Gravitational quantum mechanics: a theory for explaining spacetime. This a seminar on several scientific papers about quantum gravity Phenomenology which has been gathered several important outcomes.
The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.
Rethinking the dynamics of an accelerated charge from classical concepts. From the idea that radiation
comes from kinetic energy and managing the problem of the auto-energy of a point charge, a system of
non-linear dynamic equations are found and results amenable to experimental verification. In developing, a
relationship between the principle of causality, which affects the direction of time, and the constancy of
mass appears. Another consequence are the fluctuations in the motion of particles, compatible with
Brownian motion and Heisenberg´s indeterminacy principle. The case of gravitational acceleration is also
analyzed, concluding that no electromagnetic radiation is possible and there is no electric field that can
produce a constant acceleration on a point charge. Thus the constant acceleration is an exclusive feature of
gravity.
Rethinking the dynamics of an accelerated charge from classical concepts. From the idea that radiation
comes from kinetic energy and managing the problem of the auto-energy of a point charge, a system of
non-linear dynamic equations are found and results amenable to experimental verification. In developing, a
relationship between the principle of causality, which affects the direction of time, and the constancy of
mass appears. Another consequence are the fluctuations in the motion of particles, compatible with
Brownian motion and Heisenberg´s indeterminacy principle. The case of gravitational acceleration is also
analyzed, concluding that no electromagnetic radiation is possible and there is no electric field that can
produce a constant acceleration on a point charge. Thus the constant acceleration is an exclusive feature of
gravity.
1) The document analyzes the classical dynamics of an accelerated electric charge, exploring the idea that electromagnetic radiation comes from the charge's kinetic energy.
2) It derives equations for the motion of an accelerated charge while accounting for the possibility of fluctuations in the charge's mechanical mass due to radiation.
3) Applying Poynting's theorem, it shows that the rate of change of electromagnetic energy within a volume around the charge is equal to the rate of change of the charge's kinetic energy plus the outgoing electromagnetic energy flux due to radiation.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
This document discusses the incompatibility between classical mechanics and electromagnetism. It shows that under a Galilean transformation, the wave equation governing electromagnetic waves takes on a different form in different reference frames, violating Galilean invariance. This means that the laws of electromagnetism depend on the choice of reference frame. As such, classical mechanics and electromagnetism cannot be unified without modifications to account for this issue.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.
Rethinking the dynamics of an accelerated charge from classical concepts. From the idea that radiation
comes from kinetic energy and managing the problem of the auto-energy of a point charge, a system of
non-linear dynamic equations are found and results amenable to experimental verification. In developing, a
relationship between the principle of causality, which affects the direction of time, and the constancy of
mass appears. Another consequence are the fluctuations in the motion of particles, compatible with
Brownian motion and Heisenberg´s indeterminacy principle. The case of gravitational acceleration is also
analyzed, concluding that no electromagnetic radiation is possible and there is no electric field that can
produce a constant acceleration on a point charge. Thus the constant acceleration is an exclusive feature of
gravity.
Rethinking the dynamics of an accelerated charge from classical concepts. From the idea that radiation
comes from kinetic energy and managing the problem of the auto-energy of a point charge, a system of
non-linear dynamic equations are found and results amenable to experimental verification. In developing, a
relationship between the principle of causality, which affects the direction of time, and the constancy of
mass appears. Another consequence are the fluctuations in the motion of particles, compatible with
Brownian motion and Heisenberg´s indeterminacy principle. The case of gravitational acceleration is also
analyzed, concluding that no electromagnetic radiation is possible and there is no electric field that can
produce a constant acceleration on a point charge. Thus the constant acceleration is an exclusive feature of
gravity.
1) The document analyzes the classical dynamics of an accelerated electric charge, exploring the idea that electromagnetic radiation comes from the charge's kinetic energy.
2) It derives equations for the motion of an accelerated charge while accounting for the possibility of fluctuations in the charge's mechanical mass due to radiation.
3) Applying Poynting's theorem, it shows that the rate of change of electromagnetic energy within a volume around the charge is equal to the rate of change of the charge's kinetic energy plus the outgoing electromagnetic energy flux due to radiation.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
This document discusses the incompatibility between classical mechanics and electromagnetism. It shows that under a Galilean transformation, the wave equation governing electromagnetic waves takes on a different form in different reference frames, violating Galilean invariance. This means that the laws of electromagnetism depend on the choice of reference frame. As such, classical mechanics and electromagnetism cannot be unified without modifications to account for this issue.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
The document proposes the most general form of deformation of the Heisenberg algebra motivated by the generalized uncertainty principle (GUP). This generalized deformation contains arbitrary fractional powers of momentum and can produce fractional derivative terms in higher dimensions. The document analyzes a specific limit of this deformation for one-dimensional systems, where the Hamiltonian is modified by correction terms scaling as p^3 and p^4. Fractional derivative terms occurring for dimensions greater than one can be handled using the harmonic extension of functions.
