The momentum operator may be generalized for the correspondence of the limit of quantum commutators of observables with the classical commutators. The substitution of the momentum operator into the nonrelativistic equation for the conservation of energy yields a generalization of the stationary wave equation. Replacing also the energy operator, a generalization of the time-dependent Schrödinger equations, there exists a special class of wavefunctions that satisfies the nonrelativistic wave equation in quantum theory.
The momentum operator may be generalized for the correspondence of the limit of quantum commutators of observables with the classical commutators. The substitution of the momentum operator into the nonrelativistic equation for the conservation of energy yields a generalization of the stationary wave equation. Replacing also the energy operator, a generalization of the time-dependent Schrödinger equations, there exists a special class of wavefunctions that satisfies the nonrelativistic wave equation in quantum theory.
Many-body quantum-mechanical stationary states that have real valued wavefunctions are shown to satisfy a classical conservation of energy equation with a kinetic energy function. The terms in the equation depend on the probability distribution, and, in addition, pressure and velocity functions, but these functions also depend on the probability distribution. There are two possible directions of the velocity that satisfy the energy equation. A linear momentum function is defined that integrates to zero, and this property is consistent with the expectation value of the linear momentum for stationary states with real-valued wave functions. The energy equation is integrated to obtain a version of the well known energy equation involving reduced density matrices, where the kinetic energy functional of the one-particle density matrix is replaced by a function of the electron density and a velocity function. Also, the noninteracting kinetic energy functional from the Hohenberg--Kohn theorem is given as an explicit functional of the orbital densities. For the purpose of describing the behavior of particles in a stationary state, a model based on the energy equation is constructed. The model is evaluated for the two different velocity directions using the grounds state of the particle in a one-dimensional box and the hydrogen atom. For one velocity direction, equations of motions with contradictory properties are obtained, and, in the other, an unstable system is found. A discussion is given with suggestions of additional elements that might improve the model.
The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy operators are defined. Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well.
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The time-independent Schrödinger equation represents a stationary-state equation. The stationary wave functions obtained so far are time independent. Their time-dependence is obtained by means of rather general arguments. Then, stationarystate functions are found. The next step in Newton’s and Euler’s representations of classical mechanics is (to derive) the equation of change of stationary-states. Here, Euler’s principles of stationary-state change are generalized to quantum-mechanical systems. This enables us to derive the quantum-mechanical equation of change of stationary-states. The time-independent Schrödinger equation, i.e, the equation of motion will follow in the next chapter.