Classical to Quantum Mechanics

Blackbody Radiation
Reference: Disturbance Theory

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  1. The Classical Mechanics made a transition into Quantum Mechanics at the beginning of 20th century when the interactions between field and matter were studied. The first field-matter interaction was encountered in the Black Body Radiation. The classical equipartition theory failed to account for the energy of the emitted electromagnetic spectrum.

  2. There was a thermodynamic equilibrium observed between the temperature of the body and the spectrum of the electromagnetic field surrounding the body. In other words, the agitation of atoms (temperature) was in equilibrium with the absorption and emission of thermal electromagnetic radiation (spectrum).

  3. The formulae based on classical thermodynamics could either explain the low frequency part of the spectrum (Raleigh-Jean formula), or the high frequency part of the spectrum (Wien’s Distribution formula), but not the entire spectrum at once. Planck found the formula, which could replicate the entire spectrum by ingeniously interpolating between the above two formulae. This was purely an empirical effort based on mathematics. He came up with the explanation for his formula later.

  4. From Derivation of Planck’s radiation law:

    In order to reproduce the formula which he had empirically derived and presented in October 1900, Planck found that he could only do so if he assumed that the radiation was produced by oscillating electrons, which he modelled as oscillating on a massless spring (so-called “harmonic oscillators”). The total energy at any given frequency would be given by the energy of a single oscillator at that frequency multiplied by the number of oscillators oscillating at that frequency.

    However, he had to assume that

    1. The energy of each oscillator was not related to either the square of the amplitude of oscillation or the square of the frequency of oscillation (as it would be in classical physics), but rather just to the frequency,
      E α ν
    2. The energy of each oscillator could only be a multiple of some fundamental “chunk” of radiation, hν, so En = nhν
      where n = 0, 1, 2, 3, 4
    3. The number of oscillators with each energy En was given by the Boltzmann distribution, so

      Nn = N0e–nhν/kT

      where N0 is the number of oscillators in the lowest energy state.

      By combining these assumptions, Planck was able in November 1900 to reproduce the exact equation which he had derived empirically in October 1900. In doing so he provided, for the first time, a physical explanation for the observed blackbody curve.

  5. The frequency of the radiation matched the frequency of the “oscillators” in the body. The high frequency oscillators could be activated only when energy proportional to their frequency was available. Therefore, lesser numbers of oscillators were activated at higher frequencies. Planck thus resolved the Ultraviolet catastrophe.

  6. We may postulate that the kinetic and potential states of oscillators produce the electric and magnetic states of radiation respectively. Therefore, the electric state may be related to magnetic state the way the kinetic state is related to potential state. The magnetic state could be a concentrated electric state; and the electric state could be a flowing magnetic state.

  7. Thus an electromagnetic cycle consists of a pulse of energy of magnitude ‘h’. A three-dimensional electromagnetic field is made up of such dynamic pulses.

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November 13, 2017 – 7:44 AM Categories: Quantum | Comments (1)

Black-body radiation (Notes)

Black_body
Reference: Disturbance Theory

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Black body

  1. A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

  2. A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation.

  3. The radiation has a spectrum that is determined by the temperature alone, not by the body’s shape or composition.

  4. It is extremely difficult to realize a perfect black body, for which, the absorption of radiation is 100%. Transmission and reflection is zero.

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Thermodynamic equilibrium

  1. In thermodynamic equilibrium all kinds of equilibrium hold at once.

  2. It is characterized by no net macroscopic flows of matter or of energy.

  3. Any microscopic exchanges are perfectly balanced.

  4. The temperature is spatially uniform.

  5. Entropy maximizes with equilibrium.

Thermodynamic state

  • A thermodynamic system is a macroscopic object, the microscopic details of which are not explicitly considered in its thermodynamic description.

Internal energy

  • It excludes the kinetic energy of motion of the system as a whole and the potential energy of the system as a whole due to external force fields.

Boltzmann constant

  • The Boltzmann constant (kB or k), which is named after Ludwig Boltzmann, is a physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas. It is the gas constant R divided by the Avogadro constant NA.

