1. The document discusses key concepts in quantum physics including wave-particle duality, quantum states as linear combinations or superpositions, and the uncertainty principle.
2. It outlines the quantum postulates including that states are elements of a state space, physical quantities are represented by operators with real eigenvalues, and measurements change the system state.
3. Examples are provided of solving the Schrodinger equation for a rectangular potential barrier, potential well, and quantum harmonic oscillator to illustrate quantum mechanical solutions.
This document discusses applications of second-order differential equations, including mechanical vibrations, electric circuits, and forced vibrations. It provides examples of spring-mass systems, describing them with second-order differential equations. It also examines damped and undamped vibrations, as well as free and forced vibrations. Electrical circuits are modeled using differential equations for current or charge. The document analyzes cases of underdamped, critically damped, and overdamped systems.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
1) Ordinary differential equations relate a dependent variable to one or more independent variables by means of differential coefficients. They can be classified based on order, degree, whether they are linear or non-linear, and type (exact, separable variables, homogeneous).
2) First order differential equations can sometimes be solved by separation of variables, or by finding an integrating factor. Homogeneous equations can be transformed by substitution.
3) Second order linear differential equations can be reduced to a system of two first order equations. The complementary function and particular solutions combine to form the general solution. Unequal or equal roots of the characteristic equation determine the form of the complementary function.
This document provides an introduction to tensor calculus. It begins with definitions of tensors and coordinate systems. Section 1 defines tensors and contravariant and covariant indices. Section 2 focuses on Cartesian tensors and introduces tensor notation rules. It defines tensor terms, expressions, and operations. Section 3 will cover general curvilinear coordinates and covariant differentiation. The document establishes the foundation for working with tensors and their transformation properties.
Scalars represent physical quantities at a point, like pressure. Vectors track magnitude and direction of quantities like force and velocity. Tensors track three pieces of information, like stress which has magnitude, direction, and acting plane. A scalar has one value, a vector has three components, and a tensor has nine components. The continuity equation relates the accumulation of mass in a region to the net flux through the boundary, and for incompressible fluids reduces to the divergence of velocity being zero.
Tensor algebra and tensor analysis for engineersSpringer
This document discusses vector and tensor analysis in Euclidean space. It defines vector- and tensor-valued functions and their derivatives. It also discusses coordinate systems, tangent vectors, and coordinate transformations. The key points are:
1. Vector- and tensor-valued functions can be differentiated using limits, with the derivatives being the vector or tensor equivalent of the rate of change.
2. Coordinate systems map vectors to real numbers and define tangent vectors along coordinate lines.
3. Under a change of coordinates, components of vectors and tensors transform according to the Jacobian of the coordinate transformation to maintain geometric meaning.
applications of second order differential equationsly infinitryx
1) Second-order differential equations are used to model vibrating springs and electric circuits. They describe oscillations, vibrations, and resonance.
2) Springs obey Hooke's law, resulting in a second-order differential equation relating position to time. The solutions describe simple harmonic motion.
3) Damping forces can be added, resulting in overdamped, critically damped, or underdamped systems with different behavior.
Solution of Fractional Order Stokes´ First EquationIJRES Journal
Fractional sine transform and Laplace transform are used for solving the Stokes` first problem with
ractional derivative, where the fractional derivative is defined in the Caputo sense of orderm1 m.
The solution of classical problem for Stokes` first problem has been obtained as limiting case.
This document discusses applications of second-order differential equations, including mechanical vibrations, electric circuits, and forced vibrations. It provides examples of spring-mass systems, describing them with second-order differential equations. It also examines damped and undamped vibrations, as well as free and forced vibrations. Electrical circuits are modeled using differential equations for current or charge. The document analyzes cases of underdamped, critically damped, and overdamped systems.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
1) Ordinary differential equations relate a dependent variable to one or more independent variables by means of differential coefficients. They can be classified based on order, degree, whether they are linear or non-linear, and type (exact, separable variables, homogeneous).
2) First order differential equations can sometimes be solved by separation of variables, or by finding an integrating factor. Homogeneous equations can be transformed by substitution.
3) Second order linear differential equations can be reduced to a system of two first order equations. The complementary function and particular solutions combine to form the general solution. Unequal or equal roots of the characteristic equation determine the form of the complementary function.
This document provides an introduction to tensor calculus. It begins with definitions of tensors and coordinate systems. Section 1 defines tensors and contravariant and covariant indices. Section 2 focuses on Cartesian tensors and introduces tensor notation rules. It defines tensor terms, expressions, and operations. Section 3 will cover general curvilinear coordinates and covariant differentiation. The document establishes the foundation for working with tensors and their transformation properties.
Scalars represent physical quantities at a point, like pressure. Vectors track magnitude and direction of quantities like force and velocity. Tensors track three pieces of information, like stress which has magnitude, direction, and acting plane. A scalar has one value, a vector has three components, and a tensor has nine components. The continuity equation relates the accumulation of mass in a region to the net flux through the boundary, and for incompressible fluids reduces to the divergence of velocity being zero.
Tensor algebra and tensor analysis for engineersSpringer
This document discusses vector and tensor analysis in Euclidean space. It defines vector- and tensor-valued functions and their derivatives. It also discusses coordinate systems, tangent vectors, and coordinate transformations. The key points are:
1. Vector- and tensor-valued functions can be differentiated using limits, with the derivatives being the vector or tensor equivalent of the rate of change.
2. Coordinate systems map vectors to real numbers and define tangent vectors along coordinate lines.
3. Under a change of coordinates, components of vectors and tensors transform according to the Jacobian of the coordinate transformation to maintain geometric meaning.
applications of second order differential equationsly infinitryx
1) Second-order differential equations are used to model vibrating springs and electric circuits. They describe oscillations, vibrations, and resonance.
2) Springs obey Hooke's law, resulting in a second-order differential equation relating position to time. The solutions describe simple harmonic motion.
3) Damping forces can be added, resulting in overdamped, critically damped, or underdamped systems with different behavior.
Solution of Fractional Order Stokes´ First EquationIJRES Journal
Fractional sine transform and Laplace transform are used for solving the Stokes` first problem with
ractional derivative, where the fractional derivative is defined in the Caputo sense of orderm1 m.
The solution of classical problem for Stokes` first problem has been obtained as limiting case.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
This document summarizes applications of differential equations to real world systems including cooling/warming, population growth, radioactive decay, electrical circuits, survivability with AIDS, economics, drug distribution in the human body, and a pursuit problem. Examples are provided for each application to illustrate solutions to related differential equations. Key concepts covered include Newton's law of cooling, population models, carbon dating, series circuits, survival models, supply and demand models, compound interest, drug concentration in the body over time, and a mathematical model for a dog chasing a rabbit.
application of first order ordinary Differential equationsEmdadul Haque Milon
The document provides information about Group D's presentation which includes 10 members. It lists the members' names and student IDs. It then outlines 3 topics that will be covered: 1) Applications of first order ordinary differential equations, 2) Orthogonal trajectories, and 3) Oblique trajectories. For the first topic, it provides examples of applications in fields like physics, statistics, chemistry, and engineering. It also shows sample problems and solutions related to population growth, curve determination, cooling/warming, carbon dating, and radioactive decay. For the second topic, it defines orthogonal trajectories and provides an example of finding the orthogonal trajectories of a family of parabolas.
