Evolution of Quantum Theory

Notes

• • Refer to Evolution of Quantum Chemistry

Newton's laws of motion and Universal gravitation are explained with mathematical equations, and interestingly everyday observations completely agree with them. For an example, Newton's Mathematics can precisely explain why the moon stays on its orbit around the earth, and can predict the trajectory of astronauts landing on the moon. However, Newton was unable to explain about the agent that tugs every objects nearby. After 300 years, Einstein came up with the theory of relativity - Special Theory of Relativity (or simply special relativity) of 1905, and General Theory of Relativity (or general relativity) of 1916, to explain that agent. 

The General Theory of relativity describes gravity as a property of geometry (curvature) of space-time as a result of the mass's existence in space i.e. interaction between space-time and matter - a relativistic theory of gravity. So, the agent was the space-time warp or geodetic effect. The ball rolling past the warp or gravity well in the trampoline created by a person, spiral towards his/her mass The geometry of the trampoline depends on the mass of a person; heavier the person, more the bending of the space. Also, the warp lessens farther away from the mass towards the edge of the trampoline. See the conceptual image of curved space-time around the earth (Fig. 1) through gravity probe B Mission. The probe did confirm the effect.

Fig. 1: Conceptual image of curved space-time around the earth through gravity probe B Mission [1].

Since the space-time is distorted, even on those straight paths, particles accelerate as if they were under the influence of gravitational force as Newton describes. However, there is no tug that masses exerts on other masses. Orbiting objects follow the path that is shortest and that requires the least amount of energy. The planets therefore move in the ellipses. the most energy-efficient path in the gravity well of the sun.

The space-time distortions between the earth and galaxy cause light waves to be bent into different paths. So, multiple images of the galaxy appear on the same telescope. This phenomena gives the evidence of the existence of black holes and worm holes. Black hole is a region of highly dense space in which the gravitational field is so powerful that nothing, including electromagnetic radiation can escape its pull. At its center lies an infinitely small, infinitely dense singularity, where the normal laws of physics completely break down. The worm hole or bridges are the hypothetical tunnel or short-cut path through space-time that connect widely distant regions.

Only gargantuan changes in mass as a result of orbiting  neutron stars, colliding black holes, or supernovas, can cause a perceivable ripple  which squeezes and stretches space travelling at the speed of light in all direction called the gravitational waves. LIGO (Laser Interferometer Gravitational-wave Observatory) is being used to measure infinitesimal distances or disturbances. The gravitational waves not only carry energy but also the information about the events at which they were formed. 

Einstein's relativistic effects become prominent only when objects are at speeds close to the speed of light. So, there were no problems in Newton's theory before 1905. In Newtonian mechanics, if you keep applying a force to a material object, it will eventually go faster than the speed of light. This statement is false according to Einstein's theory of relativity - the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source and is the absolute observation having upper limit on speed of any particle. This postulation was the result of the successful Michelson-Moreley experiment in 1887 which measured the speed of light to be "c" regardless of the source and the observer. it also falsifies the theory of the "quasi-rigid" luminiferous ether physicists were bound to arrive as a medium of propagation for the electromagnetic (EM) waves discovered by James Clerk Maxwell in 1864. 

The High Enrgy Particle (HEP) accelerator are able to achieve electrons speed slightly more than 99.9999 per cent of the speed of light. The closer the object's speed (v) is to light speed (c), the greater the increase in inertial mass (m) and it will require an infinitely strong force acting on the body. The increase in inertial mass is part of a more general phenomenon, the relativistic equivalence of mass and energy.

Einstein uses Lorentz factor for calculating also the dilation of time and the contraction of length. which is the result of Lorentz transformation, the transformation equations between inertial frames relative to each other, derived before special relativity by Hendrik Lorentz. Galileo Galilei actually stated the Principle of Relativity in 1632 which was overlooked by Newton in 17th century in generalizing his laws of motion in different reference frames.

In spite of the complete experimental confirmation of the theory as applied to diffraction, reflection, refraction, dispersion, etc, when the theory of light  was  applied to the phenomena of emission and transformation of light, scientists experienced contradictions. In one such case, Rayleigh–Jeans attempted to describe the spectral radiance of electromagnetic radiation at all wavelengths from a black body at a given temperature through classical arguments. His laws agreed with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe.