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Equipartition theorem

  1. It relates the temperature of a system to its average energies in thermal equilibrium.

  2. It assumes that energy is shared equally among all of its various modes. For example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

  3. It gives the average values of individual components of the energy, such as, the kinetic energy of a particular particle, or the potential energy of a single spring. For example, it predicts that every atom in a monatomic ideal gas has an average kinetic energy of (3/2) kBT in thermal equilibrium.

  4. When the thermal energy kBT is smaller than the quantum energy spacing in a particular degree of freedom (such as at lower temperatures), the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition.

  5. Such decreases in heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required.

  6. Along with other evidence, equipartition’s failure to model black-body radiation—also known as the ultraviolet catastrophe—led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.

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Rayleigh–Jeans law

  1. The Rayleigh–Jeans law revealed an important error in physics theory of the time.

  2. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity).

  3. Measurements of the spectral emission of actual black bodies revealed that the emission agreed with the Rayleigh–Jeans law at low frequencies but diverged at high frequencies; reaching a maximum and then falling with frequency, so the total energy emitted is finite.

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Ultraviolet catastrophe

  1. The ultraviolet catastrophe was the prediction of classical physics that an ideal black body at thermal equilibrium will emit more energy as the frequency increases.

  2. A blackbody would release an infinite amount of energy, contradicting the principles of conservation of energy.

  3. The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of (1/2)kT. It assumes that vibrating modes can increase infinitely.

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Black-body radiation

  1. Black-body radiation is the thermal electromagnetic radiation within or surrounding a body.

  2. It has a specific spectrum and intensity that depends only on the body’s temperature.

  3. As its temperature increases the peak of the spectrum shifts from infra-red toward higher frequencies of visible light.

  4. Black-body radiation has a characteristic, continuous frequency spectrum.

  5. If each Fourier mode of the equilibrium radiation in an otherwise empty cavity with perfectly reflective walls is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode.

  6. Since there are an infinite number of modes this implies infinite heat capacity (infinite energy at any non-zero temperature), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe.

  7. Instead, in quantum theory the occupation numbers of the modes are quantized, cutting off the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.

Explanation

  1. The radiation from matter represents a conversion of a body’s thermal energy into radiative energy. At thermal equilibrium, matter emits and absorbs radiative substance. The radiative substance has a characteristic frequency distribution that depends on the temperature only.

  2. At thermodynamic equilibrium the amount of every wavelength in every direction of radiative energy emitted by a body at temperature T is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature T.

  3. The black-body curve is characteristic of thermal light, which depends only on the temperature of the body. The principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

  4. By making changes to Wien’s radiation law consistent with thermodynamics and radiation, Planck found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer multiples of some quantity.

  5. Einstein built on this idea and proposed the quantization of radiative energy itself in 1905 to explain the photoelectric effect.

  6. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. These quanta were called photons and the black-body cavity was thought of as containing a gas of photons.

  7. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to a different class of particles, fermions and bosons.

Also see: Classical to Quantum Mechanics

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November 13, 2017 – 6:35 AM Categories: Quantum | Post a comment

Comments on Quantization

feynman-1

Reference: Disturbance Theory

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Quantization – Wikipedia

In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the procedure for building quantum mechanics from classical mechanics. One also speaks of field quantization, as in the “quantization of the electromagnetic field”, where one refers to photons as field “quanta” (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics.

The concept of quantization starts with cycles that make up the field, where each cycle has the same amount of energy. See Energy and Cycle. The phenomena of cycle leads to the quantization of more complex sub-atomic properties observed within the atom. This does not necessarily mean that these properties are completely discrete. However, the “action at a distance” approach explains all atomic and sub-atomic phenomena with mathematical discreteness quite successfully. This approach has given us Quantum Mechanics.

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Quantum – Wikipedia

In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. The fundamental notion that a physical property may be “quantized” is referred to as “the hypothesis of quantization”. This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum.

The hypothesis of quantization is a mathematical one that uses probability statistics. This hypothesis comes from the belief in “action at a distance”.