This document contains 4 physics problems involving quantum mechanics concepts:
1) Calculating the commutator of angular momentum operators and relating the time derivative of angular momentum to torque.
2) Determining spherical harmonic functions using angular momentum operators.
3) Solving for energy levels and eigenfunctions of a rigid rotor system of two particles attached at either end of a massless rod.
4) Verifying commutation relations for spin and angular momentum operators and showing that the cross product of an angular momentum operator with itself is non-zero.
This document provides an overview of second order ordinary differential equations (ODEs). It discusses the method of variation of parameters for finding particular solutions to inhomogeneous second order linear ODEs. It provides examples of applying this method, including a spring-mass system and an electric circuit. It also discusses applications of second order ODEs to modeling spring-mass systems and electric circuits.
Necessary and Sufficient Conditions for Oscillations of Neutral Delay Differe...inventionjournals
In this paper, we discuss the oscillatory behavior of all solutions of the first order neutral delay difference equations with several positive and negative coefficients ( ) ( ) ( ) ( ) 0 , i i j j k k I J K x n p x n r x n q x n , o n n (*) where I, J and K are initial segments of natural numbers, pi , rj , qk are positive numbers, i , j are positive integers and k is a nonnegative integer for iI, jJ and kK. We establish a necessary and sufficient conditions for the oscillation of all solutions of (*) is that its characteristic equation ( 1) 1 0 j i k i j k I J K p r q has no positive roots . AMS Subject Classifications : 39A10, 39A12.
This document discusses the use of Airy stress functions to solve problems in two-dimensional elasticity. It provides the equations relating stresses to the Airy stress function in plane stress, plane strain, and polar coordinates. Examples are given to show how specific Airy stress functions can be used to solve problems like bending of beams, stresses in curved beams with end moments, and stresses in a quarter-circle beam under an end load.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
The document discusses the calculus of variations and derives Euler's equation. It explains that the fundamental problem is to find an unknown function that makes a functional (function of a function) an extremum. Euler's equation provides the necessary condition for a functional to be extremized. As an example, it derives the Euler-Lagrange equation for the shortest distance between two points in Cartesian coordinates, showing the path is a straight line.
The document summarizes an honours project that calculates the pressure of a gluon gas using statistical mechanics and thermal field theory. It introduces the quark-gluon plasma system and describes how both theories predict the same result for an ideal gas case, but field theory provides a simpler way to incorporate interactions. The project uses quantum mechanics path integral formalism to calculate the pressure via the statistical mechanics and field theory approaches, showing they give identical results for an ideal gas.
The document provides details on a course calendar and lecture plan for hidden Markov models (HMM).
1) The course calendar covers topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms over 14 weeks.
2) The HMM lecture plan introduces discrete-time HMMs and their applications. It covers the three main problems of HMMs - evaluation, decoding, and learning. Evaluation calculates the probability of an output sequence, decoding finds the most probable hidden state sequence, and learning estimates model parameters from training data.
3) The trellis diagram and forward algorithm are described for solving the evaluation problem, while the Viterbi and forward-backward algorithms are mentioned
This document discusses differential equations, which are equations involving derivatives of an unknown function. It provides examples of first and second order differential equations. Differential equations have applications in science and engineering, such as modeling Newton's laws of cooling, the rate of decay of radioactive materials, Newton's second law of dynamics, the Schrodinger wave equation, RL circuits, and the heat equation in thermodynamics. The document also covers the order, degree, and types of differential equations, as well as their use in modeling natural growth/decay, free falling objects, springs, and Jacobian properties.
Two mutually delay-coupled Rössler oscillators are studied to characterize their synchronized states. For small delays, the oscillators exhibit isochronal synchronization where their trajectories evolve in unison. For larger delays, they can exhibit chaotic anti-phase synchronization where they are always out of phase. Stable periodic orbits are also observed in transitions between synchronized states and for delays near integer multiples of the oscillators' mean period. Increasing the delay decreases the size of stability regions and can produce complex anomalous behavior where the oscillators interchange leader and follower roles.
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
This document discusses differential equations and their origins and applications. It begins by defining differential equations as equations containing derivatives of dependent variables with respect to independent variables. It notes that differential equations involving ordinary derivatives are called ordinary differential equations. Examples are provided of first order, second order, linear and non-linear ordinary differential equations. The document also discusses the physical applications of differential equations, such as modeling simple harmonic motion, oscillations of springs, and rotational dynamics of shafts.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Quantum mechanics describes physical systems using state functions and operators. The four main postulates are:
1) Every physical system is described by a state function containing all information. Combinations of state functions are also valid states.
2) The probability of measurement outcomes can be calculated from the state function.
3) Observables are represented by operators, and applying operators to the state function provides measurable properties.
4) The time-dependent Schrodinger equation (TDSWE) governs how isolated quantum systems evolve over time, with the Hamiltonian operator determining evolution.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
This document summarizes applications of differential equations to real world systems including cooling/warming, population growth, radioactive decay, electrical circuits, survivability with AIDS, economics, drug distribution in the human body, and a pursuit problem. Examples are provided for each application to illustrate solutions to related differential equations. Key concepts covered include Newton's law of cooling, population models, carbon dating, series circuits, survival models, supply and demand models, compound interest, drug concentration in the body over time, and a mathematical model for a dog chasing a rabbit.
application of first order ordinary Differential equationsEmdadul Haque Milon
The document provides information about Group D's presentation which includes 10 members. It lists the members' names and student IDs. It then outlines 3 topics that will be covered: 1) Applications of first order ordinary differential equations, 2) Orthogonal trajectories, and 3) Oblique trajectories. For the first topic, it provides examples of applications in fields like physics, statistics, chemistry, and engineering. It also shows sample problems and solutions related to population growth, curve determination, cooling/warming, carbon dating, and radioactive decay. For the second topic, it defines orthogonal trajectories and provides an example of finding the orthogonal trajectories of a family of parabolas.
This document contains 4 physics problems involving quantum mechanics concepts:
1) Calculating the commutator of angular momentum operators and relating the time derivative of angular momentum to torque.
2) Determining spherical harmonic functions using angular momentum operators.
3) Solving for energy levels and eigenfunctions of a rigid rotor system of two particles attached at either end of a massless rod.
4) Verifying commutation relations for spin and angular momentum operators and showing that the cross product of an angular momentum operator with itself is non-zero.
This document provides an overview of second order ordinary differential equations (ODEs). It discusses the method of variation of parameters for finding particular solutions to inhomogeneous second order linear ODEs. It provides examples of applying this method, including a spring-mass system and an electric circuit. It also discusses applications of second order ODEs to modeling spring-mass systems and electric circuits.
Necessary and Sufficient Conditions for Oscillations of Neutral Delay Differe...inventionjournals
In this paper, we discuss the oscillatory behavior of all solutions of the first order neutral delay difference equations with several positive and negative coefficients ( ) ( ) ( ) ( ) 0 , i i j j k k I J K x n p x n r x n q x n , o n n (*) where I, J and K are initial segments of natural numbers, pi , rj , qk are positive numbers, i , j are positive integers and k is a nonnegative integer for iI, jJ and kK. We establish a necessary and sufficient conditions for the oscillation of all solutions of (*) is that its characteristic equation ( 1) 1 0 j i k i j k I J K p r q has no positive roots . AMS Subject Classifications : 39A10, 39A12.