Max Planck solved this great problem of blackbody radiation in 1900 by applying the statistical mechanics of the Maxwell-Boltzmann velocity distribution law for particles to the distribution of energy in a radiation field. The distribution of radiant energy in a black body at any temperature is empirically obtained by an expression:

Bν (v, T) = (2hν3 / c2) (1 / ( e hν / kT - 1) )

Where, "v" is the frequency, "h" is the Planck's constant, "K" is the Boltzman's constant, "c" is the speed of light, and "T" is the absolute temperature. At very high frequency (kT << hv), the radiant energy becomes zero as showed in Fig. 2 which holds the principles of conservation of energy that black body releases. It was a foundational aspect of the development of quantum mechanics in the early 20th century.

Fig. 2: The Ultra violet Catastrophe - black curve showing classical theory and the measure blue curve predicted and corrected by Plank's Laws [1].

in 1899, Plank had noted that the energy of photons could only take on certain discrete values which were always a full integer multiple of a certain constant, known as the “Planck constant” i.e. E = hv. Light and other waves were emitted in discrete packets of energy that he called "quanta". It was however refined by Albert Einstein in 1905 with his photoelectric effect paper in which he proposed "light-quantum hypothesis" based on Heinrich Hertz's discovery of electric sparks from electrodes illuminated with UV light in 1887. Einstein assumed that light exists in a particle -like state or packets of energy called photons. According to him, the photon whose rest mass is zero should have momentum to knock out the electron from the material and the momentum should depend on its wavelength given by h/λ . So, in the photoelectric experiment conducted by Millikan, red photons having longer wavelengths or lower frequency were unable to knock out electrons or absorb the entire quantum of energy on impact irrespective of their intensity unlike UV photons which verified Einstein's law on Photoelectric effect, a particle nature of photon. This led to the concept of the quantum nature of light and electrons and wave-particle duality.

The wave behavior of light can be demonstrated by passing it through a double-slit to produce alternating bright and dark interference patterns on the screen. In 1924, Louis-Victor de Broglie formulated the de Broglie hypothesis, claiming that all matter, not just light, has a wave-like nature. The relation between the wavelength (denoted as λ, de Broglie's wavelength) and momentum (denoted as p) is defined by de Broglie's wavelength equation:

This is a generalization of Einstein's equation from relativity and photoelectric effect. So, in his hypothesis everything propagates like a wave, and that everything interacts like a particle. In 1927, De Broglie's formula was confirmed for electrons, that unlike photon have rest mass, through two independent experiments at the University of Aberdeen by George Paget Thomson and  at Bell Labs by Clinton Joseph Davisson and Lester Halbert Germer observing the predicted interference patterns. 

The stream of electrons through the double-slits should have just piled-up in two locations behind the slits but they showed the double-slit interference patterns similar to the ones produced by light waves. For massive objects, wave-like behavior is not noticeable. For an example, a baseball of weight 0.15 Kg thrown at about 90 miles/hour results in a shorter wavelenght of 0.12nm using de Broglie's wavelength equation which is much smaller than the shortest visible spectrum of 390nm for our eyes to see the wave nature of baseball. 

If the electromagnetic waves radiate into space in all directions, how can the energy instantaneously collect itself together to be absorbed in a single electron. So, the radiated waves are just the probabilities of finding a single electron and electron can be found anywhere where the probability amplitude in not zero. This indeterminacy makes quantum theory an irreducibly discontinuous and statistical theory against classical deterministic theory in which all physical variables are completely and locally determined by the four-dimensional field of space-time. So, quantum theory is a theory with alternative possibilities. When the electron is detected at the screen in a double-slit experiment, the probability simply disappears instantly. It is the similar concept that if one horse crosses the finish line in a horse race, the winning probability goes to certainty, and all finite probabilities of other horses, including the one in the rear, instantaneously drop to zero. It is the probabilistic density that is waving and the interference pattern is produced by the superposition of possible indeterministic paths the electron could take.

This disappearance of the probability i.e.the process of measurement invoking a collapse of a wave function from super-positional state into a single state as interpreted by Heisenberg with a probability determined by Born's rule, is known as Copenhegan Interpretation or Collapse approach to Quantum Mechanics. There is no way to predict at what state a superposition will collapse into other than assigning each possibility a probability. It appears to contradict Einstein's theory of Relativity of absolute speed of light as collapse happens instantly, faster than the speed of the light. However, the sum of all probabilities of measuring anywhere on the screen is not a physical quantity but immaterial information i.e. nothing physical is moving from one place to the other. This led to the idea of nonlocality - something measured in one place "influencing" measurements far away. 

