In principle this should remain stationary forever. In the space of ... configuration together with a photon that carries away the energy difference and has constant ...
Deterministic Dynamical Theory of the Hydrogen Atom Daniel Crespin Facultad de Ciencias Universidad Central de Venezuela First two pages of sketch of a paper in production Abstract Starting from Schr¨ odinger self adjoint Hamiltonian operator, this paper builds up Realism, a classical non-linear wave dynamics for the hydrogen atom. States are pairs [ψ], φ with [ψ] and electron configuration and φ a conjugate momentum of [ψ]. The space of states SH is the cotangent bundle, SH = T ∗ P EH , of the configuration space P EH =projective space associated to the linear space EH of wave functions. The total energy of a state is hHψ, ψi/hψ, ψi + hφ, φi/hψ, ψi. The deterministic evolution equations are then the Hamiltonian equations for this energy function. An initial excited stationary state ([ψj ], 0) radiates energy by means of a continuous transition toward a state ([ψk ], φj,k ) consisting of a stationary configuration [ψk ] and a constant momentum φj,k that is the carrier of the energy difference. The causeless, indetermined, discontinuous, probabilistic, random, observer dependent, contradictory collapse of wave packets, or disentanglement, or quantum jumping (“verdammte Quantenspringerei”) and other constructs of Quantism can then be replaced with a deterministic, causal, continuous, predictable, rational, common sense and objective dynamics that provides a different understanding of protonelectron-photon interaction.
1
Realism
Quantism postulates a unitary evolution equation, (ı~)∂ψ/∂t = Hψ, that requires the introduction of complex valued wave functions. Only real valued functions will be presently needed, suggesting the short name “Realism” for our deterministic theory.
2
Sketch of Realism
A brief description of energy radiation is as follows: The initial state is ([ψj ], 0) ∈ SH . Here [ψj ] is a motionless electronic cloud. In principle this should remain stationary forever. In the space of states and under the deterministic evolution equation of Realism, it is a hyperbolic critical point. But then a perturbation moves the system to a nearby unstable state, say ([ψj + δψ], δφ). This ha a trajectory that converges to a state ([ψk ], φj,k ) where [ψk ] is a stationary configuration with energy −λk and the photon momentum φj,k is stationary with energy λk − λj . In other words the final state consists of a new stationary configuration together with a photon that carries away the energy difference and has constant momentum. The system is Hamiltonian, hence time reversible. Therefore for absorption we reverse momentum and start at ([ψk ], −φj,k ), that means a lower (than −λj ) energy stationary
configuration together with an incoming photon moving in the correct direction and having enough energy to reach ([ψj ], 0). The following are, in approximate order of apparition, some of the formal objects involved in the just described deterministic physical processes • Coulomb potential V • Laplace operator ∇2 • Wave functions ψ(x, y, z) • Traditional Schr¨odinger self adjoint operator H • Eigenfunctions ψj , eigenspaces Ej and eigenvalues −λj of H • Space of wave functions EH • Electron configurations [ψ] • Space of electron configurations P EH =projective space of EH • Photon momentum φ=conjugate variable of [ψ] • States ([ψ], φ) • Space of states SH = T ∗ P EH =cotangent space of P EH • Configuration energy econ ([ψ]) = hHψ, ψi/hψ, ψi • Kinetic energy eki (([ψ], φ)) = hφ, φi/hψ, ψi • Total energy f = econ + eki • Classical Hamiltonian evolution equations d[ψ]/dt = ∂f /∂φ, dφ/dt = −∂f /∂[ψ] on SH • Hamilton equations are non-linear • Hamiltonian flow [Ut ] : SH → SH • SH is a submanifold of P (E × E ∗ ) definable by one holonomic constraint • Infinitesimal generator XH : SH → T SH of [Ut ] • Lifting of XH to a linear operator C σ : E × E ∗ → E × E ∗ • Spectral structure of the lifting C σ • Stationary manifolds of [Ut ] with hyperbolic flow on their normal bundles • Stationary states ([ψj ], 0) ∈ SH • Perturbation of ([ψj ], 0) to a nearby unstable state ([ψj + δψ], δφ) • Perturbation evolves to ([ψk ], φj,k ) 2