This document discusses the use of Airy stress functions to solve problems in two-dimensional elasticity. It provides the equations relating stresses to the Airy stress function in plane stress, plane strain, and polar coordinates. Examples are given to show how specific Airy stress functions can be used to solve problems like bending of beams, stresses in curved beams with end moments, and stresses in a quarter-circle beam under an end load.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
The document discusses the calculus of variations and derives Euler's equation. It explains that the fundamental problem is to find an unknown function that makes a functional (function of a function) an extremum. Euler's equation provides the necessary condition for a functional to be extremized. As an example, it derives the Euler-Lagrange equation for the shortest distance between two points in Cartesian coordinates, showing the path is a straight line.
The document summarizes an honours project that calculates the pressure of a gluon gas using statistical mechanics and thermal field theory. It introduces the quark-gluon plasma system and describes how both theories predict the same result for an ideal gas case, but field theory provides a simpler way to incorporate interactions. The project uses quantum mechanics path integral formalism to calculate the pressure via the statistical mechanics and field theory approaches, showing they give identical results for an ideal gas.
The document provides details on a course calendar and lecture plan for hidden Markov models (HMM).
1) The course calendar covers topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms over 14 weeks.
2) The HMM lecture plan introduces discrete-time HMMs and their applications. It covers the three main problems of HMMs - evaluation, decoding, and learning. Evaluation calculates the probability of an output sequence, decoding finds the most probable hidden state sequence, and learning estimates model parameters from training data.
3) The trellis diagram and forward algorithm are described for solving the evaluation problem, while the Viterbi and forward-backward algorithms are mentioned
This document discusses differential equations, which are equations involving derivatives of an unknown function. It provides examples of first and second order differential equations. Differential equations have applications in science and engineering, such as modeling Newton's laws of cooling, the rate of decay of radioactive materials, Newton's second law of dynamics, the Schrodinger wave equation, RL circuits, and the heat equation in thermodynamics. The document also covers the order, degree, and types of differential equations, as well as their use in modeling natural growth/decay, free falling objects, springs, and Jacobian properties.
Two mutually delay-coupled Rössler oscillators are studied to characterize their synchronized states. For small delays, the oscillators exhibit isochronal synchronization where their trajectories evolve in unison. For larger delays, they can exhibit chaotic anti-phase synchronization where they are always out of phase. Stable periodic orbits are also observed in transitions between synchronized states and for delays near integer multiples of the oscillators' mean period. Increasing the delay decreases the size of stability regions and can produce complex anomalous behavior where the oscillators interchange leader and follower roles.
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
This document discusses differential equations and their origins and applications. It begins by defining differential equations as equations containing derivatives of dependent variables with respect to independent variables. It notes that differential equations involving ordinary derivatives are called ordinary differential equations. Examples are provided of first order, second order, linear and non-linear ordinary differential equations. The document also discusses the physical applications of differential equations, such as modeling simple harmonic motion, oscillations of springs, and rotational dynamics of shafts.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Quantum mechanics describes physical systems using state functions and operators. The four main postulates are:
1) Every physical system is described by a state function containing all information. Combinations of state functions are also valid states.
2) The probability of measurement outcomes can be calculated from the state function.
3) Observables are represented by operators, and applying operators to the state function provides measurable properties.
4) The time-dependent Schrodinger equation (TDSWE) governs how isolated quantum systems evolve over time, with the Hamiltonian operator determining evolution.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
Lagrangian formulation provides an alternative but equivalent way to derive equations of motion compared to Newtonian mechanics.
The document provides examples of deriving equations of motion for simple harmonic oscillators, Atwood's machine, and a spring pendulum using the Lagrangian formulation. It also shows the equivalence between Lagrange's equations and Newton's second law.
Specifically, it demonstrates that for a conservative system using generalized coordinates, Lagrange's equations reduce to F=ma, where the generalized forces are equal to the negative gradient of the potential energy.
1) Relativistic quantum mechanics aims to satisfy the requirements of special relativity theory, which non-relativistic quantum mechanics does not. Two approaches are proposed: using a relativistic Hamiltonian or a first-order wave equation.
2) Klein and Gordon developed the Klein-Gordon equation using the relativistic Hamiltonian for a free particle. This equation describes spin-0 particles. Dirac developed his relativistic wave equation using a first-order approach, describing spin-1/2 particles like electrons.
3) Both equations satisfy continuity equations, but the Klein-Gordon equation allows for negative probability densities, which was resolved by interpreting the densities as electric charge/current densities.
1) The document discusses quantizing the electromagnetic field by treating it as a collection of harmonic oscillators, one for each mode of the field (determined by the wave vector and polarization).
2) For each mode, the electric and magnetic fields are represented by creation and annihilation operators that satisfy the commutation relations of quantum harmonic oscillators.
3) The total quantum Hamiltonian for the electromagnetic field is the sum over all modes of the individual mode Hamiltonians, with each mode Hamiltonian resembling that of a quantum simple harmonic oscillator.
This document provides an introduction to quantizing the electromagnetic field. It begins with a classical description of the electromagnetic field using Maxwell's equations. It then shows that the classical electromagnetic field can be described as an infinite collection of independent harmonic oscillators. The document proceeds to quantize these harmonic oscillators by promoting the classical variables to quantum operators. This leads to a description of the electromagnetic field in terms of photon creation and annihilation operators. The quantized electromagnetic field gives rise to phenomena like zero-point energy and the Casimir effect that cannot be explained classically.
This document discusses approximate methods for determining natural frequencies of structures, including Rayleigh's method and Dunkerley's method. Rayleigh's method involves estimating the mode shape and using the Rayleigh quotient to calculate an upper bound for the fundamental frequency. Dunkerley's method provides a lower bound by assuming the structure vibrates as separate components. Examples are provided to illustrate both methods and how they can provide good estimates of natural frequencies.
This document provides an overview of classical and quantum mechanics concepts relevant to statistical thermodynamics. It begins with an introduction to classical mechanics using Lagrangian and Hamiltonian formulations. It then discusses limitations of classical mechanics and introduces key concepts in quantum mechanics, including the Schrodinger equation. Examples are provided of a particle in a box and harmonic oscillator to illustrate differences between classical and quantum descriptions of particle behavior.
This document summarizes key concepts about the particle in a rigid one-dimensional box:
1. It finds the energy eigenstates and discusses the wave functions and their properties like orthogonality.
2. It calculates the probability and expected values for the particle's position and discusses the physical interpretation of the wave function and coefficients when expanding an arbitrary function in the eigenstates.
3. It addresses several questions about normalized wave functions, time-dependent wave functions, energy measurements, and the wave function after a measurement.
This chapter discusses systems of two first order differential equations. It introduces linear systems with constant coefficients, which can be solved using eigenvalues and eigenvectors. The chapter presents methods to find the general solution of homogeneous systems and the solution satisfying initial conditions. Graphical approaches are described, including direction fields and phase portraits to visualize solutions. An example of a two-equation model of a rockbed heat storage system is provided and transformed into matrix notation.
