# Telling the Wave Function: An Electrical Analogy

## Abstract

**:**

## 1. Introduction

_{1}) in which the particle emitted by source S

_{1}hits detector D

_{1}, and the particle emitted by source S

_{2}hits detector D

_{2}, or there may be another situation (H

_{2}) in which the particle emitted by source S

_{1}hits detector D

_{2}, and the particle emitted by source S

_{2}hits detector D

_{1}. The absence of trajectories leads to the undecidability between H

_{1}and H

_{2}, and therefore to an interference connected with this undecidability [8,9]. This interference is a particular effect of a general phenomenon correlated with the non-factorizability of the wave function of the system consisting of the two particles. This phenomenon is the entanglement [10], and it constitutes a further aspect of the quantum domain that does not seem to admit a classical representation.

## 2. Basic Ideas

**x**, whose coordinates in reference to the rest of the particle are (x, y, z, ct), and we consider the two regions of the light cone (past and future) having vertices in

**x**and extension ±L/c in t. There are no stringent indications on the value of L; we will assume that L = ħ/mc is the Compton length of the particle. This assumption seems plausible because it is below this spatial scale that the particle is dissociated into particle–antiparticle pairs, and therefore, the polarization effects are manifested [15]; however, any other physically reasonable choice of L is just as good.

_{1}(

**x**) and −Q

_{1}(

**x**) will be induced in the future light cone of

**x**(t < t’ <t + L/c), while two opposite charges of +Q

_{2}(

**x**) and −Q

_{2}(

**x**) will be induced in the past light cone of

**x**(t −L/c < t’ <t). We pose Q

_{1}, Q

_{2}≥ 0 (the opposite choice is just as good). If we admit the existence of “vacuum capacity” C, dependent only on the type of the particle (electron, muon, etc.), these two charges correspond to two energies Q

_{i}

^{2}/2Cn, i = 1, 2. The total energy (Q

_{1}

^{2}+ Q

_{2}

^{2})/2Cn = Q

^{2}/2Cn is that of a group of n(

**x**) capacitors in parallel to the same capacity C brought to the common voltage V. The voltage V is assumed to be independent of

**x**. The charge q = CV then depends on the type of the particle. From the usual formalism of capacitors in parallel [16], we have Q

^{2}/2Cn = nCV

^{2}/2, from which the relation Q = nq follows.

**x**. The key assumption is that this group contributes, with a part of its charge Q, to the total charge of an actuator (the particle), which we will assume to be q.

**x**, its charge would be integrally supplied by the capacitor group present in

**x**. In this case, the total energy of the group would vary by an amount of ±q

^{2}/2Cn. The negative sign corresponds to the transfer, by the group, of charge q to the particle; the positive sign corresponds to the transfer, by the particle, of charge q to the group. In the hypothesis of the perfect localization of the particle in

**x**, the number of possible energy elements that can be exchanged between the group and the particle is given by the ratio (Q

^{2}/2Cn)/(q

^{2}/2Cn) = (Q/q)

^{2}= n

^{2}.

**x**)dxdydz = dn

^{2}(

**x**)/A as the probability of the presence of the particle in the neighborhood dxdydz of

**x**. The dimensionless normalization constant A can be determined according to the relation:

^{2}of the element dxdydz around

**x**is then q

^{2}ρ(

**x**)dxdydz = q

^{2}dn

^{2}/A = dQ

^{2}. Each point of space contributes, with its own group of capacitors in parallel, to the total charge q of the particle. This result constitutes a description of the delocalization of the particle in classical terms, and we will return to it later. At the Compton scale (the minimum scale at which the wave function is defined [15]), the relation q

^{2}ρ = dQ

^{2}/(dxdydz) becomes q

^{2}ρ~Q

^{2}/L

^{3}. Since q does not depend on

**x**, the density (ρ) is locally proportional to Q

^{2}.

## 3. Positional and Impulse Representation

^{2}= Q

_{1}

^{2}+ Q

_{2}

^{2}since q is independent of

**x**. Therefore, ρ is proportional to ψψ*. We interpret functions (2) and (3) as the two wave functions, retarded and advanced, of the particle. We note that:

- (1)
- If charges Q
_{1,2}are multiplied by a real common factor (k), the probability density (not normalized) is multiplied by k^{2}, while (2) and (3) are multiplied by k; - (2)
- If k is complex, the (non-normalized) probability density is multiplied by kk* when (2) and (3) are multiplied by k and k*, respectively. This implies that the modulus of (2) and (3) is multiplied by the modulus of k, while the two functions are rotated around the origin of the complex plane by an angle equal to the argument of k, k*;
- (3)
- From both the proportionality of (2) and (3) to the capacitor charges and the additive nature of the charges, it follows that functions of this type can be summed generating interference effects;
- (4)
- The time inversion t→ −t implies the exchange Q
_{1}↔Q_{2}, and then ψ↔ψ*; - (5)
- Functions (2) and (3) have, of course with reference to the representation discussed here, a clear ontic meaning as charge states of the network of groups of capacitors associated with the particle.

