Phase transition and the Casimir effect are studied in the complex scalar field with one spatial dimension to be compactified. It is shown that the phase transition is of the second order and the Casimir effect behaves quite differently
depending on whether it’s under periodic or anti-periodic boundary conditions
This document provides exercises on Hawking radiation using scalar field theory in the Kruskal spacetime. It asks the student to find the radial equation for the scalar field and show that near the horizon, the field takes the form of ingoing and outgoing waves that are analytic in different coordinate systems. The document then derives the Klein-Gordon equation in Schwarzschild coordinates and uses separation of variables to obtain approximate solutions near the horizon. It shows that ingoing waves are regular on the future horizon but outgoing waves are not, and vice versa for the past horizon.
1) The document discusses the mathematical analysis of a basic AC circuit consisting of a resistor and inductor connected in series and driven by an external sinusoidal voltage source.
2) Kirchhoff's voltage law is applied to derive the differential equation governing the circuit and the forced steady-state response is shown to be a sinusoidal current lagging the driving voltage by a phase angle.
3) Expressions are derived relating the phase lag to the circuit properties and defining the real and reactive power consumed based on the circuit response.
This document is a physics problem set from MIT's 8.044 Statistical Physics I course in Spring 2004. It contains 5 problems related to statistical physics and probability distributions. Problem 1 considers the probability distribution and properties of the position of a particle undergoing simple harmonic motion. Problem 2 examines the probability distribution of the x-component of angular momentum for a quantum mechanical system. Problem 3 analyzes a mixed probability distribution describing the energy of an electron. Problem 4 involves finding and sketching the time-dependent probability distribution for the position of a particle given its wavefunction. Problem 5 concerns Bose-Einstein statistics and calculating properties of the distribution that describes the number of photons in a given mode.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
This document discusses statistical mechanics and the distribution of energy among particles in a system. It provides 3 main types of statistical distributions based on the properties of identical particles: Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Maxwell-Boltzmann statistics applies to distinguishable particles, while Bose-Einstein and Fermi-Dirac apply to indistinguishable particles (bosons and fermions respectively), with the key difference being that fermions obey the Pauli exclusion principle. The document also discusses applications of these distributions, including the Maxwell-Boltzmann distribution law for molecular energies in an ideal gas.
This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could help solve issues in the Standard Model such as the hierarchy problem. SUSY introduces new "quantum" dimensions beyond the usual 3 spatial and 1 time dimension. SUSY generators called Q transform fermions into bosons and vice versa. The SUSY algebra involves the generators Q satisfying anticommutation relations in addition to the usual commutation relations of generators like momentum P and angular momentum M. While experimental evidence for SUSY is still lacking, it is an attractive theoretical idea that may be discovered at energy scales below 1 TeV.
This document provides exercises on Hawking radiation using scalar field theory in the Kruskal spacetime. It asks the student to find the radial equation for the scalar field and show that near the horizon, the field takes the form of ingoing and outgoing waves that are analytic in different coordinate systems. The document then derives the Klein-Gordon equation in Schwarzschild coordinates and uses separation of variables to obtain approximate solutions near the horizon. It shows that ingoing waves are regular on the future horizon but outgoing waves are not, and vice versa for the past horizon.
1) The document discusses the mathematical analysis of a basic AC circuit consisting of a resistor and inductor connected in series and driven by an external sinusoidal voltage source.
2) Kirchhoff's voltage law is applied to derive the differential equation governing the circuit and the forced steady-state response is shown to be a sinusoidal current lagging the driving voltage by a phase angle.
3) Expressions are derived relating the phase lag to the circuit properties and defining the real and reactive power consumed based on the circuit response.
This document is a physics problem set from MIT's 8.044 Statistical Physics I course in Spring 2004. It contains 5 problems related to statistical physics and probability distributions. Problem 1 considers the probability distribution and properties of the position of a particle undergoing simple harmonic motion. Problem 2 examines the probability distribution of the x-component of angular momentum for a quantum mechanical system. Problem 3 analyzes a mixed probability distribution describing the energy of an electron. Problem 4 involves finding and sketching the time-dependent probability distribution for the position of a particle given its wavefunction. Problem 5 concerns Bose-Einstein statistics and calculating properties of the distribution that describes the number of photons in a given mode.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
This document discusses statistical mechanics and the distribution of energy among particles in a system. It provides 3 main types of statistical distributions based on the properties of identical particles: Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Maxwell-Boltzmann statistics applies to distinguishable particles, while Bose-Einstein and Fermi-Dirac apply to indistinguishable particles (bosons and fermions respectively), with the key difference being that fermions obey the Pauli exclusion principle. The document also discusses applications of these distributions, including the Maxwell-Boltzmann distribution law for molecular energies in an ideal gas.
This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could help solve issues in the Standard Model such as the hierarchy problem. SUSY introduces new "quantum" dimensions beyond the usual 3 spatial and 1 time dimension. SUSY generators called Q transform fermions into bosons and vice versa. The SUSY algebra involves the generators Q satisfying anticommutation relations in addition to the usual commutation relations of generators like momentum P and angular momentum M. While experimental evidence for SUSY is still lacking, it is an attractive theoretical idea that may be discovered at energy scales below 1 TeV.