1. The document discusses principles of quantum chemistry including classical mechanics and its inadequacies in explaining phenomena at the atomic level, Planck's quantum theory, and properties of electromagnetic radiation.
2. Key concepts covered include de Broglie's equation describing the wave-like nature of matter, Heisenberg's uncertainty principle, explanations of photoelectric effect and blackbody radiation.
3. The document also introduces quantum numbers, Hund's rule, Pauli's exclusion principle, and Aufbau's principle, which describe allowable electron configurations in atoms and molecules.
1. The document discusses key concepts in quantum physics including Planck's quantum theory, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's time-independent wave equation.
2. It provides details on experiments that verified the wave-like properties of matter including electron diffraction experiments by Davisson and Germer.
3. The document derives expressions for the energy levels of particles confined in one-dimensional potential wells and boxes in terms of Planck's constant and other variables.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
This document reviews experimental approaches to analyze spin wave dynamics in ferromagnetic nanoscale structures. It describes recent developments in frequency- and field-swept spectroscopy to determine the resonant response of nanoscale ferromagnets. It also describes time-resolved measurements in the GHz frequency and picosecond time domains to analyze the relaxation of magnetization after microwave excitation. Examples are presented for analyzing and manipulating different mechanisms for the relaxation of magnetization into its ground state.
The chapter contains fundamentals of Modern physics, the Quantumtheory explanation of Black body radiation photoelectric effect and Compton effect, and the beginning of the de-Broglie hypothesis, wave-like properties of matter, and its proof explained in detail. It is highly useful for first-year B.Tech and BE students.
Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
Conversion of the vacuum energy of electromagnetic zero point oscillations in...PublicLeaker
This document provides a summary of a scientific work that proposes a method for converting vacuum energy into classical mechanical energy. It begins with an introduction and outlines the philosophical and theoretical background. It then describes experiments conducted to test a concept of an electrostatic rotor that successfully demonstrated the conversion of vacuum energy into rotational mechanical energy. The document concludes by discussing potential applications and opportunities for future development of the energy conversion method.
Universal constants like Planck's constant h, the speed of light c, gravitational constant G, and Boltzmann's constant k can be used to structure theoretical physics. They lead to three main theories: quantum field theory (h and c), general relativity (c and G), and quantum statistics (h and k). While not fully unified, these theories underlie the standard models of particle physics and cosmology. Fundamental metrology provides reliable standards by determining the values of dimensionless constants that depend on h, c, k, and G. Metrology exists at the intersection of fundamental physics described by these theories and emergent physics involving statistical mechanics.
This document discusses the Schrodinger wave equation for hydrogen atoms. It begins by presenting the time-independent 3D Schrodinger wave equation and explains how it is converted to polar coordinates due to the radial symmetry of hydrogen atoms. The wave function is assumed to separate into three parts, leading to three equations involving the principal, azimuthal, and magnetic quantum numbers. Quantum numbers and their relationships to orbital shapes are also described. Finally, atomic orbitals are defined as regions of high probability of finding electrons based on the Schrodinger wave equation solution.
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
Polarization bremsstrahlung on atoms, plasmas, nanostructures and solidsSpringer
This document discusses the quantum electrodynamics approach to describing bremsstrahlung, or braking radiation, of a fast charged particle colliding with an atom. It derives expressions for the amplitude of bremsstrahlung on a one-electron atom within the first Born approximation. The amplitude has static and polarization terms. The static term corresponds to radiation from the incident particle in the nuclear field, reproducing previous results. The polarization term accounts for radiation from the atomic electron and contains resonant denominators corresponding to intermediate atomic states. The full treatment allows various limits to be taken, such as removing the nucleus or atomic electron, reproducing known results from quantum electrodynamics.
The meaning of quantum mechanics becomes clearer when we restate Planck's constant and the gravitational constant in natural Planck units. These units reveal hidden structure that improves our understanding of physics and gives new meaning to fundamental ideas.
1) The document discusses the classical theory of electromagnetic radiation confined within an isothermal enclosure and the discrepancies with experimental observations.
2) It analyzes the temperature dependence of the energy density and pressure of the radiation using thermodynamic considerations.
3) This leads to the derivation of the Stefan-Boltzmann law relating the emissive power of a black body to the fourth power of its temperature.
4.1 The Atomic Models of Thomson and Rutherford
4.2 Rutherford Scattering
4.3 The Classic Atomic Model
4.4 The Bohr Model of the Hydrogen Atom
4.5 Successes and Failures of the Bohr Model
4.6 Characteristic X-Ray Spectra and Atomic Number
4.7 Atomic Excitation by Electrons
This document provides an overview of quantum mechanics concepts including the Schrödinger wave equation, expectation values, infinite and finite square well potentials, the three-dimensional infinite potential well, the simple harmonic oscillator, barriers and tunneling. Key topics covered include the quantization of energy, boundary conditions, normalization, penetration depth, degeneracy, reflection and transmission probabilities, and an explanation of tunneling using the uncertainty principle. Real world examples of these concepts like alpha particle decay are also discussed.
