One Electron Coherence | Biocommerce©

 

 To mathematically represent the quantum phase alterations due to gravitational fields, we must integrate quantum mechanics with the curvature of space-time described by general relativity.

 In the context of quantum mechanics, the phase of a wavefunction is a central feature that determines the state of coherence. When considering gravitational fields, especially those around a mini black hole, the phase of a particle's wavefunction could be influenced by the curvature of space-time. This effect can be described by the following formalism...

1. General Relativistic Quantum Phase: The phase φ of a quantum particle in a gravitational field can be affected by the space-time metric gμν. The phase shift Δφ due to gravity can be expressed as:

   Δφ = (1/ħ) ∫ gμν pᵈˣⁿᵘ

   where pᵈˣⁿᵘ is the four-momentum of the particle and dˣⁿᵘ is the differential displacement in space-time.

2. Schrödinger Equation in Curved Space-Time: The Schrödinger equation can be modified to include the effects of gravity by incorporating the metric tensor:

   iħ ∂ψ/∂t = [-ħ²/(2m) gᵛμ ∇ᵤ ∇ᵤ + mc²] ψ

   where ∇ᵤ is the covariant derivative and ψ is the wavefunction of the particle.

3. Coherence Time Estimation: The coherence time τ can be estimated by considering the rate of dephasing due to the gravitational field, which can be related to the energy uncertainty ΔE caused by the gravitational potential Φ:

   τ ≈ ħ/(ΔE) = ħ/(mc²ΔΦ)

4. Entanglement in Gravitational Fields: The entanglement between two particles can be affected by the difference in gravitational phase shift experienced by each particle. This can be described by the entanglement entropy S, which would now depend on the gravitational field:

   S = -Tr(ρ log ρ)

   where ρ is the density matrix of the entangled system, and the trace is taken over the space-time dependent states.

These equations and formalisms provide a theoretical foundation for understanding how the gravitational fields of mini black hole electrons could influence quantum coherence and entanglement in a conspansive universe.

- PhiloFi Inc. Research

  Biocommerce©