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
This document describes a procedure for obtaining maximum likelihood estimates of parameters for a generalized exponential distribution of earthquake intensity and magnitude. It presents the maximum likelihood parameter values for initial and extremal distributions of earthquakes in Zagreb, Croatia. Specifically:
- It describes an iterative procedure to determine the maximum likelihood values of the parameters a', k', and xo' that characterize the initial distribution, and parameters a, k, and xo that characterize the extremal distribution.
- When applied to earthquake intensity data from Zagreb, the procedure yields maximum likelihood parameter estimates of k' = 0.23820, a' = 1.9387, xo' = 1 for the initial distribution, and k = 0.
- The Stern-Gerlach experiment in 1922 showed that a beam of silver atoms passed through an inhomogeneous magnetic field split into two beams, providing early evidence that angular momentum is quantized.
- The development of quantum mechanics helped explain phenomena like the fine structure of hydrogen emission spectra, but failed to account for observed splittings until the concept of intrinsic "spin" angular momentum was introduced.
- Angular momentum operators like Jx, Jy, and Jz are defined based on classical angular momentum expressions, with momentum terms replaced by operators involving partial derivatives. These operators obey specific commutation relationships and do not commute with one another.
This document discusses statistical thermodynamics and the partition function. It introduces the concept of microscopic configurations and their weights. The Boltzmann distribution relates the probability of a configuration to its weight, which depends on the energy levels and temperature. The partition function allows calculating thermodynamic properties like internal energy, entropy, and heat capacity from knowledge of the energy levels and degeneracies alone. It provides a statistical mechanical approach to thermodynamics.
This document introduces key concepts in statistical mechanics, including the idea that macroscopic properties are thermal averages of microscopic properties. It discusses common statistical ensembles like the microcanonical ensemble (isolated systems with constant energy) and the canonical ensemble (systems in equilibrium with a heat reservoir). The canonical partition function Z relates microscopic quantum mechanics to macroscopic thermodynamics and can be used to calculate thermodynamic variables. Properties like heat capacity can be derived from fluctuations in energy calculated from the partition function.
This document provides an overview of field theory concepts related to constructing a toy Standard Model using an SU(2) x U(1) gauge symmetry. It discusses how scalar and spinor fields can be incorporated into a Lagrangian that respects this symmetry. It describes how the gauge fields transform under subgroups like the diagonal subgroup, and how this relates to the masses of the Z and W bosons. It also discusses the Higgs potential and how it gives mass to the Higgs boson while the Goldstone bosons are eliminated via a gauge condition.
The document describes three models of photons with physical extent beyond the traditional point particle model: a KdV particle, a normal probability classical packet, and a sinc function quantum packet. The sinc function model is identified as most suitable, describing a photon peaked at its origin that converges to ±∞. In this model, the photon has a disk shape with radii ranging from 10-17m for gamma rays to unlimited sizes for long radio wavelengths. The photon is proposed to have internal magnetic fields and a possible rest mass upper limit of 2×10-69kg.
Sweeping discussion on_dirac_fields_securedfoxtrot jp R
- The document presents a Lagrangian for a basic electron theory that can be outlined from an SU(2)XU(1) construction, excluding additional fermions of the full electroweak theory.
- The Lagrangian includes an effective Lagrangian for the electron and an interaction Lagrangian for the left-handed electron and left-handed neutrino.
- Various forms of the Lagrangian and action are presented and manipulated using integration by parts, arriving at Dirac's first order equation of motion for the electron spinor.
1) The document discusses solving the differential equation R(r) with vanishing mass M and angular frequency ω by transforming it into Legendre form.
2) Two regular singular points are identified that correspond to physical radii, and a power series solution is proposed.
3) Recurrence relations are derived for the coefficients in the power series, allowing one solution to be written as a Legendre polynomial.
1) The document reviews Seiberg-Witten duality by first discussing N=2 supersymmetric Yang-Mills (SYM) theory and the topics needed to understand it, such as SUSY algebra, massless multiplets, massive multiplets, chiral and vector superfields, and N=1 SYM.
2) It then briefly discusses Olive-Montonen duality from 1977 before reviewing Seiberg-Witten duality from 1994.
3) The objective is to work out the form of the low energy effective action for N=2 SYM theory, which involves finding the prepotential term.
To the Issue of Reconciling Quantum Mechanics and General RelativityIOSRJAP
The notion of gravitational radiation as a radiation of the same level as the electromagnetic radiation is based on theoretically proved and experimentally confirmed fact of existence of stationary states of an electron in its gravitational field characterized by the gravitational constant K = 1042G (G is the Newtonian gravitational constant) and unrecoverable space-time curvature Λ. If the numerical values of K 5.11031 Nm2 kg-2 and =4.41029 m -2 , there is a spectrum of stationary states of the electron in its own gravitational field (0.511 MeV ... 0.681 MeV).Adjusting according to the known mechanisms of broadening does not disclose the broadening of the registered portion of the emission spectrum of the micropinch. It indicates the presence of an additional mechanism of broadening the registered portion of the spectrum of the characteristic radiation due to the contribution of the excited states of electrons in their own gravitational field. The energy spectrum of the electron in its own gravitational field and the energy spectra of multielectron atoms are such that there is a resonance of these spectra. As obvious, the consequence of such resonant interaction is appearance, including new lines, of electromagnetic transitions not associated with atomic transitions. The manuscript is the review of previously published papers cited in the references.
Sweeping discussion on_dirac_fields_update1foxtrot jp R
(1) The document presents a Lagrangian for a basic electron theory using a Dirac field ψ. The Lagrangian includes an effective Lagrangian ffeL )( 2ψ for the electron and an interaction Lagrangian tni
LL
L ),( 21 ψψ for the left-handed neutrino and electron.
(2) The action for the effective electron theory can be written as an action for the adjoint spinor field ψ using integration by parts. Variations of the two actions are equivalent imposing boundary conditions.
(3) Varying the action with respect to the adjoint spinor ψ yields the Dirac equation of motion, the first order Euler-Lagrange equation for ψ.
Uniformity of the Local Convergence of Chord Method for Generalized EquationsIOSR Journals
This document summarizes research on the uniform convergence of the Chord method for solving generalized equations. The Chord method is an iterative method for finding solutions to equations of the form y ∈ f(x) + F(x), where f is a function and F is a set-valued mapping. The authors prove that under certain conditions, including F being pseudo-Lipschitz and the derivative of f being continuous, the Chord method converges uniformly for small variations in the parameter y. They obtain this result in two different ways. The document also provides relevant definitions and preliminaries on generalized equations, set-valued mappings, and convergence properties.
This chapter discusses the theory of angular momentum in quantum mechanics and its applications. Eigenvectors of the angular momentum operator J satisfy certain eigenvalue equations involving the quantum numbers j and m. Specific cases of spin-1/2 and spin-1 systems are then derived. The chapter covers topics like coupling of angular momentum systems and angular momentum matrix elements.
This chapter discusses spin in strong interactions like pion-nucleon and nucleon-nucleon scattering. It introduces the density matrix and reaction matrix to describe mixed spin states. The density matrix is determined by the mean values of spin operators and completely characterizes the spin state. The reaction matrix relates the initial and final density matrices. This allows calculating observables in the final state given the initial state parameters and scattering matrix. Pauli matrices provide a complete spin operator basis for pion-nucleon reactions. The density matrix is expressed in terms of the target or beam polarization vector. Constraints from rotational, parity and time reversal symmetries on the nucleon-nucleon scattering matrix are also discussed.
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.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
This document summarizes research on Casimir torque in the weak coupling approximation. It examines manifestations of Casimir torque between planar objects characterized by delta function potentials. The key findings are:
1) An exact calculation of the Casimir torque between a finite rectangular plate above a semi-infinite plate is presented and agrees well with the proximity force approximation when the plate separation is small compared to their sizes.
2) Cusps in the torque arise when the corners of the finite plate pass over the edge of the semi-infinite plate.
3) A similar calculation is done for a disk above a semi-infinite plate, again finding good agreement with the proximity force approximation.
1) The Fermi-Dirac distribution describes the number of electrons in a given state at finite temperature. The chemical potential μ shifts slightly downward as temperature increases to keep the total number of electrons constant.
2) Specific heat and Pauli paramagnetism calculations using the free electron model are incorrect due to an effective mass m*. Transport properties calculations are the same as in the Drude model when using the Fermi velocity and density of states at the Fermi energy.
3) Electron-electron collisions conserve energy and momentum but can change the heat current, contributing to the thermal conductivity relaxation time τQ. They do not affect the electrical conductivity relaxation time τe`.
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
This document describes a procedure for obtaining maximum likelihood estimates of parameters for a generalized exponential distribution of earthquake intensity and magnitude. It presents the maximum likelihood parameter values for initial and extremal distributions of earthquakes in Zagreb, Croatia. Specifically:
- It describes an iterative procedure to determine the maximum likelihood values of the parameters a', k', and xo' that characterize the initial distribution, and parameters a, k, and xo that characterize the extremal distribution.
- When applied to earthquake intensity data from Zagreb, the procedure yields maximum likelihood parameter estimates of k' = 0.23820, a' = 1.9387, xo' = 1 for the initial distribution, and k = 0.
- The Stern-Gerlach experiment in 1922 showed that a beam of silver atoms passed through an inhomogeneous magnetic field split into two beams, providing early evidence that angular momentum is quantized.
- The development of quantum mechanics helped explain phenomena like the fine structure of hydrogen emission spectra, but failed to account for observed splittings until the concept of intrinsic "spin" angular momentum was introduced.
- Angular momentum operators like Jx, Jy, and Jz are defined based on classical angular momentum expressions, with momentum terms replaced by operators involving partial derivatives. These operators obey specific commutation relationships and do not commute with one another.