_{1,2}= c∂

_{μ}Q

_{1,2}, where c is the maximal speed, and μ = 0,1,2,3 is the spacetime coordinate index, are currents. In functions (2) and (3), those that are the eigenfunctions of c∂

_{μ}are also the eigenfunctions of the four momentums (iħ∂

_{0}, -iħ∂

_{i}), i = 1,2,3. These eigenfunctions can be superposed, thus generating generic wave packets. It, therefore, becomes possible to replace (2) and (3) with analogous complex functions containing currents instead of charges, thus passing to the momentum representation:

_{1}

^{2}+ I

_{2}

^{2})

^{1/2}flows. The energy of the single inductor is Mi

^{2}/2, with i = I/n as the current flowing in it. The total inductance of the system is M

_{T}= M/n. We, therefore, have [16]:

## 4. Multi-Particle Systems

**x**of functions Q

_{1}(

**x**) and Q

_{2}(

**x**) labels a group of capacitors connected in parallel. Therefore, there is a continuous quadruple infinity of these groups. In a quantum jump, functions (2) and (3) are zeroed, and new ϕ functions of the same type are generated at the output. This means that the capacitors associated with the labels

**x**= (x, y, z, ct), with t = instant of the jump, are discharged and new capacitors associated with new labels of the same type are charged according to the functions ϕ. It is possible to represent the quantum jump ψ→ϕ with the electrical diagram in Figure 1.

**x**

_{i}) paired with groups ϕ(

**y**

_{i}), as in Figure 1, where the labels

**x**,

_{i}**y**(i = 1, 2, …) represent the same point-event, i.e., the same spatial position and same instant in time coinciding with those of the jump.

_{i}**x**

_{i}) are connected to the corresponding two ends of switch a. Closing a involves short-circuiting all the groups of capacitors associated with function ψ. The opposite charges are recombined, and ψ is canceled. The closing of a also implies, as an automatic consequence, the closing of other switches such as b. The latter, in turn, implies the charging of all the groups of capacitors associated with function ϕ.

**x**(

**y**) argument represents a mesh and, therefore, a discharge (charge) line. These labels are unique to the network of groups of capacitors that are discharged or charged. If the network corresponds to a particle, in the sense that it exchanges charge with that particle (actuator) only, then the label is shared by that particle. Two distinct particles, A and B, then have distinct spacetime labels of

**x**

_{A}and

**x**

_{B}. The wave functions associated with them are, respectively, ϕ(

**x**

_{A}) and φ(

**x**

_{B}), and each describes the state of the charge of the network associated with the corresponding particle. The overall state of the charge of the two quadruple infinity of the groups of capacitors associated with the two particles will be represented by the product ϕ(

**x**

_{A})φ(

**x**

_{B}). Normally, in this product, the two functions are considered at the same instant in time, in such a way that the square modulus of the wave function provides the compound probability density of the two particles at that instant.

**x**

_{A})φ(

**x**

_{B}) ± ϕ(

**x**

_{B})φ(

**x**

_{A}). This entanglement describes the contributions of the two networks to the opening of the two switches a, inserted, respectively, on network A and network B.

## 5. Corpuscle–Wave Dualism

^{2}/2C, with the meaning of the symbols already seen in the previous sections. In this hypothesis, this is the minimum energy required to keep switch a open and thus allow the propagation of ψ. It is, therefore, natural to suppose that, for a particle with mass m, the rest energy of the particle is q

^{2}/2C = mc

^{2}. This is, in fact, the minimum energy required for an interaction to create an outgoing state containing that particle. The delocalization of the particle described in Section 2 and Section 3then corresponds, physically, to the delocalization of its rest energy; the energy stored in each capacitor group represents the local contribution to the rest energy. The relation q

^{2}/2C = mc

^{2}in turn implies C = 2πε

_{0}r

_{cl}, where ε

_{0}and r

_{cl}are, respectively, the dielectric constant of the vacuum and the classical radius of the particle; r

_{cl}/L is the fine structure constant.

^{−18}s) [17]. On this time scale, it is possible to resolve the temporal evolution of atomic orbitals during a transition, but it is not yet possible to resolve the quantum jumps that terminate this transition. The duration of the jumps—if actually finite—must, therefore, be much shorter.

_{cl}/c = 0.937·10

^{−23}s, and then C = 2πε

_{0}r

_{cl}= 1.56·10

^{−25}F; R = τ/C = 60 Ω; i = e/τ = 17,100 A. These results are derived from the currently accepted value for the classical electron radius, r

_{cl}= 2.81·10

^{−15}m. However, we do not elaborate here on this aspect, due to its speculative character.

