Electron Trajectories in Molecular Orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations, since they predict that the electrons in a ground state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the non-relativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.

Electron Trajectories in Molecular Orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations, since they predict that the electrons in a ground state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the non-relativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.

Electron Trajectories in Molecular Orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations, since they predict that the electrons in a ground state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the non-relativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.

Planck Gravitation Theory

The author proposes a new gravitation theory, the Planck gravitation theory. The theoretical name is in honor of Planck proposing the Planck Length. Gravitation is a force in 4-dimensional space, or a force in 5-dimensional space-time. Every quantum particle produces gravitation. Gravitation is not actually related to the mass of particles. Every quantum particle produces the same gravitation, regardless of the type of particle. The origin of gravitation is quantum, and gravitation is a quantum force. In 4-dimensional space, gravitation is inversely proportional to the cubic of distance, not square of distance. The strength of gravitation is entirely determined by the Planck Length. The Planck Length is the identification constant of gravitation. The author's earlier projection gravitation theory is only a derivation of the Planck gravitation theory. From the theory, we deduce the gravitation of photons. Planck gravitation is separated into two different patterns in 3-simensional space. For particles with rest mass, Planck gravitation translates into projection gravitation, which is inversely proportional to the square of distance. For particles with zero rest mass, gravitation is inversely proportional to the cubic of distance. Every quantum particle is an empty hole in space, the radius of empty hole is Planck Length. This brings the effect of the quantization of space-time. Planck gravitation theory can solve the problem of ultraviolet divergence in quantum field theory without the need for renormalization. If the Planck gravitation theory is true, human need to rethink the gravitation, and need to rethink the way of gravitation quantization. The author finally discusses the projection action, it is the key to human understanding of the truth about gravitation.

A first step towards quantum energy potentials of electron pairs

A first step towards the construction of a quantum force field for electron pairs in direct space is taken.