This document discusses statistical thermodynamics and the partition function. It introduces the concept of microscopic configurations and their weights. The Boltzmann distribution relates the probability of a configuration to its weight, which depends on the energy levels and temperature. The partition function allows calculating thermodynamic properties like internal energy, entropy, and heat capacity from knowledge of the energy levels and degeneracies alone. It provides a statistical mechanical approach to thermodynamics.
This document introduces key concepts in statistical mechanics, including the idea that macroscopic properties are thermal averages of microscopic properties. It discusses common statistical ensembles like the microcanonical ensemble (isolated systems with constant energy) and the canonical ensemble (systems in equilibrium with a heat reservoir). The canonical partition function Z relates microscopic quantum mechanics to macroscopic thermodynamics and can be used to calculate thermodynamic variables. Properties like heat capacity can be derived from fluctuations in energy calculated from the partition function.
This document provides an overview of field theory concepts related to constructing a toy Standard Model using an SU(2) x U(1) gauge symmetry. It discusses how scalar and spinor fields can be incorporated into a Lagrangian that respects this symmetry. It describes how the gauge fields transform under subgroups like the diagonal subgroup, and how this relates to the masses of the Z and W bosons. It also discusses the Higgs potential and how it gives mass to the Higgs boson while the Goldstone bosons are eliminated via a gauge condition.
The document describes three models of photons with physical extent beyond the traditional point particle model: a KdV particle, a normal probability classical packet, and a sinc function quantum packet. The sinc function model is identified as most suitable, describing a photon peaked at its origin that converges to ±∞. In this model, the photon has a disk shape with radii ranging from 10-17m for gamma rays to unlimited sizes for long radio wavelengths. The photon is proposed to have internal magnetic fields and a possible rest mass upper limit of 2×10-69kg.
Sweeping discussion on_dirac_fields_securedfoxtrot jp R
- The document presents a Lagrangian for a basic electron theory that can be outlined from an SU(2)XU(1) construction, excluding additional fermions of the full electroweak theory.
- The Lagrangian includes an effective Lagrangian for the electron and an interaction Lagrangian for the left-handed electron and left-handed neutrino.
- Various forms of the Lagrangian and action are presented and manipulated using integration by parts, arriving at Dirac's first order equation of motion for the electron spinor.
1) The document discusses solving the differential equation R(r) with vanishing mass M and angular frequency ω by transforming it into Legendre form.
2) Two regular singular points are identified that correspond to physical radii, and a power series solution is proposed.
3) Recurrence relations are derived for the coefficients in the power series, allowing one solution to be written as a Legendre polynomial.
1) The document reviews Seiberg-Witten duality by first discussing N=2 supersymmetric Yang-Mills (SYM) theory and the topics needed to understand it, such as SUSY algebra, massless multiplets, massive multiplets, chiral and vector superfields, and N=1 SYM.
2) It then briefly discusses Olive-Montonen duality from 1977 before reviewing Seiberg-Witten duality from 1994.
3) The objective is to work out the form of the low energy effective action for N=2 SYM theory, which involves finding the prepotential term.
To the Issue of Reconciling Quantum Mechanics and General RelativityIOSRJAP
The notion of gravitational radiation as a radiation of the same level as the electromagnetic radiation is based on theoretically proved and experimentally confirmed fact of existence of stationary states of an electron in its gravitational field characterized by the gravitational constant K = 1042G (G is the Newtonian gravitational constant) and unrecoverable space-time curvature Λ. If the numerical values of K 5.11031 Nm2 kg-2 and =4.41029 m -2 , there is a spectrum of stationary states of the electron in its own gravitational field (0.511 MeV ... 0.681 MeV).Adjusting according to the known mechanisms of broadening does not disclose the broadening of the registered portion of the emission spectrum of the micropinch. It indicates the presence of an additional mechanism of broadening the registered portion of the spectrum of the characteristic radiation due to the contribution of the excited states of electrons in their own gravitational field. The energy spectrum of the electron in its own gravitational field and the energy spectra of multielectron atoms are such that there is a resonance of these spectra. As obvious, the consequence of such resonant interaction is appearance, including new lines, of electromagnetic transitions not associated with atomic transitions. The manuscript is the review of previously published papers cited in the references.
Sweeping discussion on_dirac_fields_update1foxtrot jp R
(1) The document presents a Lagrangian for a basic electron theory using a Dirac field ψ. The Lagrangian includes an effective Lagrangian ffeL )( 2ψ for the electron and an interaction Lagrangian tni
LL
L ),( 21 ψψ for the left-handed neutrino and electron.
(2) The action for the effective electron theory can be written as an action for the adjoint spinor field ψ using integration by parts. Variations of the two actions are equivalent imposing boundary conditions.
(3) Varying the action with respect to the adjoint spinor ψ yields the Dirac equation of motion, the first order Euler-Lagrange equation for ψ.
Uniformity of the Local Convergence of Chord Method for Generalized EquationsIOSR Journals
This document summarizes research on the uniform convergence of the Chord method for solving generalized equations. The Chord method is an iterative method for finding solutions to equations of the form y ∈ f(x) + F(x), where f is a function and F is a set-valued mapping. The authors prove that under certain conditions, including F being pseudo-Lipschitz and the derivative of f being continuous, the Chord method converges uniformly for small variations in the parameter y. They obtain this result in two different ways. The document also provides relevant definitions and preliminaries on generalized equations, set-valued mappings, and convergence properties.