This document provides an overview of quantum mechanics topics including:
1) The Schrödinger wave equation and its time-dependent and time-independent forms.
2) Expectation values and how they are used to calculate probabilities, momentum, position, and energy.
3) Specific quantum systems like infinite and finite square wells and simple harmonic oscillators. It also discusses quantization, degeneracy, and other concepts.
4) Barrier penetration and tunneling, where particles can pass through barriers that would be forbidden classically.
The document covers many fundamental aspects of quantum mechanics through examining various quantum systems and potentials.
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Publisher’s note: Dr. Beckwith: I am honored that you have seen fit to acknowledge me in your presentations of today, September 3, 2011 at the G999 Conference in Philadelphia, and Friday, September 9, at the San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.
In my view, these are rather extraordinary postulations, especially the probability of extra-universal black hole gravitational origins, along with expansion beyond Hawking of Quantum Wave theory and the “quantizing” of gravity. The latter may very well lead to a thorough reexamination of our concept of space and time, and that the latter might not be so unidirectional afterall. The potential is breathtaking, as it represents steps forward in proving the existence of subspace, the possibility of concomitant multi-universiality and 5D dimensionality of black holes.
Can time manipulation and Biefeld-Brown suggested electro-gravitics as a spacecraft propulsive methodology be far behind? I think not…
This document presents new ideas in loop quantum gravity, including:
1. Deriving a relationship between time and vorticity using wavefunction continuity.
2. Introducing a new "Eikonal constraint" and showing how it removes acausality by gauging time to light cones.
3. Proposing a new form of the Hamiltonian constraint in the style of the Dirac equation, which reduces state fuzziness in line with Penrose's ideas.
The document proposes the most general form of deformation of the Heisenberg algebra motivated by the generalized uncertainty principle (GUP). This generalized deformation contains arbitrary fractional powers of momentum and can produce fractional derivative terms in higher dimensions. The document analyzes a specific limit of this deformation for one-dimensional systems, where the Hamiltonian is modified by correction terms scaling as p^3 and p^4. Fractional derivative terms occurring for dimensions greater than one can be handled using the harmonic extension of functions.
1. The document discusses principles of quantum chemistry including classical mechanics and its inadequacies in explaining phenomena at the atomic level, Planck's quantum theory, and properties of electromagnetic radiation.
2. Key concepts covered include de Broglie's equation describing the wave-like nature of matter, Heisenberg's uncertainty principle, explanations of photoelectric effect and blackbody radiation.
3. The document also introduces quantum numbers, Hund's rule, Pauli's exclusion principle, and Aufbau's principle, which describe allowable electron configurations in atoms and molecules.
1. The document discusses key concepts in quantum physics including Planck's quantum theory, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's time-independent wave equation.
2. It provides details on experiments that verified the wave-like properties of matter including electron diffraction experiments by Davisson and Germer.
3. The document derives expressions for the energy levels of particles confined in one-dimensional potential wells and boxes in terms of Planck's constant and other variables.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
This document reviews experimental approaches to analyze spin wave dynamics in ferromagnetic nanoscale structures. It describes recent developments in frequency- and field-swept spectroscopy to determine the resonant response of nanoscale ferromagnets. It also describes time-resolved measurements in the GHz frequency and picosecond time domains to analyze the relaxation of magnetization after microwave excitation. Examples are presented for analyzing and manipulating different mechanisms for the relaxation of magnetization into its ground state.
The chapter contains fundamentals of Modern physics, the Quantumtheory explanation of Black body radiation photoelectric effect and Compton effect, and the beginning of the de-Broglie hypothesis, wave-like properties of matter, and its proof explained in detail. It is highly useful for first-year B.Tech and BE students.
Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
Conversion of the vacuum energy of electromagnetic zero point oscillations in...PublicLeaker
This document provides a summary of a scientific work that proposes a method for converting vacuum energy into classical mechanical energy. It begins with an introduction and outlines the philosophical and theoretical background. It then describes experiments conducted to test a concept of an electrostatic rotor that successfully demonstrated the conversion of vacuum energy into rotational mechanical energy. The document concludes by discussing potential applications and opportunities for future development of the energy conversion method.
Universal constants like Planck's constant h, the speed of light c, gravitational constant G, and Boltzmann's constant k can be used to structure theoretical physics. They lead to three main theories: quantum field theory (h and c), general relativity (c and G), and quantum statistics (h and k). While not fully unified, these theories underlie the standard models of particle physics and cosmology. Fundamental metrology provides reliable standards by determining the values of dimensionless constants that depend on h, c, k, and G. Metrology exists at the intersection of fundamental physics described by these theories and emergent physics involving statistical mechanics.
