The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

Quantum Theory: The Wave 

Quantum Theory was the logical consequence of three discoveries. In 1900 the German physicist Max Planck solved the mystery of radiation emitted by heated objects (that Newton’s physics failed to explain): he realized that atoms can emit energy only in discrete amounts. Nature seemed to forbid exchanges of energy in between those discrete values. According to Planck, therefore, the energy of light is proportional to the frequency: E=hv (where “h” is Plack’s constant). In 1905 Einstein, to explain the photoelectric effect,  argued that light must be physically made of packets (“photons”) whose energy is proportional to the frequence.

In 1913 the Danish physicist Niels Bohr solved another mystery, the structure of the atom: electrons turn around the nucleus and are permitted to occupy only some orbits (or, better, the angular momentum of an electron occurs only in integer multiples of a constant, which happens to be proportional to Planck’s constant). Again, Nature seemed to forbid existence in between orbits. The electron “jumps” from one orbit to another orbit without ever being in the space in between the two orbits (as if it stopped existing in the old orbit and was suddenly created again in the next orbit).

In 1925 George Uhlenbeck  and Samuel Goudsmit discovered that each electron “spins” with an angular momentum of one half Planck constant. (Particles actually don’t spin, but interact as if they were spinning, hence the property that defines how they interact is called “spin” and particles are said to be “spinning”). The “spin” does not vary: the electron always rotates with the same “spin”. It would turn out that every particle has its own spin, and the spin for any kind of particle is always the same.

The fundamental assumption of Quantum Theory is that any field of force manifests itself in the form of discrete particles (or “quanta”). Forces are manifestations of exchanges of discrete amounts of energy. For example, electromagnetic waves carry an energy which is an integer multiple of a fundamental constant, the "Planck constant".

A way to solve the apparent paradox of Bohr’s electrons was discovered by the French physicist Louis de Broglie (“Waves and Quanta”, 1923) after Einstein had made the same assumption regarding light: if an electron is viewed as a wave spreading over many orbits, the electron does not need to “jump” from one orbit to another. The electron “is” in all orbits at the same time, to some degree. De Broglie proved that the equation for a standing wave matched the behavior of the electron. Each particle is associated with a wave whose wavelength is inversely proportional  the particle’s momentum. That equation expressed a relationship between quantities of matter (e.g., speed, momentum, energy) and quantities of waves (e.g., wavelength and frequency). Thus he concluded that waves and particles are dual aspects of the same phenomena: every particle behaves like a wave. One can talk of energy and mass (quantities previously associated only to matter), or one can talk of frequency and wavelength (quantities previously associated only to waves). The two descriptions are equivalent, or, better, one complements the other. It didn’t take long to observe “interference patterns” (typical of waves) among streams of electrons, and therefore confirm de Broglie’s theory. Einstein’s Relativity had shown that energy and matter were dual aspects of the same substance. De Broglie showed that particles and waves were dual aspects of the same phenomenon.

The character of this relationship was defined by Werner Heisenberg in Germany ("Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations", 1925) and Erwin Schroedinger in Austria  ("An Undulatory Theory of the Mechanics of Atoms and Molecules", 1926). Both devised equations that replaced the equations of Newton's physics, but both equations had unpleasant consequences: Heisenberg's equation (based on matrix algebra) implied that the result of a physical experiment depends on the order in which the calculations were performed, and Schroedinger's equation (based on wave mechanics) implied that each particle could only really be considered a wave. Schroedinger wanted to remove the discrete jumps (that were inherent in Heisenberg’s formulation) and restore the continuum of classical Physics. His equation, after all, simply replaces Newton's (or, better, Hamilton's) equations and predicts the state of the system at a later time given the current state; except that his "system" is not a confined object but a wave. He thought of the wave as describing the location of the object (i.e., the object being spread out in space). However, experiments showed that the object (e.g., the electron) was a very confined object (just like in classical Physics) while Schroedinger's equation described it as a wave spread out in space. Max Born (“On the quantum mechanics of collisions”, 1926) realized the implications of the wave-particle duality: the wave associated to a particle turns out to be a “wave of probabilities”, that accounts for the alternative possibilities that open up for the future of a particle. In other words, the wave summarizes the possible values for the electron’s attributes (e.g., position, energy, spin) and how those values may evolve over time (the square of the wave’s amplitude represents the probability of finding a given value for an attribute when measuring that attribute). In particular, Schroedinger’s wave is not  a representation of where the object is spread out but of all the places where the object could possibly be, each to a certain degree of probability. This meant that the position of a particle cannot be known for sure: we can only guess it from a distribution of probability. We only know the probability of finding a particle in a given position.

The state of a particle is described by this “wave function” which summarizes (and superposes) all the alternatives and their probabilities. The wave function contains all the information there is about the particle (or, in general, about a system). It contains the answers to all the questions that can be asked about the particle.

The reason this is a "wave" of probabilities and not just a set of probabilities is that Schroedinger’s equation that describes it is the equation of an electromagnetic wave.

Schroedinger's equation describes how this wave function evolves in time, and is therefore the quantum equivalent of Hamilton's equations. The Schroedinger equation fixes, deterministically, the temporal development of the state of the universe. But at every point in time the wave function describes a set of possibilities, not just one actuality. The particle’s current state is actually to be thought of as a “superposition” of all those alternatives that are made possible by its wavelike behavior. A particle's current state is, therefore, a number of states: one can view the particle as being in all of those states at the same time. This is a direct consequence of a particle not being just a particle but being also a wave.

John von Neumann realized that, mathematically speaking, a classical system is represented in Newton’s Physics by a point in a six-dimensional phase space (three coordinates for the position and three for the velocity), whereas quantum systems are represented by vectors in a vector space.

As Born put it, the motion of particles follows the law of probabilities, but the probability itself follows the law of causality.

In 1927 Bohr stated the ultimate paradox of the wave-particle duality: everything is both particle and wave, but one must choose whether to measure one or the other aspect of nature, and then stick to it. If you try to mix the two, you run into contradictions.

A particle is described by a function of probabilities. When it is observed by an instrument, the function “collapses” to one specific value. The transition from the world of the wave to the world of the particle is traumatic. The measurement that collapses the wave function creates an irreversible arrow of time. The fact that a measurement causes the collapse of the wave function (also called “state-vector reduction”) is de facto a natural law that has to be added to the classical ones.

In Thermodynamics the microscopic laws of Physics were still Newtonian and therefore reversible. In Quantum Mechanics the microscopic laws of Physics are already irreversible, because nothing can undo the measurement: once you measure the position or the momentum of a particle, you have forever changed the state of the universe.


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