It is further shown that such a formulation will provide the exact correspondence between classical and quantum statistical mechanics with the Liouville theorem being interpreted as a conservation law, which is derivable from the equation of motion only in the quantum case.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.