This document discusses the Schrodinger wave equation for hydrogen atoms. It begins by presenting the time-independent 3D Schrodinger wave equation and explains how it is converted to polar coordinates due to the radial symmetry of hydrogen atoms. The wave function is assumed to separate into three parts, leading to three equations involving the principal, azimuthal, and magnetic quantum numbers. Quantum numbers and their relationships to orbital shapes are also described. Finally, atomic orbitals are defined as regions of high probability of finding electrons based on the Schrodinger wave equation solution.
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
Polarization bremsstrahlung on atoms, plasmas, nanostructures and solidsSpringer
This document discusses the quantum electrodynamics approach to describing bremsstrahlung, or braking radiation, of a fast charged particle colliding with an atom. It derives expressions for the amplitude of bremsstrahlung on a one-electron atom within the first Born approximation. The amplitude has static and polarization terms. The static term corresponds to radiation from the incident particle in the nuclear field, reproducing previous results. The polarization term accounts for radiation from the atomic electron and contains resonant denominators corresponding to intermediate atomic states. The full treatment allows various limits to be taken, such as removing the nucleus or atomic electron, reproducing known results from quantum electrodynamics.
The meaning of quantum mechanics becomes clearer when we restate Planck's constant and the gravitational constant in natural Planck units. These units reveal hidden structure that improves our understanding of physics and gives new meaning to fundamental ideas.
1) The document discusses the classical theory of electromagnetic radiation confined within an isothermal enclosure and the discrepancies with experimental observations.
2) It analyzes the temperature dependence of the energy density and pressure of the radiation using thermodynamic considerations.
3) This leads to the derivation of the Stefan-Boltzmann law relating the emissive power of a black body to the fourth power of its temperature.
4.1 The Atomic Models of Thomson and Rutherford
4.2 Rutherford Scattering
4.3 The Classic Atomic Model
4.4 The Bohr Model of the Hydrogen Atom
4.5 Successes and Failures of the Bohr Model
4.6 Characteristic X-Ray Spectra and Atomic Number
4.7 Atomic Excitation by Electrons
This document provides an overview of quantum mechanics concepts including the Schrödinger wave equation, expectation values, infinite and finite square well potentials, the three-dimensional infinite potential well, the simple harmonic oscillator, barriers and tunneling. Key topics covered include the quantization of energy, boundary conditions, normalization, penetration depth, degeneracy, reflection and transmission probabilities, and an explanation of tunneling using the uncertainty principle. Real world examples of these concepts like alpha particle decay are also discussed.
This document provides an overview of quantum mechanics topics including:
1) The Schrödinger wave equation and its time-dependent and time-independent forms.
2) Expectation values and how they are used to calculate probabilities, momentum, position, and energy.
3) Specific quantum systems like infinite and finite square wells and simple harmonic oscillators. It also discusses quantization, degeneracy, and other concepts.
4) Barrier penetration and tunneling, where particles can pass through barriers that would be forbidden classically.
The document covers many fundamental aspects of quantum mechanics through examining various quantum systems and potentials.
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Publisher’s note: Dr. Beckwith: I am honored that you have seen fit to acknowledge me in your presentations of today, September 3, 2011 at the G999 Conference in Philadelphia, and Friday, September 9, at the San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.
In my view, these are rather extraordinary postulations, especially the probability of extra-universal black hole gravitational origins, along with expansion beyond Hawking of Quantum Wave theory and the “quantizing” of gravity. The latter may very well lead to a thorough reexamination of our concept of space and time, and that the latter might not be so unidirectional afterall. The potential is breathtaking, as it represents steps forward in proving the existence of subspace, the possibility of concomitant multi-universiality and 5D dimensionality of black holes.
Can time manipulation and Biefeld-Brown suggested electro-gravitics as a spacecraft propulsive methodology be far behind? I think not…
This document presents new ideas in loop quantum gravity, including:
1. Deriving a relationship between time and vorticity using wavefunction continuity.
2. Introducing a new "Eikonal constraint" and showing how it removes acausality by gauging time to light cones.
3. Proposing a new form of the Hamiltonian constraint in the style of the Dirac equation, which reduces state fuzziness in line with Penrose's ideas.
Excitons, lifetime and Drude tail within the current~current response framew...Claudio Attaccalite
We compare the optical absorption of extended systems calculated starting from the density-density and current-current linear response formalisms within the equilibrium many-body perturbation theory(MBPT). We show how, using the latter, one can incur in errors due to quasiparticle lifetimes, electron-hole interaction or the presence of a Drude tail. We present a solution for each one of these problems.