Chemical dynamics and rare events in soft matter physicsBoris Fackovec
Talk for the Trinity Math Society Symposium. First summarises the approximations leading from Dirac equation to molecular description and then the synthesis towards non-equilibrium statistical mechanics. The relaxation approach to projection of a molecular system to a Markov jump process is discussed.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Decay Property for Solutions to Plate Type Equations with Variable CoefficientsEditor IJCATR
In this paper we consider the initial value problem for a plate type equation with variable coefficients and memory in
1 n R n ), which is of regularity-loss property. By using spectrally resolution, we study the pointwise estimates in the spectral
space of the fundamental solution to the corresponding linear problem. Appealing to this pointwise estimates, we obtain the global
existence and the decay estimates of solutions to the semilinear problem by employing the fixed point theorem
Beams are slender structural members that support transverse loads. They are analyzed based on their cross-sectional properties and how bending stresses are distributed. The document discusses beam theory, including equations for normal stress, strain, deflection, and bending moment. It also provides an overview of the finite element method applied to beam analysis, describing the discretization of beams into elements, use of shape functions, and derivation of the element stiffness matrix.
This document summarizes a mechanical engineering student project analyzing the kinematics and dynamics of a forging manipulator. It includes:
1) Modeling the hydraulic actuators as spring-damper systems and computing kinematics using vector equations of the degrees of freedom.
2) Computing static preloads on the actuators to achieve equilibrium.
3) Linearizing the equations of motion around the equilibrium position to determine natural frequencies and mode shapes.
4) Calculating frequency response functions by solving the linearized equations with an external forcing function.
This document derives the energy of the first excited state of the harmonic oscillator using the Schrödinger equation. It first shows that the wave function for the first excited state is a1xe−ax2, where a is a constant. It then substitutes this wave function and its derivatives into the Schrödinger equation for the harmonic oscillator to obtain an expression for the energy E1 in terms of the angular frequency ω. Solving this expression yields the result that the energy of the first excited state is E1 = (3/2)ħω.
https://arxiv.org/abs/2011.04370
A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.
- The atom consists of a small, dense nucleus surrounded by an electron cloud.
- Electrons can only exist in certain discrete energy levels around the nucleus. Their wavelengths are determined by the principal quantum number.
- The Bohr model improved on earlier models by introducing energy levels and quantization, but had limitations. The quantum mechanical model treats electrons as waves and uses Schrodinger's equation.
Dynamic analysis considers mass and acceleration effects that are neglected in static analysis. When loads are suddenly applied or variable, a solid body will vibrate elastically about its equilibrium position. This periodic motion is called free vibration. The document then provides formulations for modeling dynamic systems using kinetic and potential energy and Lagrange's equations. It derives the mass matrices for various finite elements, including bars, trusses, plane stress/strain elements, axisymmetric elements, beams, and frames. Consistent mass matrices are developed using the elements' shape functions and densities.
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1. 1
2. Quantum physics
2.1. Quantum behaviour
Consequences of the above chapter:
i) Microsystems behave as waves and as particles as well –
wave-particle dualism
ii) States of microsystems may be expressed as linear combinations
of other states – combination or superposition principle
iii) Energies, angular momenta and perhaps other physical quantities
may be quantized – quantification
iv) Measurements modify the state of a system in a non-controlled
manner; conjugated quantities can not be defined with arbitrary
precision – uncertainty principle. These conjugated quantities
are e.g. the coordinate and the associated linear momentum.
2.2. Quantum postulates
Remark: the following postulates do not form a complete family of logically consistent
propositions from which all the theory could be deduced. They are rather a way to construct
quantum world and are based on experiments as well as on inferences tested a posteriori.
1. States of a system are elements of a linear space called the state space. They
are symbolized: by Greek letters φ, ψ, Ψ, Φ; by Greek letters with variable
specifications, Ψ(x, y, z, t); by symbols in brackets, zyx ,, , ),( tr
, 0,2,3 . In
the last form they are known as bras and kets.
The interpretation of the wave function: its modulus squared gives the
probability density for the system to be in a certain region, or to have a certain
momentum:
*2
r
= density of probability to have the system around the position vector
r
, i.e. between zzzyyyxxx dand,dand,dand . (QM2.1)
The state functions are usually normalized:
1ddd
2
zyxr
(QM2.2)
State functions must be finite, continuous, with continuous derivatives (with one
exception, points in which the potential has infinite discontinuities).
2. To each physical quantity A is associated a certain Hermitian operator Â. The
measurement of the physical quantity A in the state is represented by the action of
2. 2
the operator  on this state. Such operators have real eigenvalues and their
eigenvectors form a base of the space. They are called observables.
As the result of the measurement the state of the system changes in a chaotic
way. A new measurement completed immediately after the first leaves the state
unchanged. The results of the measurements are the eigenvalues of the observables.
3. The limit of quantum relations should be the classical results
(correspondence principle)
4. Fundamental commutators:
ipx x ˆ,ˆ , ipy y ˆ,ˆ , ipz z ˆ,ˆ (QM2.3)
All the other are zero.
2.3. The Schrödinger equation
If the system evolves freely, without any perturbation – such as measurements –
the state at a later moment tt d is perfectly defined by the state at a previous
moment t. We get the state at the moment tt d using an evolution operator tT ˆ
which must keep the same norm and the superposition relations:
ttTdtt ˆ (QM2.4)
To simplify notations we have not indicate the spatial dependence of the function,
which should be written in full-form tr,
.
As ttttTtTtdttdtt *** ˆˆ ,
ItTtT ˆˆˆ
, and tT ˆ is unitary.
Assuming t small, we develop
tiItT ˆˆˆ and tiItT
ˆˆˆ
The above condition shows that ˆ is hermitian. Denote ˆˆ H . (QM2.4)
gives:
tHt
i
tttH
i
Idtt
ˆˆˆ
Eventually we get the Schrödinger equation:
tH
t
t
i
ˆ
d
d
(QM2.5)
3. 3
One may show that UEH k
ˆˆˆ is the Hamiltonian of the system. Assume
trn ,
is an eigenfunction of Hˆ , belonging to the eigenvalue nE :
trEtrH nnn ,,ˆ
. For such a state Eq. (QM2.54) gives:
trEtrH
t
tr
i nnn
n
,,ˆ
d
,d
with a simple solution
rtE
i
tr nnn
exp, (QM2.6)
The temporal dependence factorizes out. Such states are known as stationary states,
because their modulus squared is constant. They are the eigenfunctions of the
equation with eigenvalues for the Hamilton operator. The time-independent
Schrödinger equation EH ˆ is written as: EU
m
p
ˆ
2
ˆ 2
, or, for potential
energy depending only on coordinates:
ErU
m
2
2
2
(QM2.7)
2.4. One-dimensional examples
2.4.1. The rectangular potential barrier
Fig. 1. 1D square potential barrier
The Schrödinger equation (QM2.7) writes
0
2
2
xUE
m
In the three regions we write:
4. 4
0
2
2
II E
m
with solution ikxikx
I eBeAx
11)( (QM2.8’)
0
2
02
IIII UE
m
with solution xx
II eBeAx
22)( (QM2.8’’)
0
2
2
IIIIII E
m
with solution ikx
III eAx 3)( (QM2.8’’’)
(see lectures for details). Here
2
2 2
mE
k EU
m
02
2 2
(QM2.9)
Continuity conditions:
)0()0( III , )()( aa IIIII , )0()0( III , )()( aa IIIII
The interesting quantity is the transparency
1
*
1
3
*
3
AA
AA
D
which measures the probability to find the particle on the right if initially it moves
towards the barrier from x . If we could neglect the term proportional to a
e
with regard to the term proportional to a
e
the final result is:
aEUm
U
EUE
D 02
0
0
2
2
exp16
(QM2.10)
For the general barrier described by a function U(x), the result is (see lectures):
2
1
d2
2
exp16 02
0
0
x
x
xEUm
U
EUE
D
(QM2.10’)
The factor
2
0
0
0 16
U
EUE
D
is close to the unity.