For example, a photon is a single quantum of light (or of any other form of electromagnetic radiation), and can be referred to as a “light quantum”. Similarly, the energy of an electron bound within an atom is also quantized, and thus can only exist in certain discrete values. The fact that electrons can only exist at discrete energy levels in an atom causes atoms to be stable, and hence matter in general is stable.

A photon represents the energy equivalent of a certain number of cycles taking part in the photoelectric effect. Since each cycle has the same amount of energy, the energy of cycles that take part in this interaction at the sub-atomic level appears to be discrete. This leads to the perception of a discrete energy particle. We call such an energy particle a photon. Similar considerations apply to energy levels observed within an atom and the electrons.

Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of the energy and its influence on how energy and matter interact (quantum electrodynamics) is part of the fundamental framework for understanding and describing nature.

Such quantization is obvious in the interactions between field and matter at sub-atomic and atomic levels, where a mass particle breaks into energy particles called “quanta”.

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November 10, 2017 – 7:16 AM Categories: Quantum | Post a comment

Elementary particle – Wikipedia

Elementary Particle

Reference: Disturbance Theory

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Elementary particle – Wikipedia

In particle physics, an elementary particle or fundamental particle is a particle whose substructure is unknown; thus, it is unknown whether it is composed of other particles. Known elementary particles include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are “matter particles” and “antimatter particles”, as well as the fundamental bosons (gauge bosons and the Higgs boson), which generally are “force particles” that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.

The concept of elementary particles is based on “action at a distance” ideology, which is primarily mathematical. According to this ideology particles have “space” (emptiness) between them that is not filled by any field. On the other hand, in Faraday’s field concept, a particle is simply a high frequency region within the field. There is no emptiness between particles.

Everyday matter is composed of atoms, once presumed to be matter’s elementary particles—atom meaning “unable to cut” in Greek—although the atom’s existence remained controversial until about 1910, as some leading physicists regarded molecules as mathematical illusions, and matter as ultimately composed of energy. Soon, subatomic constituents of the atom were identified. As the 1930s opened, the electron and the proton had been observed, along with the photon, the particle of electromagnetic radiation. At that time, the recent advent of quantum mechanics was radically altering the conception of particles, as a single particle could seemingly span a field as would a wave, a paradox still eluding satisfactory explanation.

The difference between a particle and a wave is that a particle propagates as itself, but a wave requires a medium to propagate in. As matter gets “thinner” it spreads and assumes wave characteristics within itself.

Via quantum theory, protons and neutrons were found to contain quarks—up quarks and down quarks—now considered elementary particles. And within a molecule, the electron’s three degrees of freedom (charge, spin, orbital) can separate via the wavefunction into three quasiparticles (holon, spinon, orbiton). Yet a free electron—which is not orbiting an atomic nucleus and lacks orbital motion—appears unsplittable and remains regarded as an elementary particle.

There are no particles inside the atom, The inconsistencies described above disappear when Faraday’s field concept is considered.

Around 1980, an elementary particle’s status, as indeed elementary—an ultimate constituent of substance—was mostly discarded for a more practical outlook, embodied in particle physics’ Standard Model, what’s known as science’s most experimentally successful theory. Many elaborations upon and theories beyond the Standard Model, including the popular supersymmetry, double the number of elementary particles by hypothesizing that each known particle associates with a “shadow” partner far more massive, although all such superpartners remain undiscovered. Meanwhile, an elementary boson mediating gravitation—the graviton—remains hypothetical.

The whole particle physics is an approximation of the field concept.

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November 9, 2017 – 10:35 PM Categories: Quantum | Post a comment

The Limitation of Einstein’s Theory

disturbance

Reference: Disturbance Theory

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The Electromagnetic Spectrum happens to provide us with a universal range of physical substance. It presents to us a scale of inertia. Matter is a very special condition that exists at the upper end of this scale. The scale starts at the lower end with a theoretical state of emptiness of zero inertia.

The electromagnetic spectrum is a scale of inertia from emptiness to matter.

Like on any scale, the range of physical substance can best be understood from the reference point of zero inertia. However, this may create a great confusion because we are so used to looking at everything from the viewpoint of matter.