## 6. Spin

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- de Broglie, L. Recherches sur la théorie des quanta. Ann. Phys.
**1925**, 10, 22. [Google Scholar] [CrossRef] [Green Version] - Lévy-Leblond, J.M.; Balibar, F. Quantique (Rudiments); Interéditions/CNRS: Paris, France, 1984. [Google Scholar]
- Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1932. [Google Scholar]
- Merli, P.G.; Missiroli, G.F.; Pozzi, G. Electron interferometry with the Elmiskop 101 electron microscope. J. Phys. E Sci. Instrum.
**1974**, 7, 729–732. [Google Scholar] [CrossRef] - Merli, P.G.; Missiroli, G.F.; Pozzi, G. On the statistical aspect of electron interference phenomena. Am. J. Phys.
**1976**, 44, 306–307. [Google Scholar] [CrossRef] [Green Version] - Taylor, G.I. Interference fringes with feeble light. Proc. Camb. Philos. Soc.
**1909**, 15, 114. [Google Scholar] - Feynman, R.; Leighton, R.; Sands, M. The Feynman Lectures on Physics; California Institute of Technology: Berkeley, CA, USA, 1965; Volume 3. [Google Scholar]
- Hanbury Brown, R.; Twiss, R.Q. A new type of interferometer for use in radio astronomy. Philos. Mag.
**1954**, 45, 663–682. [Google Scholar] [CrossRef] - Hanbury Brown, R.; Twiss, R.Q. Correlation between photons in two coherent beams of light. Nature
**1956**, 177, 27–29. [Google Scholar] [CrossRef] - Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K. Quantum entanglement. Rev. Mod. Phys.
**2009**, 81, 865. [Google Scholar] [CrossRef] [Green Version] - Gray, J. Bernhard Riemann, Posthumous Thesis ‘On the Hypotheses Which Lie at the Foundation of Geometry’ (1867). In Landmark Writings in Western Mathematics 1640–1940; Grattan-Guinness, I., Corry, L., Guicciardini, N., Cooke, R., Crépel, P., Eds.; Elsevier: Amsterdam, The Netherlands, 2005; pp. 50–520. [Google Scholar]
- Lobačevskij, N.I. New Principles of Geometry with Complete Theory of Parallels. Complete Collected Works; Kagan, V.F., Ed.; GITTL: Moscow, Russia, 1951; Volume 2. [Google Scholar]
- Bolyai, J. Appendix (The Theory of Space); Kárteszi, F., Ed.; Wiley: Amsterdam, The Netherlands, 1987. [Google Scholar]
- Non Euclidean Geometry. Britannica. Available online: https://www.britannica.com/science/non-Euclidean-geometry (accessed on 30 May 2022).
- Davydov, A.S. Quantum Mechanics; Elsevier: Amsterdam, The Netherlands, 1965. [Google Scholar]
- Halliday, D.; Resnick, R.; Walker, J. Fundamental of Physics; Wiley: Hoboken, NJ, USA, 1996. [Google Scholar]
- Corkum, P.; Krausz, F. Attosecond science. Nat. Phys.
**2007**, 3, 381–387. [Google Scholar] [CrossRef] - Ballentine, L.E. Quantum Mechanics. A Modern Development; World Scientific: Singapore, 1998. [Google Scholar]
- Fock, V.A. Fundamentals of Quantum Mechanics; MIR: Moscow, Russia, 1978. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory; Butterworth-Heinemann: Oxford, UK, 1981. [Google Scholar]
- Aerts, D.; Sassoli de Bianchi, M. Do spins have directions? Soft Comput.
**2017**, 21, 1483–1504. [Google Scholar] [CrossRef] [Green Version] - Bohm, D. A Suggested interpretation of the quantum theory in terms of ‘hidden’ variables, I and II. Phys. Rev.
**1952**, 85, 166–193. [Google Scholar] [CrossRef] - Everett, H., III. ‘Relative state’ formulation of quantum mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef]

**Figure 1.**Electrical diagram of the quantum jump ψ→ϕ (for simplicity, only two pairs of opposing meshes are represented).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chiatti, L.
Telling the Wave Function: An Electrical Analogy. *Foundations* **2022**, *2*, 862-871.
https://doi.org/10.3390/foundations2040058

**AMA Style**

Chiatti L.
Telling the Wave Function: An Electrical Analogy. *Foundations*. 2022; 2(4):862-871.
https://doi.org/10.3390/foundations2040058

**Chicago/Turabian Style**

Chiatti, Leonardo.
2022. "Telling the Wave Function: An Electrical Analogy" *Foundations* 2, no. 4: 862-871.
https://doi.org/10.3390/foundations2040058