2.4.2. Potential well
5. 5
For 0UE the energetic spectrum is continuous. We are interested in the
situation 0UE . Inside the well 2/2/ axa the Schrödinger equation is
identical with that in (QM2.8’), outside it is the same as that from (QM2.8’’). As the
potential is symmetric, solutions may be symmetric (even) or anti-symmetric (odd).
Even solutions: Odd solutions
x
Ce
)(
1
x
Ce
)(
1
kxAcos)(
2
kxBsin)(
2
x
Ce
)(
3
x
Ce
)(
3
with notations from (QM2.9). Continuity conditions in –a/2 and a/2 give
eventually:
for even solutions –
k
ka
2
tan
for odd solutions –
k
ka
2
tan
Both relations may be written in the form 22
2
tan
k
k
ka . But k and are
both functions of the energy E. The last equation gives the possible values for this
energy, which is quantified. The program below shows the graphs:
ClearAll[x,k1,k2,e,u0]
a=10;u0=5;
k1[e_]=Sqrt[e]
k2[e_]=Sqrt[u0-e]
f[e_]=Tan[k1[e]*a]
g[e_]=2*k1[e]*k2[e]/((k1[e])^2-(k2[e])^2)
Plot[f[e],{e,0,u0}]
6. 6
1 2 3 4 5
-20
-10
10
20
Plot[g[e],{e,0,u0}]
1 2 3 4 5
-60
-40
-20
20
40
60
Show[%,%%]
Points give the allowed energies. In the example there are only 7 discrete energy
levels in the well. They are called bound states.
2.4.3. Quantum harmonic oscillator
7. 7
xEx
xm
x
m
xH
22
ˆ
222
With notations
2
and
0xm
x
:
0)(
d
d 2
2
2
x
Finite solutions depend on a quantum number ,3,2,1,0n :
2/2
eHC nnn (QM2.11)
with
...2
!2
)3)(2(1
2
!1
1
2 42
nnn
n
nnnnnn
H (QM2.12)
the Hermite polynomials.
Energies are quantified as
2
1
nEn (QM2.13)
To compute the normalization constants, use the normalization condition:
1
2
d (QM2.14)
One finds 2/1
!2
nC n
n , so the first three normalized eigenfunctions are:
2
0
2/1
0
0
2
1
exp
1
0
x
x
x
x
2
00
2/1
0
1
2
1
exp
2
2
1
1
x
x
x
x
x
x
2
0
2
0
2/1
0
2
2
1
exp24
8
1
2
x
x
x
x
x
x
2.5. Electron in a central field
2.5.1. The Scrödinger equation in spherical coordinates
8. 8
rErrUr
m
)(
2 0
2
(QM2.15)
Go to spherical variables and get for the Laplace operator:
,2
2
2
22
2
2
1
sin
1
sin
sin
111
r
rr
r
rr
r
(QM2.16)
(15) becomes: 0,,)(,,
1 2
,2
rrkr
r
r (QM2.15’)
with
)(
2
)( 2
02
rUE
m
rk
Separation of variables: )()()(),()(,, rRYrRr .
The simplest equation is the one for )( :
02
2
2
lm
d
d
with solutions
im
Ce . The periodicity condition 2 gives
...3,2,1,0 lm (QM2.17)
The number m is called the magnetic quantum number.
The equation in is difficult:
0
sin
sin
sin
1
2
2
m
d
d
d
d
, (QM2.18)
with m the magnetic quantum number and another quantum number, coming from
equation:
0,,, YY (QM2.19)
Conditions for continuous, single-valued and finite solutions imply
1 ll , l=0, 1, 2, … (QM2.20)
Another condition is lml (QM2.21)
Solutions of Eq. (QM2.19) with the above limitations are known as spherical
harmonics. Some expressions of the spherical functions are:
4
1
00 Y
9. 9
cos
4
3
10 Y
i
Y
esin
8
3
11
1cos3
16
5 2
20
Y
i
Y
ecossin
8
15
12
i
Y 22
22 esin
32
15
It turns out that spherical functions Ylm are eigenfunctions of the operator square of the
angular momentum 2222 ˆˆˆˆ
zyx llll . The exact eigenvalue equation is:
,1,ˆ 22
llmlm YllYl , (QM2.22)
with l=0, 1, 2, … and lml ,...,2,1,0 (QM2.23)
The magnitude of the angular momentum takes only quantized values equal to
1ll l=0, 1, 2, … (QM2.24)
For the operator zlˆ one finds the eigenvalue equation
zz ll ˆ
with eigenvalues lz ml , lml ,...,2,1,0 (QM2.25)
2.5.2. The Hydrogen atom (and hydrogen-like atoms)
For more graphs see e.g. http://undergrad-ed.chemistry.ohio-state.edu/H-AOs/
The potential energy is
r
Ze
rU
0
2
4
. The radial equation could be solved
analytically and the finite, continuous solutions are the Laguerre polynomials. They
depend on a single principal quantum number n=1, 2, 3, …. If we put together all the
preceding results, we see that the eigenfunctions lnlm mlnrl
,,,, depend on
three different quantum numbers:
The principal q.n. n=1, 2, 3, ….
The azimuthal q.n. l=0, 1, 2, …, n-1
The magnetic q.n. lml ,...,2,1,0
See attached files H eigenfunc 1 & 2.
2.5.3. Characterization of stationary states of H atoms.
Each state of a H atom is described by three quantum numbers, n, l, and ml.
Therefore the notation mlnrnlm ,,,, . These q.n. fix the quantified values of
the energy (see below), the magnitude of the angular momentum and of its projection
on the Oz axis, Eqs. (24, 25). Energy is given by the expression from Bohr’s theory:
10. 10
222
0
2
4
0
22
4
00
2
-eV
6.13
42
1
2
1
n
Ry
n
em
n
em
n
En
(QM2.26)
Here
2
0
2
2
0
2
2
02
0
2
4
0
2
1
42
1
42
cm
c
e
cm
em
Ry
(QM2.26’)
is the rydberg. The factor in the bracket is a dimensionless quantity called the fine
structure constant:
137
1
4 0
2
c
e
(QM2.27)
Because the energy does not depend on l and ml each level is degenerated and to
each value of the principal q.n. n correspond n2 different states with l=0, 1, 2, …, n-1
and lml ,...,2,1,0 .