Our view of universe as “material” has been very narrow and upside down.

When we consider the substance as a whole range of field along with matter we start to have a much broader view. When we view from the theoretical state of EMPTINESS, instead of matter, the thinking reorients to right side up. We start to see the evolution of physical substance.

The physical substance evolving from emptiness is the correct view.

The fundamental substance appears as disturbance in emptiness.  We postulate this substance to be energy. This energy evolves as the electromagnetic field of increasing frequency. The electromagnetic field may be described in terms of disturbance levels (DL) as base 2 logarithm of frequency. For example, the disturbance level of yellow light is DL 49 because its frequency is 5.8 x 1014 (249) Hz. This makes it possible to conveniently map the whole range of physical substance.

We may map the whole range of physical substance as DISTURBANCE LEVELS.

The disturbance levels on the electromagnetic spectrum may be listed as follows (see appendix below for the method of calculation):

Emptiness …………………………………. 0

Radio Waves (3 Hz – 3 GHz) ……………. 1.6 – 31

Microwaves (3 GHz – 300 GHz) ………… 31 – 38

Infrared (300 GHz – 300 THz) …………… 38 – 48.5

Visible (400 THz – 800 THz) …………….. 48.5 – 49.5

Ultraviolet (800 THz – 30 PHz) ………….. 49.5 – 54.7

X-Rays (30 PHz – 30 EHz) ……………….. 54.7 – 64.7

Gammy Rays (> 30 EHz) …………………. 64.7 and greater

Electron ……………………………………… 66.7

Proton ………………………………………… 77.6

Neutron ………………………………………. 77.6

Earth ………………………………………….. 235.6

Sun ………..………………………………….. 256.6

 

We may now compare this reference point to the reference point used in the theory of relativity by Einstein.

Inertial Frame of Relativity

Einstein borrowed the inertial frame from Galileo and Newton and applied it to Relativity. This frame of reference views light (DL 49) from the reference of matter (DL 138.4 minimum). It is an upside down view.

The inertial frame of relativity views light from the reference of matter.

The speed of light is very close to the universal constant ‘c’, which is essentially a fixed ratio of space to time.  Einstein correctly assumed ‘c’ to be a universal constant. Because of this constant we can treat space-time as a single entity.

Space-time is a single property because time is related to space by ‘c’.

But even as a single entity, space-time scales up and down with disturbance levels, or inertia. The space-time at the level of matter is not the same space-time at the level of light. The inertia of light is many orders of magnitude lower than matter, but it is not zero because it has a disturbance level.

The inertia of light is not zero.

By saying that the “speed of light” is constant in all inertial frames, Einstein is basically assuming that the inertia of light is either zero, or insignifant to the inertial frames based on matter.

The theory of relativity ignores the inertia of light.

There is no doubt that Einstein’s theory of relativity has been very successful, but this success has occurred only where the phenomena has been material (DL > 138.4). For phenomemon of disturbance levels, such as, when considering quantum or electromagnetic phenomena, the inertia of light cannot be ignored.

The inertia of light cannot be ignored for electromagnetic and quantum phenomena.

An approach based on the zero inertia at the lower end of electromagnetic spectrum shall apply to the whole range of phenomena from electromagnetic to quantum to material.

The reference point of zero inertia at the lower end of the electromagnetic spectrum applies universally to the whole range of physical substance.

This is the reference point used by the Disturbance Theory.

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APPENDIX

If the frequency is ‘f’ then the disturbance level is “log f / log 2”.

The frequency associated with a mass object is calculated as follows:

De Broglie Equation,       λ = h/p,

where h is Plank’s constant, and p is momentum

Frequency,                       f = c/λ = (c/h) p = 4.528 x 1041 p

Disturbance level,          DL = (log f) / (log 2) = 138.4 + 3.322 log p

For earth,

ME = 5.972 x 1024 kg, and VE = 3 x 104 m/s

Hence, p = ME VE = 1.79 x 1029

Therefore, DL (earth) = 235.6

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November 7, 2017 – 3:11 PM Categories: Physics | Comments (4)