This chapter discusses the theory of angular momentum in quantum mechanics and its applications. Eigenvectors of the angular momentum operator J satisfy certain eigenvalue equations involving the quantum numbers j and m. Specific cases of spin-1/2 and spin-1 systems are then derived. The chapter covers topics like coupling of angular momentum systems and angular momentum matrix elements.
This chapter discusses spin in strong interactions like pion-nucleon and nucleon-nucleon scattering. It introduces the density matrix and reaction matrix to describe mixed spin states. The density matrix is determined by the mean values of spin operators and completely characterizes the spin state. The reaction matrix relates the initial and final density matrices. This allows calculating observables in the final state given the initial state parameters and scattering matrix. Pauli matrices provide a complete spin operator basis for pion-nucleon reactions. The density matrix is expressed in terms of the target or beam polarization vector. Constraints from rotational, parity and time reversal symmetries on the nucleon-nucleon scattering matrix are also discussed.
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.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
This document summarizes research on Casimir torque in the weak coupling approximation. It examines manifestations of Casimir torque between planar objects characterized by delta function potentials. The key findings are:
1) An exact calculation of the Casimir torque between a finite rectangular plate above a semi-infinite plate is presented and agrees well with the proximity force approximation when the plate separation is small compared to their sizes.
2) Cusps in the torque arise when the corners of the finite plate pass over the edge of the semi-infinite plate.
3) A similar calculation is done for a disk above a semi-infinite plate, again finding good agreement with the proximity force approximation.
1) The Fermi-Dirac distribution describes the number of electrons in a given state at finite temperature. The chemical potential μ shifts slightly downward as temperature increases to keep the total number of electrons constant.
2) Specific heat and Pauli paramagnetism calculations using the free electron model are incorrect due to an effective mass m*. Transport properties calculations are the same as in the Drude model when using the Fermi velocity and density of states at the Fermi energy.
3) Electron-electron collisions conserve energy and momentum but can change the heat current, contributing to the thermal conductivity relaxation time τQ. They do not affect the electrical conductivity relaxation time τe`.
This document discusses atomic structure, beginning with the hydrogen atom and one-electron atoms. It then discusses the Hamiltonian and solutions of the Schrodinger equation for these systems. It introduces quantum numbers and describes the orbitals and energy levels. For polyelectronic atoms, it discusses separating the Schrodinger equation and introduces Hartree-Fock self-consistent field approximations. It describes Slater determinants which satisfy the Pauli exclusion principle for many-electron wavefunctions.
Modern theory of magnetism in metals and alloysSpringer
This document provides an introduction to magnetism in solids. It discusses how magnetic moments originate from electron spin and orbital angular momentum at the atomic level. In solids, electron localization determines whether magnetic properties are described by localized atomic moments or collective behavior of delocalized electrons. The key concepts of metals and insulators are introduced. The document then presents the basic Hamiltonian used to describe magnetism in solids, including terms for kinetic energy, electron-electron interactions, spin-orbit coupling, and the Zeeman effect. It also discusses how atomic orbitals can be used as a basis set to represent the Hamiltonian and describes the symmetry properties of s, p, and d orbitals in cubic crystals.
Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.
Surface Polaritons in GAAS/ALGAAS/LH Hetrojunction Structure in a High Magnet...ijrap
The surface polaritons (SP) variation in Ga As/ Al Ga As/ LH hetrojunction composition in the presence of
a strong transverse quantized magnetic field is estimated using the quantum Hall effect case. The
dispersion characteristics of the SPs are investigated using the dielectric constants values of the Ga As and
the Alx Ga 1-x As media and the defined thickness, the Alx Ga 1-x As medium. The dispersion behaviours
calculated results are listed for considered cases. It was shown that the frequency values against the wave
vector values are affected in a strong manner by changing thickness, of the Alx
Ga 1-x- As media and by
changing the variation of the dielectric constants of Ga As against the Alx Ga 1-x As. The significance
effects of the use of the left-handed (LH) medium as an upper layer of the proposed composition was
demonstrated; the frequency values are remarkably increased using LH material as an upper layer. It was
noticed that at certain conditions of the LH upper layer composition, similar results have been obtained
such as found by using dielectric upper layer.
Nonlinear Electromagnetic Response in Quark-Gluon PlasmaDaisuke Satow
The document discusses nonlinear electromagnetic response in quark-gluon plasma, specifically focusing on quadratic induced currents. It first outlines collision-dominant and collisionless cases. For the collision-dominant case, it lists possible forms of quadratic currents using CP symmetry properties and derives the Boltzmann equation in relaxation time approximation to calculate induced currents order-by-order in electromagnetic fields. The linear terms reproduce known results while quadratic terms are most sensitive to quark chemical potential at high temperature.
This document summarizes elementary particles in physics. It describes how particles are classified into leptons and hadrons. Leptons include electrons, muons, taus and their neutrinos. Hadrons include baryons like protons and neutrons, and mesons. Interactions are also classified, including the electromagnetic, weak, and strong interactions. The electromagnetic interaction between charged leptons and photons is described based on local gauge invariance, resulting in a theory of quantum electrodynamics that agrees well with experiments.