Spectral notation: l=0 s-states
l=1 p-states
l=2 d-states
l=3 f-states
Possible states: 1s; 2s, 2p; 3s, 3p, 3d; 4s, 4p, 4d, 4f;
The three observables zllH ˆandˆ,ˆ 2
form a complete set of commutative
operators and their eigenvalues give the maximum information we could have
on the state.
See the applet http://www.falstad.com/qmatom/ for atomic orbitals and 1-D periodic
potentials.
2.6. Emission and absorption of the radiation
2.6.1 Quantum transitions. Eigenstates of the Hamiltonian described above are
stationary, i.e. a system being in such a state will remain there. But quantum systems
experience transitions from one stationary state to another and doing so they emit or
absorb energy, usually under the form of photons. That means the microsystem has
not the simple Hamiltonian we used before; it is perturbed by some extra terms
describing the interaction with other systems. As a result transitions occur. Let’s study
a system which at t=0 is in a stationary state 0
n of a Hamiltonian 0
ˆH :
000
0
ˆ
nnn EH . During a finite time t the system is acted upon by a perturbation
11. 11
described by the operator for the interaction energy Wˆ . The system evolves following
the Schrödinger equation:
WH
t
i ˆˆ
d
d
0 . is no more an eigenstate of 0
ˆH .
But because 0
ˆH is an observable, one can develop any state function 0
kkc .
Assume we measure the state of the system at the moment t. What is the transition
probability mnP to find the system in the state 0
m if it was initially in the state
0
n ? The result is given by the Fermi golden rule:
2
00
2
ˆ1
nmmn WP
(QM2.28)
There is a supplementary condition: the perturbation must contain a frequency
given by Bohr’s condition:
00
nm
mn
EE
(QM2.28’)
For each perturbation we have to compute the square of the matrix element
dxdydzWWW nmnmmn 0*0000 ˆˆ . If this element is zero, the transition is
forbidden. If it is non-zero, the transition is allowed. These computations lead to
selection rules controlling the variations of the quantum numbers.
2.6.2. Selection rules. The most encountered perturbations are given by electric
or magnetic interactions. They could be classified following the type of field: dipole,
quadri-pole,... The most intense elmgn transitions in atoms and molecules are the
dipolar-electric transitions. The selection rules for such transitions are:
1l 1,0 lm (QM2.29)
There are no selection rules for the principal q.n. Selection rules explain thoroughly
atomic and molecular spectra, as well as nuclear gamma spectra.
2.6.3. Einstein’s theory of emission and absorption. Assume identical atoms in
an external elmgn field. To make simple, atoms have but two energy levels with
energies Em and En. Levels have degeneracy gm and gn. The number of atoms are Nm
12. 12
and Nn The field may have many frequencies but we suppose that the Bohr frequency
of the transition (QM2.28’) is present. The spectral density of energy of the elmgn
field is mnuTu )(
Three processes: spontaneous and stimulated emission and stimulated
absorption. Denote by Amn the probability for an atom to suffer a spontaneous
emission in a time t . For stimulated processes such a probability should be
proportional to )( mnu and also to the corresponding Einstein coefficient Bmn or Bnm.
The detailed balance equation is:
tNuBtNuBA nmnnmmmnmnmn (*)
Hence
nm
mn
mnmn
mn
B
mn
m
n
B
B
uB
A
TkN
N
exp
This relation holds even for T , hence nmmn BB .
At equilibrium use the Boltzmann distribution:
Tkg
g
Tk
EE
g
g
N
N
B
nm
n
m
B
nm
n
m
n
m
expexp
The final result is the Planck relation:
1exp
1
)(
Tkg
gB
A
u
B
mn
m
nmn
mn
mn
It follows also that 32
3
cB
A mn
mn
mn
(QM2.30)
13. 13
The spontaneous coefficient A has s-1 as MU. If during dt a number dNm atoms suffer
transitions from the state m to the state n, one can write: dtNAdN mmnm , hence
/
)0()( t
mm eNtN (QM2.31)
The duration is the average time the atom stays in the excited state, the lifetime of
the state.
2.7. Magnetic interaction. Electronic spin
2.7.1. The orbital magnetic moment of the electron. If an atom is put in en
external magnetic field its spectrum becomes more complex. Many spectral lines split
into several closed components with different polarizations. This effect was
discovered by Zeeman in 1897. Soma lines split in just three components; this is the
normal effect and is explained classically. Indeed in classical electromagnetism each
magnetic moment interacts with a magnetic field through an energy term
BEmgn
(QM2.32)
Here
is the magnetic moment defined in analogy with the same quantity for a small
loop as Ai
(see figure below)
The vector nAA
is associated with the sense of the current i by the usual rule. If the
current is given by an electron revolving around the nucleus at a distance r with speed
v, its magnetic moment would be:
00
02
222/2 m
el
m
vremevr
r
vr
e
, with
l the angular momentum. One may write also:
14. 14
ll
m
e
orb
02
(QM2.33)
The quantity orb is the orbital gyromagnetic factor. The magnetic energy is
BlBl
m
e
BE orbmgn
02
(QM2.34)
In quantum mechanics:
BlBl
m
e
BH orbmgn
ˆˆ
2
ˆˆ
0
(QM2.34’)
Assume the magnetic field along Oz and use results from 2.5 to find:
B
m
e
B
m
e
mBlH zorbmgn
00 2
..,2,1,0
2
ˆˆ
(QM2.35)
The quantity (taken from CODATA, a site giving last values for physical constants)
1-24-
0
J·T10×(23)9.27400915
2
m
e
B
(QM2.36)
is the Bohr magneton. For a H atom the quantified energy of a level becomes
Bm
n
Ry
EEE Blmgnnn 2
(QM2.37)
The splitting of atomic levels and of spectral lines is shown below for a sp
transition.
15. 15
The deviation B
m
e
BB
02
is the value found experimentally.
Exercise: show that even for more complex spectra (e.g. for a pd transition) one
get a three-fold splitting of spectral lines.
2.7.2. Electronic spin. So in magnetic field each spectral line should split in
three components. But this is not true: one finds doublets, quartets … This is the
anomalous Zeeman effect. The explanation is given by the existence of electronic
spin. This hypothesis set forth by Uhlenbeck and Goudsmit in 1925 is based on
spectroscopy and on the experiment of Stern and Gerlach made in 1922. The sketch of
the experiment is shown below (from Wikipedia).
Exercise: read and learn the explanation in
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SternGerlach/SternGerlach.
html, or http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html#c2
16. 16
The inhomogeneous magnetic field exercise upon a beam of atoms a vertical force
along the z direction given by B
z
B
z
B
F atomatomatom
,cos
. Atoms (Ag)
were chosen to have no magnetic moment, so the force ought to be zero. However
deviations were observed. Assumptions were made to take into account some nonzero
orbital magnetic moment. Classical theory allows any angle between the relevant
vectors hence the classical prediction in the drawing. An orbital magnetic moment
should conduct to an odd number of lines. What was actually observed was just a pair
of spots as if the magnetic moment may have just two orientations, up and down. The
corresponding value for the intrinsic – or spin – quantum number have to be
2
1
s , so
as to get 212 s possible orientation. The values oh the corresponding magnetic
moment are B . The spin hypothesis says:
a) The electron has an intrinsic angular momentum, called spin, described by
operators zyx ssss ˆ,ˆ,ˆ,ˆ2
. They fulfill the usual commutation relations of angular
moments and commute with all coordinate and momentum operators.
b) The operator 2
ˆs has but one eigenvalue
2
1
s : 222
4
3
1ˆ sss (QM2.38’)
Its projection along Oz axis has two eigenvalues:
sz ms ˆ with
2
1
sm (QM2.38’’)
17. 17
c) To this spin angular momentum is associated a magnetic moment with
quantified projections Bsz .