When two electrons are fired in opposite directions from a central source with equal velocities, the electrons got entangled, i.e. the position of second electron is instantaneously known if the position of the first electron is measured. This EPR (Einstein, Podolsky, and Rosen) Paradox was first practically experimented by David Bohm. Bohm proposed using two electrons that are prepared in an initial state of known total spin. If one electron spin is +1/2 in the up direction and the other is spin down or -1/2, the total spin is zero. The underlying physical law of importance is a second conservation law - the conservation of angular momentum. So, if either electron is measured with spin +1/2, the two-particle wave function collapses and the other will always measured with spin down.

A particle (electron or photon) is seen to show interference patterns even when one slit is blocked in a double-slit experiment. So, the wave itself is the superposition of many waves i.e. it spreads out with different wavelengths. Hence, the particle can be anywhere and its momentum based on the wavelength of the wave appear indefinite. This situation is stated by Heisenberg in his Uncertainty Principle which allows only one of each pair of non-commuting (QP = PQ + iħ ) observables (i.e. momentum or position) to be known with arbitrary accuracy.

Δx Δp ≥

.

The order in which observables are measured matters. Hence, Heisenberg showed that it is physically impossible to design an experiment that allows you to measure both of these properties at once. He proved this using a new mathematical method called matrix mechanics developed with the assistance of Max Born and Pascual Jordan. The equivalent to Matrix Mechanics, the Wave Mechanics, apparently incompatible with Matrix Mechanics, was discovered by Erwin Schrödinger in 1925-1926. In Wave Mechanics, the state of a system is described by a wave function, the solution to Schrödinger's equation.

Heisenberg's indeterminacy principle disprove causality which explains why a particular atom will decay at one moment and not the next. it is impossible to predict when a particular atom will decay regardless of how long the atom has existed. So, the classical physics offered no explanation for the incredible range of half-lives of particles decaying via alpha emission, which extends from less than a microsecond to trillions of years, over many orders of magnitude. According to Quantum Theory, a particle partially bound within a finite potential well, a stronger nuclear force related to strong nuclear binding energy, has a certain probability, upon each encounter with the barrier, of appearing as a free particle on the other side favoring the theory of quantum tunneling discovered by George Gamow in 1928. This also explains Schrödinger's cat Paradox wherein the cat's life or death, a state of quantum superposition, is depended on the state of a radioactive atom. 

Classical Physics also could not explain the Macroscopic Quantum Phenomena - superconductivity and superfluidity, that happens at lower temperature approaching absolute zero (-273.15oC or 0K). The statistical mechanics, Maxwell-Boltzman, is therefore applicable in higher temperature only. In 1911, Heike Kamerlingh Onnes observed the resistance of mercury suddenly disappearing at 4.2K and also observed the superfluidity of Helium at 2.2K. The theory of superfluidity, where matter like helium-4 (boson particle) behaves like liquid with zero viscosity, was developed by Satyendra Nath Bose in 1924-1925 and later extended by Einstein in collaboration with Bose, and is called Bose–Einstein statistics. 

In the superfluid state, atoms moved in a coordinated manner loosing its randomness. The liquid now lacks all inner friction, and therefore can overflow the container, flow out through its pores, and exhibits other non-classical effects. Superfluidity is not possible in helium-3 (Fermion particle) i.e. it can't be condensate in the lowest energy state as it follows Fermi-Dirac Statistics. The possibility of superfluidity in fermions is proposed in BCS (Bardeen-Cooper-Schreieffer) theory for superconductivity in metals in 1957. The electrons in greatly cooled metals can combine in twos to form Copper pairs which are responsible for superconductivity, and behave as bosons. These pairs then can undergo Bose-Einstein condensation to form a Bose-Einstein condensate. 

An electric current flowing through a loop of superconducting wire can persist indefinitely with no power source. Meissner and Ochsenfeld in 1933 discovered that superconductors expelled applied magnetic fields, a phenomenon which has come to be known as the Meissner effect. In 1935, Fritz and Heinz London showed that the Meissner effect was a consequence of the minimization of the electromagnetic free energy carried by superconducting current.

Arnold Sommerfeld revised Bohr's  model introducing another quantum number, azimuthal quantum number (l) in 1915, for electrons to take as many shapes corresponding to the shell number (principle quantum number n). In 1920, he  realized that the orbital shapes can have different orientation in space and he introduced magnetic quantum number (ml) to explain the Zeeman effect, the splitting of spectral lines in the presence of magnetic field. 