This document discusses 6-D cooling techniques needed for a muon collider. It begins by outlining the physics motivation for a muon collider, particularly for studying the Higgs boson. It then discusses the advantages and challenges of using muons, including their short lifetime and diffuse initial phase space. It introduces concepts of phase space distribution and 6-D cooling using canonical coordinates. Design objectives like luminosity requirements necessitate reducing the 6-D emittance by around 106. Ionization cooling is proposed, which uses energy loss in absorbers and RF cavities to cool muons transversely and longitudinally. Key concepts like stopping power, ionization energy loss, and stochastic effects are covered.
Part III Essay: Could the graviton have a mass?Yiteng Dang
This document discusses theories of massive gravity. It begins by introducing linearized general relativity and the linear Fierz-Pauli theory of a massive spin-2 particle. It then discusses three main challenges of constructing a theory of massive gravity: the van Dam-Veltman-Zakharov discontinuity, the presence of a ghost field, and issues with renormalizability. It reviews proposed solutions to these challenges, including the Vainshtein mechanism and de Rham-Gabadadze-Tolley construction. While these approaches resolve some of the issues, the document notes there remain unresolved problems with developing a complete theory of massive gravity.
Quantization of photonic energy and photonic wave lengthEran Sinbar
The document proposes that if space is quantized at the Planck length, then photonic energy and wavelength must also be quantized. It suggests that future experiments could detect these quantization levels in cosmic radiation or particle collisions. It also puts forward a "grid dimensions" theory that proposes extra non-local dimensions between Planck length pieces of space that could explain quantum non-local effects like entanglement. Key equations presented quantify proposed quantized limits for momentum, mass, velocity of particles if space-time is quantized.
The document describes a proposed Gravitational Spacecraft that could change space flight. It summarizes that electromagnetic fields can reduce, invert, or intensify gravitational forces by controlling gravitational mass. This allows concepts like a Gravitational Spacecraft that could have unusual trajectories without inertial impacts, as well as a Gravitational Motor that converts gravitational energy into rotation or electricity for free energy generation. Key devices mentioned are Gravity Control Cells and a Quantum Transceiver for communications.
Boris Stoyanov - Covariant and Consistent Anomalies in Gauged SupergravityBoris Stoyanov
This document summarizes properties of gauged supergravity models in four and six dimensions. It discusses covariant and consistent anomalies that arise in these theories, and how their cancellation constrains the types of fields that can be coupled to supergravity. It also presents the Pasti-Sorokin-Tonin construction for obtaining covariant actions for self-dual fields using an auxiliary scalar field. Various aspects of anomalies, field equations, and couplings in minimal supergravity theories are analyzed in detail.
This document summarizes a study on quantum phase transitions in small spinless fermion systems and their comparison to infinitely large systems. The study evaluates a one-dimensional model of spinless fermions with nearest neighbor interactions. For small system sizes, certain evolutionary behaviors were observed as the number of sites increased, though no direct phase transitions could be analyzed due to limitations of computer capabilities. The document outlines the model, methodology using Hamiltonian matrices, and presents some initial results on systems with interaction strength U=0.
Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.
The problem of radiation reaction in classical electrodynamics arises due to divergences in describing point particles, but Wheeler-Feynman electrodynamics provides a divergence-free theory by describing direct interactions between charged particles without fields, though it is difficult to solve due to its time-symmetric nature requiring knowledge of global trajectories.
General Relativity is inconsistent with quantum theory which
leaves our understanding of nature incomplete and unsatisfactory. The now 80 years old quest for a consistent theory of quantum gravity has so far almost entirely focused on mathematical consistency. But as of recently the possibility to look for observational evidence has received an increased amount of attention, as a tool to provide guidance for the construction of of the theory.
Here, I summarize recent developments in the search for
experimental signatures for quantum gravitational effects and how these help to put constraints on the theory-construction. Some of the topics that I will cover are the prospects of finding Planck scale effects in gamma ray bursts, in neutral Kaon oscillations, or with massive quantum oscillators. If time allows, I will also comment on the search for holographic noise and how to find evidence for space-time discreteness.
Quantum mechanics provides a mathematical description of the wave-particle duality of matter and energy at small atomic and subatomic scales. It differs significantly from classical mechanics, as phenomena such as superconductivity cannot be explained using classical mechanics alone. Key aspects of quantum mechanics include wave-particle duality, the uncertainty principle, and discrete energy levels determined by Planck's constant and frequency.
1) The document summarizes the 100-year history of gravitational wave detection, from Einstein's theory of general relativity predicting their existence to the recent direct detection by the LIGO experiment.
2) It describes how LIGO uses laser interferometry to extremely sensitively measure tiny distortions in spacetime caused by passing gravitational waves.
3) The first detected gravitational waves in 2015 matched predictions for the inspiral and merger of two black holes, with the signal analyzed to determine properties of the black holes such as their masses.
Vasil Penchev. Gravity as entanglement, and entanglement as gravityVasil Penchev
1. The document discusses interpreting gravity as entanglement by investigating the conditions under which general relativity and quantum mechanics can be mapped to each other mathematically.