This document discusses plane electromagnetic waves. It defines plane waves as waves whose wavefronts are infinite parallel planes of constant amplitude normal to the phase velocity vector. The electric and magnetic fields of a plane wave are perpendicular to each other and to the direction of propagation. Plane waves can be linearly, circularly, or elliptically polarized depending on the orientation and behavior of the electric field vector over time. Linear polarization occurs when the electric field is oriented along a fixed line. Circular polarization results when the electric field traces out a circle, and elliptical polarization is characterized by an elliptical trace.
This document discusses transmission line modes, beginning with TEM, TE, and TM waves. It then focuses on the TEM mode, deriving the electric and magnetic fields for a TEM wave. Next, it examines the TEM mode in more detail for a coaxial cable, finding the electric and magnetic fields and characteristic impedance. It concludes by briefly discussing surface waves on a grounded dielectric slab.
Rutherford scattering experiments bombarded thin metal foils with alpha particles. Most alpha particles passed straight through, but some were scattered at large angles. This was inconsistent with the plum pudding model of the atom, but agreed with Rutherford's nuclear model. The scattering was analyzed using classical mechanics. For head-on collisions, large-angle scattering required the target mass be much greater than the alpha particle mass, as is the case for alpha scattering off atomic nuclei but not electrons. This supported the existence of a small, massive nucleus at the center of the atom.
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Phase transition and casimir effect
1. JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 138-146
This paper is available online at http://stdb.hnue.edu.vn
PHASE TRANSITION AND THE CASIMIR EFFECT IN A COMPLEX SCALAR
FIELD WITH ONE COMPACTIFIED SPATIAL DIMENSION
Tran Huu Phat1
and Nguyen Thi Tham2
1
Vietnam Atomic Energy Institute,
2
Faculty of Physics, Hanoi University of Education No. 2, Xuan Hoa, Vinh Phuc
Abstract. Phase transition and the Casimir effect are studied in the complex scalar
field with one spatial dimension to be compactified. It is shown that the phase
transition is of the second order and the Casimir effect behaves quite differently
depending on whether it’s under periodic or anti-periodic boundary conditions.
Keywords: Phase transition, Casimir effect, complex scalar field, compactified
spatial.
1. Introduction
It is well known that a characteristic of quantum field theory in space-time with
nontrivial topology is the existence of non-equivalent types of fields with the same spin
[1, 2, 3]. For a scalar system in space-time which is locally flat but with topolog, that is
a Minkowskian space with one of the spatial dimensions compactified in a circle of finite
radius L, the non-trivial topology is transferred into periodic (sign +) and anti-periodic
(sign -) boundary conditions:
φ(t, x, y, z) = ±φ(t, x, y, z + L) (1.1)
The seminal discovery in this direction is the so-called Casimir effect [4, 5], where the
Casimir force generated by the electromagnetic field that exists in the area between two
parallel planar plates was found to be
F = −
π2
cS
240L4
,
here S is the area of the parallel plates and L is the distance between two plates fulfilling
the condition L2
≪ S. The Casimir effect was first written about in 1948 [4], but since the
Received June 30, 2013. Accepted August 27, 2013.
Contact Nguyen Thi Tham, e-mail address: nguyenthamhn@gmail.com
138
2. Phase transition and the casimir effect in a complex scalar field...
1970s this effect has received increasing attention of scientists. Newer and more precise
experiments demonstrating the Casimir force have been performed and more are under
way. Recently, the Casimir effect has become a hot topic in various domains of science
and technology, ranging from cosmology to nanophysics [5, 6, 7]. Calculations of the
Casimir effect do not exist below zero degrees. It has been seen that the strength of the
Casimir force decreases as the distance between two plates increases. At this time it is
not possible to predict the repulsive or attractive force for different objects and there is
no indication that the Casimir force is dependent on distance at finite temperatures. The
Lagrangian we consider is of the form
L = ∂µφ∗
∂φ − U, U = m2
φ∗
φ +
λ
2
(φ∗
φ)2
. (1.2)
in which φ is a complex scalar field and m and λ are coupling constants. In the present
article, we calculate the effective potential in a complex scalar field and, based on this, the
phase transition in compactified space-time is derived. We then study the Casimir effect
at a finite temperature and calculate Casimir energies and Casimir forces that correspond
to both boundary conditions (periodic and anti-periodic).
2. Content
2.1. Effective potential and phase transition
The space is compactified along the oz axis with length L. Then the Euclidian
Action is defined as
SE = i
L∫
0
dz
∫
LEdτdx⊥, dx⊥ = dxdy, (2.1)
where LE is the Euclidian form of the Lagrangian (1.2) and t = iτ. Assume that when
λ > 0 and m2
< 0, the field operator φ develops vacuum expectation value
⟨φ⟩ = ⟨φ∗
⟩ = u,
then the U(1) symmetry of the complex scalar field given in (1.2) is spontaneously broken.
