*“As above, so below”*

This phrase comes from the beginning of * The Emerald Tablet* and embraces the entire system of traditional and modern magic which was inscribed upon the tablet in cryptic wording by Hermes Trismegistus. The significance of this phrase is that it is believed to hold the key to all mysteries. All systems of magic are claimed to function by this formula.

*“That which is above is the same as that which is below”*…

Macrocosmos is the same as microcosmos.

*The following article prepared by Alex Sokolowski*

## Introduction: Wave-Particle Duality

**Wave–particle duality** is the fact that **every elementary particle or quantic entity exhibits the properties of not only particles, but also waves.** It addresses the inability of the classical concepts “particle” or “wave” to fully describe the behavior of quantum-scale objects. As Einstein wrote:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

### Wave behavior of large objects

Since the **demonstrations of wave-like properties in photons and electrons, similar experiments have been conducted with neutrons and protons.** Among the most famous experiments are those of Estermann and Otto Stern in 1929. Authors of similar recent experiments with atoms and molecules, described below, claim that** these larger particles also act like waves.** A wave is basically a group of particles which moves in a particular form of motion i.e. to and fro, if we break that flow by an object it will convert into radiants.

A dramatic series of experiments emphasizing the** action of gravity in relation to wave–particle duality** was conducted in the 1970s using the neutron interferometer. Neutrons, one of the components of the atomic nucleus, provide much of the mass of a nucleus and thus of ordinary matter. In the neutron interferometer, **they act as quantum-mechanical waves directly subject to the force of gravity.** While the results were not surprising since gravity was known to act on everything, including light (see tests of general relativity and thePound–Rebka falling photon experiment), **the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.**

In 1999, the diffraction of C_{60} fullerenes by researchers from the University of Vienna was reported. Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength is 2.5 pm, whereas the diameter of the molecule is about 1 nm, about 400 times larger. In 2012, these far-field diffraction experiments could be extended to phthalocyanine molecules and their heavier derivatives, which are composed of 58 and 114 atoms respectively. In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.

In 2003, the Vienna group also demonstrated the wave nature of tetraphenylporphyrin—a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a near-field Talbot Lau interferometer. In the same interferometer they also found interference fringes for C_{60}F_{48.}, a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms. Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain decoherence mechanisms.

In 2011, the interference of molecules as heavy as 6910 u could be demonstrated in a Kapitza–Dirac–Talbot–Lau interferometer.

In 2013, the interference of molecules beyond 10,000 u has been demonstrated.

Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle’s Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius, a scale at which current theories of physics may break down or need to be replaced by more general ones.

Recently Couder, Fort, *et al.* showed that we can use macroscopic oil droplets on a vibrating surface as a model of wave–particle duality—localized droplet creates periodical waves around and interaction with them leads to quantum-like phenomena: interference in double-slit experiment, unpredictable tunneling (depending in complicated way on practically hidden state of field), **orbit quantization** (that particle has to ‘find a resonance’ with field perturbations it creates—after one orbit, its internal phase has to return to the initial state) and Zeeman effect.

### Treatment in modern quantum mechanics

**Wave–particle duality is deeply embedded into the foundations of quantum mechanics.** In the formalism of the theory, all the information about a particle is encoded in its ** wave function**, a complex-valued function roughly analogous to the amplitude of a wave at each point in space. This function evolves according to a differential equation (generically called the Schrödinger equation). For particles with mass this equation has solutions that follow the form of the wave equation. Propagation of such waves leads to wave-like phenomena such as interference and diffraction. Particles without mass, like photons, has no solutions of the Schrödinger equation so have another wave.

**The particle-like behavior is most evident due to phenomena associated with measurement in quantum mechanics.** Upon measuring the location of the particle, the particle will be forced into a more localized state as given by the uncertainty principle. When viewed through this formalism, the measurement of the wave function will randomly “collapse”, or rather “decohere”, to a sharply peaked function at some location. For particles with mass the likelihood of detecting the particle at any particular location is equal to the squared amplitude of the wave function there. The measurement will return a well-defined position, (subject to uncertainty), a property traditionally associated with particles. It is important to note that a measurement is only a particular type of interaction where some data is recorded and the measured quantity is forced into a particular eigenstate. The act of measurement is therefore not fundamentally different from any other interaction.

Following the development of quantum field theory the ambiguity disappeared. The field permits solutions that follow the wave equation, which are referred to as the wave functions. The term particle is used to label the irreducible representations of the Lorentz groupthat are permitted by the field. An interaction as in a Feynman diagram is accepted as a calculationally convenient approximation where the outgoing legs are known to be simplifications of the propagation and the internal lines are for some order in an expansion of the field interaction. Since the field is non-local and quantized, the phenomena which previously were thought of as paradoxes are explained. Within the limits of the wave-particle duality the quantum field theory gives the same results.

### Wave-only view

At least one scientist proposes that the duality can be replaced by a “wave-only” view. In his book *Collective Electrodynamics: Quantum Foundations of Electromagnetism* (2000), Carver Mead purports to analyze the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particle-like behavior to quantization effects and eigenstates. According to reviewer David Haddon:

Mead has cut the Gordian knot of quantum complementarity. He claims that atoms, with their neutrons, protons, and electrons, are not particles at all but pure waves of matter. Mead cites as the gross evidence of the exclusively wave nature of both light and matter the discovery between 1933 and 1996 of ten examples of pure wave phenomena, including the ubiquitous laser of CD players, the self-propagating electrical currents of superconductors, and the Bose–Einstein condensate of atoms.

Albert Einstein, who, in his search for a Unified Field Theory, did not accept wave-particle duality, wrote:

This double nature of radiation (and of material corpuscles)…has been interpreted by quantum-mechanics in an ingenious and amazingly successful fashion. This interpretation…appears to me as only a temporary way out…

The many-worlds interpretation (MWI) is sometimes presented as a waves-only theory, including by its originator, Hugh Everett who referred to MWI as “the wave interpretation”.

The *Three Wave Hypothesis* of R. Horodecki relates the particle to wave. The hypothesis implies that a massive particle is an intrinsically spatially as well as temporally extended wave phenomenon by a nonlinear law.

*Source: Wikipedia*

## Wave Particle Duality and Bode’s Law

*Copyright 2015 by Alex Sokolowski*

Looking at the planetary orbits of the solar system, we discover that** they are not at random distances from the sun.
**In fact, at a closer look,

**each orbit is multiple of certain wave-length.**

Perhaps our Solar system exhibits particle-wave duality on a cosmic scale?

Perhaps our Solar system exhibits particle-wave duality on a cosmic scale?

### Bode’s Law

In 1768, Bode published his popular book, “Anleitung zur Kenntnis des gestirnten Himmels” [*Instruction for the Knowledge of the Starry Heavens*]. In this book, he described an** empirical law for planetary distances**, originally found by J.D. Titius (1729-96), now called **“Bode’s Law” or “Titius-Bode Law”.**

The **original formula** was:

**a = ( n + 4 ) / 10**

where n=0, 3, 6, 12, 24, 48, 96, 192, 384, …

The modern formulation of the **Titus-Bode law** is that the **mean distance a of the planet from the Sun** in astronomical units ( AU = 149.6 x 10^{6} km ) is:

** a = 0.4 + 0.3 x k
**where k=0, 1, 2, 4, 8, 16, 32, 64, 128

(

*sequence of*– from Pascal’s Triangle)

**powers of two** The resulting sequence is **very close to the distribution of mean distances of the planets from the Sun** expressed in Astronomical Units (AU):

Body |
Actualdistance (A.U.) |
Bode’sLaw |

Mercury | 0.39 | 0.4 |

Venus | 0.72 | 0.7 |

Earth | 1.00 | 1.0 |

Mars | 1.52 | 1.6 |

asteroid belt | 2.77 | 2.8 |

Jupiter | 5.20 | 5.2 |

Saturn | 9.54 | 10.0 |

Uranus | 19.19 | 19.6 |

Pluto | 39.44 | 38.8 |

*1 AU is approximately the mean Earth–Sun distance equal AU = 149.597 *10 ^{6} km*

**All planets (and asteroid belt) fit Titus-Bode Law** except for Neptune!

### Titus-Bode Law, Pascal’s Triangle and Cosmic Waves

Let’s have a look at the Bode’s Law from point of view of wave theory.

**Circular orbit has circumference** equal:

** 2Pi x R**

where R is planet’s average distance from the sun.

If we translate **distance from the sun of each planet** into their orbital circumferences, we discover that each orbit is a multiple of certain “wave-length” (just like electron can only exist on specific orbits – see image below):

Using Bode’s Law, we notice that each successive planetary orbit has **circumference increased by multiples of certain “wave-length”;**

For planetary orbit n+1 (n=0, 1, 2, 3, 4, 5, ….) its **circumference is longer from the circumference of the previous orbit** by exactly

2Pi x (R_{n+1} – R_{n}) = 60Pi x 2^{n }] = 20Pi x 3x 2^{n}

Notice how **circumference of each consecutive planet** is longer exactly by** 60Pi x 2 ^{n }**

Since consecutive orbits follow the above rule, we can interpret it from the wave theory point of view:

only 2^{n} multiples of certain cosmic wave-length are allowed for the planets.

Planetary orbits are not perfect circles ( one of the most circular orbits is orbit of Venus: 0.00677323 eccentricity).

Perhaps Bode’s Law could be used to define the cosmic Wavelength for the solar system ( multiples of which would appear in planetary orbits).

If we divide Mercury’s orbit into 12 “waves” than, according to Bode’s Law, Venus would have 21, Earth 30, Mars 48, asteroid belt 84 and so on….

In the drawing above, circumference of Mercury’s orbit (using R=1 unit) was divided into 12:

Wave-length L = 2Pi/12 = Pi/6 = 0.5236 ( 30,321,107.68 km or 30,667,643.51km accounting for ellipticity of the orbit )

**Note:** If the “wavelength” was simply L = **Pi** x 10^{7} km = 31,415,926.54 km

Using this value, we can estimate size of the orbits of other planets using Bode’s Law.

- Mercury: Orbit Radius (12x L)/2Pi = 60,000,000 km (difference of 3.5 %; official value 57,909,050 km km)
- Venus: Orbit Radius (21x L)/2Pi = 105,000,000 km ( difference of 3 %; official value 108,208,000 km)
- Earth: Orbit Radius (30x L)/2Pi = 150,000,000 km (difference of 0.3 %; official value 149,598,261 km)
- Mars: Orbit Radius (48x L)/2Pi = 240,000,000 km (difference of 5%; official value 227,939,100 km)
- Asteroid belt: Orbit Radius (84x L)/2Pi = 420,000,000 km (difference of 1.3%; official value 414,383,690 km)

According to Bode’s Law, using radius of Earth orbit = 1, we notice that each consecutive orbit is increased in length by

**2Pi * 0.3 * k = 1.885 * k**

where k=0,1,2,4,8,16,32,64,128,….

Therefore each consecutive orbit is longer by double of the length by which previous orbit was increased.

**Bode’s Law describes mathematically “perfect” solar system.** In reality, there are small deviations from such perfection, suggesting that certain fluctuations are allowed.

Perhaps destruction of the planet which debris now form asteroid belt, and possible capture of Neptune, destabilized originally perfect system ?