2. It outlines a strategy to interpret entanglement as inertial mass and gravitational mass, and to view gravity as another interpretation of any quantum mechanical or mechanical movement.
3. This would allow gravity to be incorporated into the standard model by generalizing the concept of quantum field to include entanglement, represented by a cyclic Yin-Yang mathematical structure.
Electromagnetically induced transparency (EIT) occurs in a three-level atomic system when a strong coupling laser drives a resonant transition, reducing absorption of a weak probe laser resonant with another transition that shares a common state. EIT was first observed experimentally in 1991 using strontium vapors. The dynamics of EIT in a rubidium-85 system are modeled using a ladder-type three-level system and solving the time-dependent density matrix equations to find the complex susceptibility, giving dispersion and absorption properties. EIT has applications such as lasing without inversion, slowing light, quantum memory, and quantum information processing.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
2. Content
The Quest for Quantum Gravity, Phenomenology
Examples of Phenomenological Models for Quantum
Gravity
Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
2
3. Content
Some Fundamental Features of this Gravitational Quantum Mechanics
Kernel Functions
• Particle-like Approach
• Wave-like Approach
Generalized Feynman Path Integrals
• Particle-like Approach
• Wave-like Approach
Generalized Feynman Path Integral and Some Thermodynamical Properties of an Ideal Gas
3
4. The Quest for Quantum
Gravity, Phenomenology
An introduction
4
5. The Quest for Quantum Gravity, Phenomenology
General theory of relativity
The standard model of particle
physics
General relativity has so far refused
to be quantized!
An entirely classical theory
The result of its quantizing is non-
renormalizable
There are three main reasons why
the present status requires a
solutions:
Superposition states
Singularities
The black hole information loss
problem
5
This does not mean that one necessarily
obtains quantum gravity by quantizing
gravity.
6. The Quest for Quantum Gravity, Phenomenology
Quantum particles can exist
in superposition states.
• These particles
respectively, carry energy
and thus gravitate.
• What their gravitational field
is: as a classical field it does
not exist in superpositions.
6
We don’t know what is the
gravitational field of a quantum
superposition.
7. The Quest for Quantum Gravity, Phenomenology
General relativity predicts
the formation of
singularities.
• Infinite energy density and
gravitational forces
• Unphysical and signal a
breakdown of the theory
• Requiring a more
fundamental theory!
7
General relativity predicts its
own breakdown: Singularities
8. The Quest for Quantum Gravity, Phenomenology
The black hole information loss
problem
• Black holes emit thermal radiation
• Any distribution with the same initial
mass that collapsed to a black hole
would eventually be converted into
the same thermal final state
• Detailed information contained in the
initial configuration would have
gotten lost
• Incompatible with quantum
mechanics
8
Black holes seem to destroy
information, and we don’t know how
that is compatible with quantum
mechanics.
9. The Quest for Quantum Gravity, Phenomenology
Planck scale
The scale at which effects of
quantum gravity are expected to
become relevant
Energy, length and time
To estimate the Planck scale:
An amount of energy, 𝐸, in a
volume of size ∆𝑥3
Via Einstein’s field equations:
Consider the energy to be localized
as good as quantum mechanics
possibly allows us
To its Compton wavelength:
This distortion will become non-
negligible when
9
10. The Quest for Quantum Gravity, Phenomenology
This mass scale, which
corresponds to the Planck mass
and the related Compton
wavelength, the Planck length:
10
11. The Quest for Quantum Gravity, Phenomenology
Quantum gravity: any approach
that is able to resolve the
apparent tension between
general relativity and quantum
field theories, and to address
the three problems mentioned
above.
11
We will refer as ‘quantum gravity’ to any
attempted solution of these problems.
12. Phenomenological Models
A phenomenological model:
An extension of a known theory
that allows one to compute effects
of specific additional features
Bridges the gap between theory
and experiment
It guides the development of the
theory by allowing to test general
properties
Not itself a fundamental theory
and therefore not entirely
self-contained
12
The first requirement for a good
phenomenological model is of course that it
be in agreement with already available data
and that it be internally consistent.
14. Examples of Phenomenological Models for Quantum Gravity
Violations of Lorentz Invariance
Does the ground state of spacetime
obey Lorentz-invariance?
If Lorentz-invariance was broken,
observer-independence would be
explicitly violated and a preferred
frame would be singled out.
The matter sector of the standard
model or the gravitational sector
In the purely gravitational sector:
by coupling a tensor field, or its
derivatives respectively, to the
metric or the curvature
Einstein-Aether theory
In the standard model sector:
additional terms to the standard
model Lagrangian breaking the
symmetry
Modified dispersion relations
14
15. Examples of Phenomenological Models for Quantum Gravity
Deformations of Special Relativity
The possibility that special relativity
may be modified in the high energy
regime
The modified Lorentz-
transformations in momentum
space have two invariants: the
speed of light and the Planck mass
Results:
Generalized uncertainty
Curved momentum space
Modified dispersion relation
An energy-dependent speed of
light
15
16. Examples of Phenomenological Models for Quantum Gravity
Minimal Length and Generalized
Uncertainty
There are several thought
experiments for probing smallest
distances that lead one to conclude
the non-negligible perturbations of
the background geometry close by
the Planck scale
Incorporating a minimal length into
quantum mechanics and quantum
field theory results in:
Generalized uncertainty principle
Prevents an arbitrarily good
localization in position space
Modified dispersion relation
Modified measure in momentum
space
Can be understood as a non-trivial
geometry of momentum space
May or may not have an energy-
dependent speed of light
Break Lorentz-invariance explicitly
16
17. Examples of Phenomenological Models for Quantum Gravity
Causal Sets
Space-time foam and granularity
Geometrogenesis
Loop Quantum Cosmology
String Cosmology
The Arkani-Hamed-Dimopoulos-Dvali Model
The Randall-Sundrum Model
17
20. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
Heisenberg uncertainty principle:
From the Heisenberg’s Electron
Microscope Gedanken Experiment
But he had disregarded the
gravitational interaction of the photons
with electron
Gravity is coupled to everything
20
21. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
The more carefully probing electron position the more
energetic photons so, the more gravitational interaction
Planck scale
So, an extra position uncertainty due to the gravitational
contribution of very high energy beams:
Then HUP turns into GUP:
21
A nontrivial assumption:
The minimal length = A nonzero
uncertainty in position measurement
22. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
The common form of GUP in the
presence of minimal length:
The modified Heisenberg algebra:
22
23. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length & Minimal
Momentum
The existence of minimum position
and minimum momentum
uncertainty leads to:
And a phase space commutator in
one dimension of the form:
23
24. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length & Maximal Momentum
The modified Heisenberg algebra in DSR theories:
By asymptotic expression to first order:
Corresponding Heisenberg algebra for this type of GUP:
24
A minimal measurable
length leads to a maximum
measurable momentum of
the order of the Planck
momentum which is an
upper bound on test particle
momentum
25. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Explicit Maxima and Minima in both
Position and Momentum Uncertainties
A GUP of the form
25
26. Some Fundamental
Features of a
Gravitational Quantum
Mechanics
Some basic notions of a gravitational quantum
mechanics
26
27. Some Fundamental Features of a Gravitational Quantum
Mechanics
Define the operators P and X:
The generalized identity operator:
The generalization of the scalar
product of momentum eigenstates:
A maximal momentum cutoff
the bounds of integrals over p
change from to
Position space representation
should be modified
Quasi-position space:
27
28. Some Fundamental Features of a Gravitational Quantum
Mechanics
Standard quantum mechanics
Absolute localization = Absolute
accuracy in the position
measurements
Commutative spacetime
With effects of gravity in the
ultraviolet regime
Maximally localized up to the
minimal length
Non-commutative structure of
spacetime:
We can not build a Hilbert space
on the position space wave
functions
because there is no longer a zero
uncertainty in position
There can not be any physical state
as a position eigenstate
Ordinary position space
representation is no longer
applicable
28
29. Some Fundamental Features of a Gravitational Quantum
Mechanics
Quasi-position space representation
Maximal localization states
Localizability limited to the minimum measurable distance
Momentum space wave functions of the states that are maximally localized:
29
30. Some Fundamental Features of a Gravitational Quantum
Mechanics
By projecting arbitrary states on maximally localized states, we can define the
state’s quasi-position wave function
probability amplitude for the particle being maximally localized around x
The transformation of a momentum space wave function into a quasi-position
space wave function:
30
31. Some Fundamental Features of a Gravitational Quantum
Mechanics
the transformation of a quasi-position space wave function into a momentum
space wave function:
Modified wave number in quasi-position space:
31
32. Some Fundamental Features of a Gravitational Quantum
Mechanics
Modified wave length in quasi-position space:
From the de Broglie relation:
Modified form of the frequency in the quasi-position space
32
33. Some Fundamental Features of a Gravitational Quantum
Mechanics
Modified Planck relation in quasi-position space:
Quasi-energy
Responsible for the time evolution of quasi-position wave functions
Time-dependent wave function in quasi-position space evolves as
33
35. Kernel Functions
A general definition of kernel
functions K
Standard quantum mechanics, a
definition of propagator:
Probability amplitude to find a
point particle in the position 𝑥𝑓 at
the time 𝑡𝑓, while initially it was in
the position 𝑥𝑖 at the time 𝑡𝑖
In the quasi-position formalism:
Generalized form of propagator in
the gravitational quantum
mechanics
Using identity operator:
35
36. Kernel Functions
The modified Kernel function depends on the form of energy 𝐸 𝑝
Particle-like
Wave-like
It is possible to construct a path integral even in the wave-like approach due to
the presence of natural cutoffs
36
37. Kernel Functions: Particle-like Approach
With the modified form of the energy inspired by the quasi-momentum
By some calculations (!!!)
