A pure quantum model for the Pollack battery based on dark protons

The development of the notion of Pollack battery as a possible model explaining the claims of Donut Lab has been rather painful process. At least formally, the notion of a pure quantum Pollack battery makes sense for dark hydrogen atoms (see this). Does it also make sense for dark protons? In the charging of the Pollack battery, the ordinary protons at monopole flux tube states are transferred to flux tubes and localized near the opposite electrode. When the Pollack battery is used, the voltage generated during charging has an opposite sign. The addition of load suggests that a localization to the electrode E1 takes place so that a reverse Pollack effect occurs.

Concerning the construction of the model of Pollack battery for dark protons, the technical problem is that the potential is repulsive if it is taken to vanish at the first electrode E1 so that Schrödinger equation does not give bound state solutions.

1. By gauge invariance it is however possible to add to the potential function a constant so that it vanishes at the second electrode E1. Schrödinger equation for the states localized E2 however gives rise to bound states since voltage with respect to the opposite electrode is negative and increases. Gravitational bound states in the gravitational potential of Earth near the surface of Earth defines a completely analogous system.
2. The Schrödinger equation appears in WKB approximation and solutions are Airy functions satisfying a simple differential equation. The solutions for ordinary value of Planck constant are discussed here) and for general Planck constant here).
3. The situation can be approximated as effectively 1-dimensional. The Schrödinger equation is

   (ℏ²_eff/2m)∂_z² + eE(z-z₀) )Ψ = E_nΨ .

   Here z denotes the coordinate along the monopole flux tube and E denotes the strength of the electric field. E_n is the energy eigenvalue. In the recent case the identification of h_eff is as the gravitational Planck constant ℏ_gr(M_E,m) = r_s(e)m/2β₀= Λ_grm for the pair formed by Earth and charged particle with mass m, in the recent case proton mass. β₀=v₀/c ∼ 1 is true for the Earth.

4. One can introduce the dimensionless coordinate variable

   x= (z-z₀)/z₁ ,

   where the scales z₀ ad z₁ are defined as

   z₀= (E_n/eE)= (E_n/eV₀) r_s ,

   z₁= (ℏ_gr²/2meE)^1/3 = (m_p/8eV₀)^1/3 r_s ,

   eV₀= eEr_s .

   1. In semiclassical approximation, z₀ has interpretation as a classical turning point for Bohr orbits and is of the same order of magnitude as the distance d between the electrodes E1 and E2 of the Pollack battery. Also r_s∼ 1 cm is expected to be of the order of magnitude as d.

      For proton and eV₀= 1 eV one z₁(p)∼ 10³r_s/2∼ 5 m and z₀/z₁∼ 2× 10^-3(eV₀/E_n). For electrons one has z₁(p)(m_em_p)^1/3 ∼ z 10³r_s/2∼ .4 m.

   2. That z₁ is larger than z₀ has interpretation as quantum tunnelling beyond the classically allowed region. One could say that the charges can get along flux tubes outside the region bounded by the electrodes of the battery.
5. Using the variable x, the Schrödinger equation reduces to a differential equation for Airy functions (see this) given by

   (∂_x² +x)Ψ=0 .

Just for fun, one can compare the situation to that for the gravitational potential of the Earth assuming gravitational Planck constant. In this case the value of z₁= (m_p/(8GMm_p/R_E))^1/3 r_s = (R_E/r_s(E))^1/3r_s∼ 1.08× 10³r_S(E)∼ 8.6 m. This is the size scale of trees and many animals and one can wonder how relevant this scale is for biology. Note that z₀ corresponds now to the classical turning point in the gravitational field.

See the article Are Pollack batteries possible? and the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.