In the tree approximation u corresponds to the minimum of U,
(
∂U
∂φ
)
φ=φ∗=0
= 0,
yielding u =
√
−
m2
λ
. At minimum the potential energy U reads U = m2
u2
+
λ
2
u4
. Let
us next decompose φ and φ∗
φ =
1
√
2
(u + φ1 + iφ2) , φ∗
=
1
√
2
(u + φ1 − iφ2) . (2.2)
Inserting (2.2) into (1.2) we get the matrix D representing the interaction betweenφ1, φ2
139
3. Tran Huu Phat and Nguyen Thi Tham
iD−1
= ∥Aik∥ , A11 = ω2
+ ⃗k2
+ 2λu2
, A22 = ω2
+ ⃗k2
, A12 = A21 = 0.
The partition function is established as
Z =
∫
Dφ∗
Dφ exp [−SE]
In a one-loop approximation we have that
Z = e−V LU
∫
Dφ1Dφ2 exp
[
−
∫
dxϕ+
iD−1
ϕ
]
where ϕ+
= (φ1, φ2) The effective potential is defined as
Ω = −
ln Z
V L
= Ωs + U, (2.3)
in which
ΩS =
T
2L
∞∑
m=−∞
∞∑
n=−∞
∫
dk⊥
(2π)2
{
ln
[
ω2
m + E2
1n
]
+ ln
[
ω2
m + E2
2n
]}
. (2.4)
with
E1n =
√
k2
⊥ + k2
3n + M2, E2n =
√
k2
⊥ + k2
3n, M2
= 2λu2
(2.5)
and
k3n = (2n + 1)
π
L
, n = 0, ±1, ±2, .... (2.6)
for anti-periodic boundary conditions, and
k3n = 2n
π
L
, n = ±1, ±2, ... (2.7)
for periodic boundary conditions. Note that En in (2.5) is exactly the gapless spectrum of
the Goldston boson in the broken phase and there is a similarity between L appearing in
(2.6) and (2.7) with T in the Matsubara formula β = 1/T, and therefore it is convenient
to impose a = 1/L for later use. Parameter a has the dimension of energy. Making use of
the formula:
T
∞∑
n=−∞
ln
(
ω2
+ E2
)
= E + T ln
(
1 − e−E/T
)
,
and taking into account (2.5), (2.6) and (2.7), we arrive at the expression for ΩS
ΩS = −a
∞∑
n=−∞
∫
dk⊥
(2π)2
{
E1n + E2n + T ln
[
1 − e−βE1n
]
+ T ln
[
1 − e−βE2n
]}
.
(2.8)
140
4. Phase transition and the casimir effect in a complex scalar field...
The two first terms under the integral in (2.8) are exactly the energy of an
electromagnetic vacuum restricted between two plates which gives rise to the Casimir
energy. Next let us study the phase transition of the complex scalar field without the
Casimir effect at various values of a. The effective potential (2.3) is rewritten in the form
Ω = ΩS(T) − ΩS(T = 0) + U
= −aT
∞∑
n=−∞
∫
dk⊥
(2π)2
{
T ln
(
1 − e−βE1n
)
+ T ln
(
1 − e−βE2n
)}
+ U (2.9)
From (2.9) we derive the gap equation
∂Ω
∂u
= 0, or
m2
+ λu2
+
aλ
π
n∑
n=−∞
∞∫
0
dk
k
√
k2 + k2
3n + 2λu2
1
eβEn − 1
= 0. (2.10)
In order to numerically study the evolution of u versus T at several values of a, the
model parameters chosen are those associated with pions and sigma mesons in the linear
sigma model:
m2
=
3m2
π − m2
σ
2
, λ =
m2
σ − m2
π
f2
π
, mσ = 500MeV, mπ = 138MeV, fπ = 93MeV.
where mσ, mπ are respectively the mass of sigma mesons and pions, and fπ is the pion
decay constant. Starting from this parameter set and the gap equation (2.10), we get the
behaviors of u as a function of temperature at several values of a, given in Figures 1a and
1b for periodic and anti-periodic cases. It is clear that the phase transitions in both cases
are of the second order.
(1a) (1b)
Figure 1. The evolution of u versus T at several values of a which correspond to
periodic (1a) and anti-periodic (1b) boundary conditions
141
5. Tran Huu Phat and Nguyen Thi Tham
2.2. Casimir effect
Let us mention that the vacuum energy caused by the electromagnetic field
restricted between two parallel planar plates is of the form
E(a) = −a
∫
dk⊥
(2π)2
∞∑
n=−∞
EnS, (2.11)
in which S is the area of planar plate, . It is evident that E(a) diverges. So we try to
renormalize it by introducing a rapid damping factor
ER (a) = −a
∫
dk⊥
(2π)3
∞∑
n=−∞
Ene−δEn
S, (2.12)
and the Casimir energy then reads
EC (a) = lim
δ→0
ER(a). (2.13)
Applying the Abel-Plana formula [8, 9]
∞∑
n=0
F(n) −
∞∫
0
F(t)dt =
1
2
F(0) + i
∞∫
0
dt
F(it) − F(−it)
e2πt − 1
,
for periodic conditions and
∞∑
n=0
F(n +
1
2
) − 2
∞∫
0
F(t)dt = i
∞∫
0
dt
∑
λ=±1
F (it) − F (−it)
e2π(t+i λ
2 ) − 1
.