The gyromagnetic ratio for the spin is twice as big as for the orbital case
(gyromagnetic spin anomaly):
02m
e
orb orbsss g
m
e
g
m
em
e
00
0
22
2
(QM.2.39)
Consequences: doublets, quadruplets … are allowed and explained; complex spectra
are explained; the periodic table is explained.
Examples of Zeeman splitting (from http://hyperphysics.phy-
astr.gsu.edu/hbase/quantum/zeeman.html )
"Normal" Zeeman effect
This type of splitting is observed with hydrogen and the zinc singlet.
This type of splitting is observed for spin 0 states since the spin does not contribute to
the angular momentum.
"Anomalous" Zeeman effect
When electron spin is included, there is a greater variety of splitting patterns.
3. Quantum statistics
3.1. Types of quantum statistics
18. 18
It turns out that every microscopic particle (elementary particles, atoms,
molecules) has spin. For a complex particle the spin builds up from the individual
spins of the components. Pauli shown that there are two very different families of
particles:
- Particles with integer spins, s=0, 1, 2, 3, … They are called bosons and obey
the Bose-Einstein statistics: at equilibrium the average number of particles
with energy i is given by:
1exp
Tk
g
n
B
EBi
i
EBi
(QM2.40)
Here gi is the degeneracy and EB the chemical potential. Examples: photons, an
even number of electrons or of protons. Bosons may occupy in any number a given
state. Their functions of state are symmetric to the particle permutations:
,...,...,2,1,...,...,2,1 jkkj
- Particles with half-integer spins, s=1/2, 3/2, 5/2, … They are called fermions
and obey the Fermi-Dirac statistics: at equilibrium the average number of
particles with energy i is given by:
1exp
Tk
g
n
B
Fi
i
DFi
(QM2.41)
Here gi is the degeneracy and F the chemical potential or the Fermi level.
Examples: electrons, protons, neutrons, an odd number of fermions. Fermions could
be at most one in a state (the Pauli exclusion principle). Their functions of state are
anti-symmetric to the particle permutations:
,...,...,2,1,...,...,2,1 jkkj
A comparison between classical (Boltzmann) and quantum statistics is given in
the following figure. At large values of the quantity at the exponent 1
TkB
Fi
both quantum statistics have the same classical limit.
19. 19
3.2. Lasers
According to (QM.2.30) the spontaneous emission is too large in the optical
field and a generator of stimulated radiation is difficult to build in this domain. For
such a generator to exist the stimulated power emitted must be larger than the
absorbed power supplemented by the dissipated power: extae PPP . The power
dissipates mainly by photons going out of the device. The rate of emitted and
absorbed photons are (see (*) p. 13)
mnmm WNBuN and nmnn WNBuN (QM2.42)
uBWW mnnmmn (QM2.43)
are the stimulated rates per atom for stimulated processes.
Each photon has the energy hence the powers in question are
mnme WNP and nmna WNP (QM2.44)
The stimulated rates per atom are related to the radiation intensity
V
cN
cuI
(QM2.45)
The effective cross-section is independent on the intensity and characterizes the pair
of levels m and n:
IWmn / (QM2.46)
The net rate of energy variation is the difference nmae NNIPP . Eventually
nm NNI
z
I
d
d
(QM2.47)
20. 20
As a rule the intensity reduces along the path because nm NN ; the radiation is
absorbed by the material. To amplify the radiation we need population inversion
nm NN . The Boltzmann relation (no degeneracy) is
Tk
EE
N
N
B
nm
n
m
exp ,
therefore the amplification is obtained for negative absolute temperatures T<0 K.
Two level-schemes for laser are presented below.
3-level scheme 4-level scheme
Laser block-scheme.
Without reaction io AII . The reaction sends back a part of the output intensity
iooio I
RA
A
IRIIAI
1
(QM2.48)
In general the amplification A and the reaction R depend on the frequency. If for a
certain frequency RA=1, the oscillation condition is satisfied and the system becomes
a generator of stimulated light.
3.3. Electrons in solids. Semiconductors.
3.3.1. Band structure of solids
21. 21
From Wikipedia:
“The electrons of a single isolated atom occupy atomic orbitals, which form a discrete
set of energy levels. If several atoms are brought together into a molecule, their
atomic orbitals split, as in a coupled oscillation. This produces a number of molecular
orbitals proportional to the number of atoms. When a large number of atoms (of order
× 1020 or more) are brought together to form a solid, the number of orbitals becomes
exceedingly large, and the difference in energy between them becomes very small, so
the levels may be considered to form continuous bands of energy rather than the
discrete energy levels of the atoms in isolation. However, some intervals of energy
contain no orbitals, no matter how many atoms are aggregated, forming band gaps.
Within an energy band, energy levels are so numerous as to be a near continuum.
First, the separation between energy levels in a solid is comparable with the energy
that electrons constantly exchange with phonons (atomic vibrations). Second, it is
comparable with the energy uncertainty due to the Heisenberg uncertainty principle,
for reasonably long intervals of time. As a result, the separation between energy levels
is of no consequence.”
Any solid has a large number of bands. In theory, it can be said to have infinitely
many bands (just as an atom has infinitely many energy levels). However, all but a
few lie at energies so high that any electron that reaches those energies escapes from
the solid. These bands are usually disregarded.
Bands have different widths, based upon the properties of the atomic orbitals from
which they arise. Also, allowed bands may overlap, producing (for practical purposes)
a single large band.
Figure 1 shows a simplified picture of the bands in a solid that allows the three major
types of materials to be identified: metals, semiconductors and insulators.
Metals contain a band that is partly empty and partly filled regardless of temperature.
Therefore they have very high conductivity.
The lowermost, almost fully occupied band in an insulator or semiconductor, is called
the valence band by analogy with the valence electrons of individual atoms. The
uppermost, almost unoccupied band is called the conduction band because only when
electrons are excited to the conduction band can current flow in these materials. The
difference between insulators and semiconductors is only that the forbidden band gap
between the valence band and conduction band is larger in an insulator, so that fewer
electrons are found there and the electrical conductivity is lower. Because one of the
main mechanisms for electrons to be excited to the conduction band is due to thermal
energy, the conductivity of semiconductors is strongly dependent on the temperature
of the material.
22. 22
This band gap is one of the most useful aspects of the band structure, as it strongly
influences the electrical and optical properties of the material. Electrons can transfer
from one band to the other by means of carrier generation and recombination
processes. The band gap and defect states created in the band gap by doping can be
used to create semiconductor devices such as solar cells, diodes, transistors, laser
diodes, and others.” (End of Wikipedia)
General statement: states above the Fermi level are mostly empty; states below are
typically filled with electrons. (see lectures)
Metals: The Fermi level is in the middle of allowed bands. Electrons easily acquire
energy from an external electric field and may participate to the conduction. The
density of carriers is -32322
cm1010 . It does not vary much with T.