Wolfgang Pauli was looking for an explanation for atomic model that assumes that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable closed shells. The 1924 paper from Edmund C. Stoner, helped Pauli realized that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers to describe various atomic spectra. He devised the quantum mechanical principle, Pauli Exclusion Principle, for electrons in 1925. Though he showed that electron states in an atom can be described by four quantum numbers, each quantum number corresponds to a degree of freedom of the electron - one representing energy w.r.t to the distance from the nucleus and other two representing  angular momentum (shape and orientation of the orbit), he could not explain physical significance for the fourth quantum number, which was needed empirically. 

The Stern-Gerlach experiment in 1922 showed that a beam of silver atoms directed through an in-homogeneous magnetic field would be forced into two beams. This indicates that electron should have magnetic moment which could classically occur if the electron were a spinning ball of charge. While studying certain details of spectral lines known as the anomalous Zeeman effect, George Uhlenbeck and Samuel Goudsmit  in 1925 realized that Pauli’s fourth quantum number must relate to electron spin, an intrinsic angular momentum independent of its orbital angular momentum. Quantum Spin, however, is not the physical rotation of electron on its axis. This accounts for why electrons interact with magnetic fields, explaining the anomalous Zeeman effect. The closely spaced splitting of fine structure observed in hydrogen spectral lines corresponds to two possibilities for the z-component ( +1/2h (up) & -1/2h (down)) of the total angular momentum (√3/2h) as showed in Fig. 3 which requires an angular momentum quantum number of 1/2.

Fig. 3: Electron Spin [2]

After the discovery of electron spin, Pauli in 1926 tried to use Heisenberg's Matrix Mechanics to derive the observed spectrum of the hydrogen atom and found that the non-relativistic Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state. Pauli introduced the 2 × 2 Pauli matrices as a basis of spin operators, thus solving the nonrelativistic theory of spin in 1927. This became the basis for the derivation of relativistic quantum mechanics (RQM) by Paul Dirac in 1928. With his spin-statistics theorem in 1940, Pauli extended his Pauli exclusion principle to all fermions. It states that particles with half-integeral intrinsic spin having anit-symmetric wave function under particle interchange, are fermions, while particles with integral or zero intrinsic spin under particle exchange are bosons. 

The relativistic quantum Mechanics that predicts the existence of anti-electrons with the same mass but a positive electric charge (positron, an antimatter) along with electron spin, spin magnetic moments of elementary spin-1/2 fermions, fine structure, and quantum dynamics of charged particles in electromagnetic fields marked the beginning of quantum field theory (QFT), the application of quantum mechanics to fields. In fact, the experimental discovery of the neutrons by Chadwick, and positrons by Anderson, confirmed the prediction of antimatter. 

The relativistic Quantum Field Theory not only represents the quantum behavior of particles but also the forces acting between them. For an example, in quantum field theories, the electric repulsion between two electrons is caused by the exchange of photons flitting back and forth i.e. exchange force arising from the exchange of photons as showed via Feynman diagram in Fig. 4. The emission and absorption of these carrier particles of the electromagnetic force are responsible for the electromagnetic interaction between one electron and another, and is referred to as Quantum Electrodynamics (QED). In classical theory of electromagnetism, it is due to electric field produced by each electron at the position of the other and the force is calculated from Coulomb's law.

Fig. 4: Feynman Diagram showing electric repulsion between two electrons. One electron emits a photon and recoils; the second electron absorbs the photon and acquires its momentum [4]. 

QFT also describes cases where the number of particles changes; for example in matter creation and annihilation as showed through Feynman diagram in Fig. 5.

Fig. 5: An electron (e-) and a positron (e+) annihilate, producing a force carrier photon (represented by the blue sine wave) that becomes a quark–antiquark pair, after which the antiquark radiates a gluon (represented by the green helix) [1].

Quarks, the sub-atomic particles, which make up the composite particles such as neutron and protons, comes in six flavors as showed in Fig. 6 that impart those particles their properties. The composite particle Hadron is categorized into baryon (made up three quarks e.g. proton and neutron) and meason (made up one quark and one antiquark e.g. pion). 

Fig. 8:  Gluon mediating interaction between two quarks - bi-colored gluon converting blue quartz to green and vice versa which is also responsible for holding protons and neutrons together to form nuclei [2]

Gluons are thought to interact with quarks and other gluons through color charges (+/− red, +/− green, +/− blue) analogous to electromagnetic charge which result in different types of force with different rules of behavior as detailed in the theory of Quantum Chromodynamics (QCD). Yukawa potential was an early example of a nuclear potential and the measons predicted by this theory were discovered experimentally in 1947. By the 1970s, the quark model had been developed which categorized the weaker force as a residual effect of the strong force.

Out of these four fundamental interactions of nature discovered in 1900s, the force of gravity, is the least understood interaction, and an attempt to incorporate it into quantum framework as if it is just like other forces have failed utterly. The two promising candidates are String Theory and Loop Quantum Theory. Loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein´s geometrical formulation. In the loop models, the basic structure of space-time turns out to be discrete. The basic constituents of String Theory are not point-like particles, but one-dimensional objects called strings which can oscillate in intricate patterns in contrast to particles. String theory promises nothing less than a complete unified description of all forces and all matter particles sometimes referred to as Theories of Everything (TOE). 

Unlike Stellar black holes which are formed when the center of a very massive star collapses in upon itself causing a Supernova or exploding star that blasts part of the star into space, primordial black holes are thought to have formed in the early universe, soon after the Big Bang. Big Bang model by Georges Lemaitre in 1927, describes how the universe expanded from a very high density and high temperature state, called Singularity, which contained all of the mass and space-time of the Universe, before quantum fluctuations caused it to rapidly expand.

The physicisyts have described a timeline of the formation and subsequent evolution of the Universe from the Big Bang (13.799 ± 0.021 billion years ago) to the present day through cosmological epochs. The earliest was the Planck epoch, the singularity state where all the four fundamental interactions - (1) Gravitation, Electroweak: (2) weak, (3) electromagnetic, and (4) strong (Fundamental & Residual), could be described through the unified quantum theory - Theory of Everything (TOE). 

When the universe cooled down to 1032 Kelvin in the Grand Unification Epoch, the force of gravity got separated and started operating on the universe but other three forces stabilized into the electronuclear force as described by Grand Unified Theories (GUT). Until Hadron Epoch, the temperature was so high that quarks would not bind together to form hadrons. A second after Bing Bang, all particles scattered off each other at high rates and electrons would not bind with protons to form an atom. The photons being widely scattered off the electrically charged protons and electrons, could not travel far. Only when the universe expanded and temperature dropped to about 1 billion degree Kelvin, atomic nuclei existed, and at about 3000K, electrons and protons bounded to form a electrically neutral hydrogen atoms. The other charged particles decayed, and universe became transparent for photons to travel longer distances.

In 1929, by analyzing galactic redshifts, Edwin Hubble concluded that galaxies are drifting apart supporting the hypothesis of an expanding universe. In 1964 the Cosmic Microwave Background Radiation (CMBR): the electromagnetic radiation (relic photons from the early universe having a microwave wavelength) left over from the time of recombination in Big Bang cosmology, was discovered favoring the Big Bang model which predicted the existence of background radiation throughout the universe before it was discovered. More recent measurements of the redshifts of supernovae also indicate that the expansion of the universe is accelerating, an observation attributed to the existence of dark energy, an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe, and also the existence dark matter, an invisible and unfamiliar matter which does not emit or interact with electromagnetic radiation. Based on cosmological measurements, Physicists believe that the Standard Model account for only about 4 percent of the mass of the universe, and the rest are dark matter (~ 23%) and dark energy (~73%), as showed in Fig. 9.

Fig. 9: Composition of the Universe - 96 percent invisible and unfamiliar mass-energy (Dark Energy and Dark Matter) [3] 

Higgs Mechanism, the gravity free model based on symmetry breaking scalar field potential, which explains the mass generation of otherwise massless gauge bosons, W+, W−, and Z bosons actually having relatively large masses of around 80 GeV/c2, unifies electromagnetic and weak interactions above the unification energy on the order of 100 GeV. The Large Hadron Collider, a particle accelerator at CERN  (French acronym for European Organization for Nuclear Research) announced results consistent with the Higgs particle (Higgs Boson) on March 14, 2013, making it extremely likely that the Higgs field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

The physicists, Attila Krasznahorkay and his group at the Hungarian Academy of Sciences’s Institute for Nuclear Research in Debrecen, Hungary, in 2015 observed an anomaly in radioactive decay while looking for a dark photon, light that only impacts dark matter. It pointed to the existence of light particle (new boson) just 34 times heavier than electron. In April 2015, a group of theoretical physicists led by Jonathan Feng at the University of California, Irvine (UCI), looked at the results and concluded that it can be a matter particle or force-carrier particle. They found that while the normal electric force acts on electrons and protons, this new found boson, they called X boson, interacts only with electrons and neutrons at an extremely limited range. With the further evidence of  this fifth fundamental force in 2016, research is currently being conducted at CERN and at the Thomas Jefferson National Accelerator Facility in the United States.

References:

[1] Wikipedia

[2] HyperPhysics

[3] Physics for the 21st Century

[4] Feynman Lectures in Physics