37
38. Kernel Functions: Wave-like Approach
With the modified form of the Planck relation:
An explicit relation for wave-like kernel function due to existence of these cutoffs
38
40. Generalized Feynman Path Integrals
The kernel function K can provide a new formulation which was presented by
Feynman; the path integral formalism:
40
41. Generalized Feynman Path Integrals
Modifications of Feynman path integral in the Planck scale
By using the maximally localized states:
We see that each expression is of the form:
Again we have particle-like and wave-like approaches
41
42. Generalized Feynman Path Integrals: Particle-like Approach
By using of the generalized energy relation with respect to quasi-momentum:
An important outcome:
42
43. Generalized Feynman Path Integrals: Particle-like Approach
Generalized form of the Feynman path integral in the presence of these cutoffs in
a gravitational quantum mechanics:
43
44. Generalized Feynman Path Integrals: Wave-like Approach
By using of the generalized energy relation with respect to quasi-energy:
Feynman path integral of a free particle in the wavelike approach where
quantum gravitational effects
44
45. Generalized Feynman
Path Integral and Some
Thermodynamical
Properties of an Ideal
Gas
Using of generalized path integral to modify
Thermodynamical parameter
45
46. Generalized Feynman Path Integral and Some
Thermodynamical Properties of an Ideal Gas
Partition function of n-particle systems in position space
The connection between the Euclidean path integral and statistical mechanics
Replacing the time variable t with the Euclidean time τ
A modified partition function from a modified path integral:
46
47. Generalized Feynman Path Integral and Some
Thermodynamical Properties of an Ideal Gas
The modified partition function for a bosonic ideal gas N particles:
47
49. Chemical Potential
Chemical Potential:
49
The solid thick curve
represents the
chemical potential in
the classical
theory
50. Entropy
Using the expression for the free energy:
Entropy of an ideal gas in the presence of natural cutoffs:
50
51. Specific Heat
Specific heat in a constant volume:
51
The solid thick curve
represents the
chemical potential in
the classical
theory
52. Internal Energy
Having the expressions for the free energy and entropy:
Internal energy U in the presence of natural cutoffs:
The value of internal energy of an ideal gas is not divergent
52
آزمایش میکروسکوپ هایزنبرگ که دیدن یک الکترون با تاباندن نور به دورن محفظهی آن است و به دلیل داشتن خطا در اندازهگیری مکان و تکانهی الکترون به رابطه عدم قطعیت میٰرسیم. ولی گرانش به همه چیز جفت شده است و پس یه ترم به دلیل برهمکنش گرانشی فوتون و الکترون باید اضافه شود.
ایکس بزرگ عملگر مکان در انرژیهای بالا و ایکس کوچیک عملگر مکان در انرژیهای کم است. چون مفهوم نقطه نداریم و عدم قطعیت کمینه داریم نمایش فضای مکان نداریم. ما به جای فضای مکان از فضای شبه-مکان استفاده میکنیم که بازتعریفی از فضای مکان است.
تابع موج هر فضا تابعی است که دارای عدم قطعیت صفر در مولفهی همان فضا است.
چشم داشتی تکانه صفر است.
یک تبدیل فوریه است.
در تبدیل عکس فوریه، ای به توان آی کا ایکس را داشتیم و پس کا به دست میآید.
از دو رابطهی تبدیلات و جایگذاری در این رابطهی آخر و استفاده از معادلهی پایین خواهیم داشت:
از رابطهی پی به توان تقسیم بر ۲ ام استفاده میکنیم و به جای انرژی میگذاریم.
در آلفا به سمت به رابطه عادی ابتدا میرسیم. میتوان رابطهی آخر را بسط داد...
رابطهی آخر به دو بخش موهومی و حقیقی با استفاده از رابطهی اویلر تبدیل میشود.
در اینجا به عنوان نتیجهای دیگر از وجود برشهای طبیعی، در این نظریه ما بر خلاف کوانتوم عادی، تابع کرنل موج-گونه داریم.
So, unlike the ordinary quantum mechanics, we were ableto construct a path integral in the wave-like approach thanks to the presence of natural cutoffs.This is another new result due to the presence of natural cutoffs.
We consider an ensemble system at thermodynamical equilibriumwith energy spectrum of microstates as {En}
Note also thatthe classical, non-deformed chemical potential is negative at sufficiently high temperature. Inthe presence of natural cutoffs this is not the case and chemical potential is always positive.Positivity of the chemical potential means that the change in Helmholtz free energy when aparticle is added to the system (here the ideal gas) is positive.
In the presence of natural cutoffs and for small valuesof β, the specific heat is not constant rather asymptotically increases to predicted value of theclassical model in low temperature.