for anti-periodic conditions to the calculation of (2.12) and (2.13), we derive the
expressions for the Casimir energy
EP
C (a) =
16π2
L3
∞∫
0
ydy
∞∫
b
dt
√
t2 − b2
e2πt − 1
,
2.14a
which correspond to periodic conditions and
EA
C (a) = −
16π2
L3
∞∫
0
ydy
∞∫
b
dt
√
t2 − b2
e2πt + 1
. (2.14)
which correspond to anti-periodic conditions. The parameters appearing in 2.14a and 2.14
are defined as
y = L
|k⊥|
2π
, b2
= y2
+ M2
∗ , M2
∗ = L2 M2
(2π)2
.
142
6. Phase transition and the casimir effect in a complex scalar field...
Next, based on equations 2.14a, 2.14 and the gap equation (2.10), let us consider
the evolution of Casimir energy versus a = 1/L at several values of T and versus T at
several values of a.
In Figure 2 is shown a dependence of Casimir energy that corresponds to periodic
conditions (2a) and anti-periodic conditions (2b). It is easily recognized that EP
C (a) and
EA
C (a) are negligible as L is large enough, while they increases rapidly in the opposite
case. Corresponding to periodic conditions and anti-periodic conditions respectively, we
show in Figures 3a and 3b the T dependence of Casimir energy at L = 6.5fm. It is seen
that EP
C (a) and EA
C (a) decrease as T increases.
(2a) (2b)
Figure 2. The behavior of Casimir energy as a function of L at several values of T.
Figure 2a and (2b) shows the periodic (anti-periodic) condition
(3a) (3b)
Figure 3. The behavior of Casimir energy as a function of T at L = 6.5fm.
Figure 3a (3b) shows the periodic (anti-periodic) condition
143
7. Tran Huu Phat and Nguyen Thi Tham
Finally, the Casimir forces FP
C (L) and FA
C (L) acting on two parallel plates are
concerned for both cases of periodic and anti-periodic conditions. They are determined
by
FP,A
C (L) = −
∂EP,A
C (L)
∂L
, (2.15)
Inserting 2.14 into 2.15 we find immediately that
FP
C (L) =
8λu2
L2
∞∫
0
ydy
∞∫
b
dt
(e2πt − 1)
√
t2 − b2
+
4
3L
EP
C (L),
(2.16)
FA
C (L) = −
8λu2
L2
∞∫
0
ydy
∞∫
b
dt
(e2πt + 1)
√
t2 − b2
+
4
3L
EA
C (L) .
Combining equations 2.14a, 2.14 and 2.16 together leads to the graphs representing
the L dependence of Casimir forces at T = 200MeV depicted in Figure 4 and the graphs
representing the T dependence of Casimir forces at L = 6.5 fm plotted in Figure 5.
Figures 4 and 5 indicate that
- The Casimir force is repulsive for periodic boundary conditions and becomes
attractive when boundary conditions are anti-periodic.
- The strength of the Casimir force decreases quickly as the distance L between the
two plates increases.
- Casimir forces depend considerably on temperature T for T < 300MeV.
(4a) (4b)
Figure 4. The evolution of Casimir forces versus L at T = 200MeV
144
8. Phase transition and the casimir effect in a complex scalar field...
(5a) (5b)
Figure 5. The evolution of Casimir forces versus T at L = 6.5fm
3. Conclusion
In the preceding sections we systematically studied the phase transition and Casimir
effect in a complex scalar field embedded in compactified space-time. We obtained the
following results:
i) The U(1) restoration phase transition at high temperature is of the second order
for both boundary conditions and the critical temperature depends on L.
ii) Based on the calculated Casimir energies and Casimir forces for periodic and
anti-periodic boundary conditions, we investigated numerically their T dependences at
several values of L, plotted in Figures 3 and 5, and their L dependences at several values
of T, plotted in Figures 2 and 4.
iii) The physical content of periodic and anti-periodic boundary conditions is
cleared up in Figure 4 which shows that the Casimir force is repulsive for periodic
conditions while it is attractive when boundary conditions are anti-periodic. Evidently
this statement is very interesting for those working in nanophysics and nanotechnology.
The revelation that the critical temperature depends on the compactified length L
reveals a new direction for the investigation of high temperature superconductors and
Bose-Einstein condensations in space with (2D + ε) dimensions.
Acknowledgments. This paper is financially supported by the Vietnam National
Foundation for Science and Technology Development (NAFOSTED) under Grant No.
103.01-2011.05.
145
9. Tran Huu Phat and Nguyen Thi Tham
REFERENCES
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[2] L. H. Ford, 1980. Instabilities in interacting quantum field theories in
non-Minkowskian spacetimes. Phys. Rev. D22, pp. 3003-3011.
[3] Modern Kaluza - Klein Theories, 1987. Edited by T. Appelquits, A. Chodos and
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