Semiconductors. The Fermi level is in the middle of the forbidden gap, whose energy
is eV1gE . Carriers are electrons in the conduction band (CB) and holes in the
valence band (VB). To participate to conduction an e- must “jump” from VB to CB
and for this it needs energy, e.g. thermal energy. At room temperature eV
40
1
TkB .
The carrier density varies mainly exponentially with temperature, between 1012 and
1020 cm-3.
Insulators. The Fermi level is in the middle of the forbidden gap, whose energy is
eV5gE . At usual T an insulator does not conduct electrical currents.
23. 23
3.3.2. Carrier statistics in semiconductors
For intrinsic scond the Fermi level is roughly in the middle of the gap. For
doped scond the Fermi level moves towards the CB (n-type) or VB (p-type) as in Fig.
above. In this figure we see that the concentrations of carriers in both bands depend
very much on the position of the Fermi level:
- they are equal in the intrinsic case
- the e- concentration in CB is much bigger that holes concentration in the VB
for n-type semiconductors
- the concentration pf holes in the VB is much bigger that e- concentration in
the CB for p-type semiconductors
Electrons in CB are “quasi-particles” which may be different from free
electrons. In particular their mass depends to a great extent on the particular structure
of the CB. We may find (sometimes even in the same material) light electrons, with
effective mass m* smaller than the mass of free electrons m0 and heavy electrons, with
m*> m0. The same is true for the holes in the VB.
In nondegenerate semiconductors: the Fermi level is well into the forbidden
band and the Fermi distribution is approximated with the Boltzmann's:
TkEETkEE BVFBFC 5,5 (QM2.49)
It can be shown that in this case the densities of electrons in the CB and holes in
the VB are given by:
24. 24
Tk
EE
TN
Tk
EE
h
Tkm
n
B
CF
C
B
CFBn
expexp
2
2
2/3
2
*
(QM2.50’)
Tk
EE
TN
Tk
EE
h
Tkm
p
B
FV
V
B
FVBp
expexp
2
2
2/3
2
*
(QM2.50’’)
In such nondegenerate semiconductors the product np is constant:
2
expexp i
B
g
VC
B
VC
VC n
Tk
E
NN
Tk
EE
NNnp
(QM2.51)
where Eg is the energy gap. Quantities NC and NV are the effective concentrations of e-
in the CB and of holes in the VB. In Si at 300 K they are of the order of 1019 cm-3.
a) Intrinsic semiconductors.
The neutrality condition: pn writes with (QM2.50):
Tk
EE
N
Tk
EE
N
B
FV
V
B
CF
C expexp . The position of the Fermi level is:
*
*
ln
4
3
ln
2 n
pB
B
CV
C
VB
B
CV
F
m
mTk
Tk
EE
N
NTk
Tk
EE
E
(QM2.52)
The position of EF when T varies is shown below:
25. 25
The intrinsic concentration is
]
2
exp[
kT
E
NNn
g
VCi (QM2.53)
Exercise: Draw the graph of )/1()ln( Tfni . What quantity could be computed from the
slope of this graph ?
b) Impurified semiconductors
The most general case: donors and acceptors compensated semiconductor.
Taking into account spin degeneration of e on the impurities levels and the fact that a
ionized donor level means the absence of an e the neutrality condition is roughly:
No. of e in CB + No. ionized acceptors = No. of holes in VB + No. ionized donors
DA NpNn (QM2.54)
(The numbers are actually concentrations)
26. 26
b1) Type-n semiconductors.
Impurities from the Vth group e.g. P, As. have extra levels in the gap close to EC as in
the drawing below:
The neutrality condition: ndn pNn
.
dN is the concentration of ionized donors.
These ionized donors have lost their electrons, so their concentration is (up to a
degeneration factor 2 in the denominator)
1exp2
Tk
EE
N
N
B
dF
d
d
Neglecting the minority concentration np one finds:
1exp
exp
kT
EE
N
kT
EE
N
DF
DCF
C .
Denoting
kT
E
x Fexp
0expexp2 2
D
C
C
CD
C Nx
kT
E
Nx
kT
EE
N , or
0exp
22
exp
2
kT
EE
N
N
x
kT
E
x DC
C
D
D
27. 27
Keeping only the positive root:
kT
EE
N
NkT
E
x DC
C
D
D
exp
8
11
4
exp
or
kT
EE
N
N
kTEE DC
C
D
DF exp
8
11
4
1
(QM2.55)
Limit Cases.
A) Very low T (a few K). Condition: 1exp
8
kT
EE
N
N DC
C
D . We neglect the
units in (QM2.55) and find
C
DDC
F
N
NkTEE
E
2
ln
22
(QM2.55')
At T=0 K the Fermi level is at half the distance between EC and ED. When T grows EF
approaches the conduction band and later drops as in the figure above.
The e- concentration is
kT
EENN
Tn DCDC
n
2
exp
2
)lowvery( (QM2.56')
The graph T
fn 1ln is virtually a straight line with the slope
kT
EE DC
2
.
This is a method to find the position of the donor level below EC.
Question: what is the situation in an intrinsic semiconductor?
B) Low temperatures (from several tens of K to room temperatures). Condition:
1exp
8
kT
EE
N
N DC
C
D , hence DC NN 8 . We develop the square root
and eventually find
C
D
CF
N
N
kTEE ln (QM2.55'')
The argument of ln is negative, so the Fermi level comes close to the middle of the
gap. The e- concentration is
28. 28
D
C
D
Cn N
N
N
NTn
lnexp)low( (QM2.56'')
This is the depletion region when all the donors are ionized and the intrinsic
generation is low. The majority carriers have an almost constant concentration; the
concentration of the minority carriers grows rapidly with T:
kT
E
NN
N
n
n
n
p
g
VC
D
i
n
i
n
2
exp
22
C) High temperatures. We go back to the neutrality condition and assume all the
donors are ionized
n
i
DnDn
n
n
NpNn
2
. The positive solution is:
2
2
4
11
2 D
iD
n
N
nN
n , and hence
2
2
2
4
11
2
D
i
D
i
n
N
n
N
n
p
The Fermi level is given by:
kT
E
N
NN
N
N
kTEE
g
D
VC
C
D
CF exp
4
11
2
ln
2
For high T but with 1
4
2
2
D
i
N
n
one find again the depletion region (show!).
Finally, if 1
4
2
2
D
i
N
n
the donor impurities are overtaken by the intrinsic generation
The majority carrier concentration as function of 1/T is given below:
29. 29
b2) Type- p semiconductors.
Impurities from the IIIrd group e.g. Al, Ga. Extra levels in the gap close to EV as in the
drawing below:
aN is the concentration of ionized acceptors, i.e. states filled with electrons. They
obey a Fermi-Dirac statistics (up to a factor 2 in front of the exp):
1exp2
Tk
EE
N
N
B
FA
a
a The neutrality condition is pap pNn
. Neglecting np
one finds the dependence )(TEF as in the figure above. The analytic results are
obtained analogous to the case of the n-type scond. E.g. (QM2.55) becomes: