This document contains a formula sheet for quantum mechanics provided by Fiziks, an institute that prepares students for physics exams. It covers various topics in quantum mechanics including wave-particle duality, mathematical tools, the Schrodinger equation, angular momentum, multidimensional problems, perturbation theory, and more. It also provides the contact information and addresses for Fiziks' head office and branch office in New Delhi, India.
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
Introduction to quantum mechanics and schrodinger equationGaurav Singh Gusain
Classical mechanics describes macroscopic objects while quantum mechanics describes microscopic objects due to limitations of classical theory. Quantum mechanics was introduced after classical mechanics failed to explain experimental observations involving microscopic particles. Some key aspects of quantum mechanics are the photoelectric effect, blackbody radiation, Compton effect, wave-particle duality, the Heisenberg uncertainty principle, and Schrodinger's wave equation. Schrodinger's equation describes the wave function and probability of finding a particle.
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
The presentation opens up by introducing Schrodinger's time dependent and independent wave equation. Then it covers the derivation of time independent wave equation, followed by its applications.
This document summarizes problems involving the Fermi-Dirac distribution function. It includes:
1) Calculating the velocity of electrons at the Fermi level for potassium, which is 2.1eV.
2) Computing the probability of an energy level 0.01eV below the Fermi level being unoccupied, which is 0.405.
3) Finding the probabilities of electronic states being occupied that are 0.11eV above and below the Fermi level, which are 0.0126 and 0.987 respectively.
4) Evaluating the Fermi function for an energy level kT above the Fermi energy, which is 0.269.
5) Demonstrating
The Wigner-Seitz cell is a Voronoi cell used to study crystalline solids in physics. It is defined as the region in space closer to a given lattice point than any other. The Wigner-Seitz cell of the reciprocal lattice is called the first Brillouin zone. In one dimension the Wigner-Seitz cell is an interval, in two dimensions it is typically a hexagon, and in three dimensions it takes the shape of the smallest polyhedron enclosing the lattice point and its nearest neighbors. Higher order Brillouin zones are identified by planes perpendicular to reciprocal lattice vectors, with the first zone being the smallest such region around the origin.
Superconductivity is a phenomenon where certain materials have zero electrical resistance and expel magnetic fields when cooled below a critical temperature. It was discovered in 1911 by Heike Kamerlingh Onnes. The Meissner effect describes how superconducting materials actively push magnetic fields out of their interior when transitioning to the superconducting state. The BCS theory explains superconductivity as electrons forming Cooper pairs that pass through the material unimpeded. Superconducting materials include various metals, metal alloys, iron-based compounds, cuprates, and organic materials. Applications include maglev trains, medical imaging, and more efficient power transmission.
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
Introduction to quantum mechanics and schrodinger equationGaurav Singh Gusain
Classical mechanics describes macroscopic objects while quantum mechanics describes microscopic objects due to limitations of classical theory. Quantum mechanics was introduced after classical mechanics failed to explain experimental observations involving microscopic particles. Some key aspects of quantum mechanics are the photoelectric effect, blackbody radiation, Compton effect, wave-particle duality, the Heisenberg uncertainty principle, and Schrodinger's wave equation. Schrodinger's equation describes the wave function and probability of finding a particle.
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
The presentation opens up by introducing Schrodinger's time dependent and independent wave equation. Then it covers the derivation of time independent wave equation, followed by its applications.
This document summarizes problems involving the Fermi-Dirac distribution function. It includes:
1) Calculating the velocity of electrons at the Fermi level for potassium, which is 2.1eV.
2) Computing the probability of an energy level 0.01eV below the Fermi level being unoccupied, which is 0.405.
3) Finding the probabilities of electronic states being occupied that are 0.11eV above and below the Fermi level, which are 0.0126 and 0.987 respectively.
4) Evaluating the Fermi function for an energy level kT above the Fermi energy, which is 0.269.
5) Demonstrating
The Wigner-Seitz cell is a Voronoi cell used to study crystalline solids in physics. It is defined as the region in space closer to a given lattice point than any other. The Wigner-Seitz cell of the reciprocal lattice is called the first Brillouin zone. In one dimension the Wigner-Seitz cell is an interval, in two dimensions it is typically a hexagon, and in three dimensions it takes the shape of the smallest polyhedron enclosing the lattice point and its nearest neighbors. Higher order Brillouin zones are identified by planes perpendicular to reciprocal lattice vectors, with the first zone being the smallest such region around the origin.
Superconductivity is a phenomenon where certain materials have zero electrical resistance and expel magnetic fields when cooled below a critical temperature. It was discovered in 1911 by Heike Kamerlingh Onnes. The Meissner effect describes how superconducting materials actively push magnetic fields out of their interior when transitioning to the superconducting state. The BCS theory explains superconductivity as electrons forming Cooper pairs that pass through the material unimpeded. Superconducting materials include various metals, metal alloys, iron-based compounds, cuprates, and organic materials. Applications include maglev trains, medical imaging, and more efficient power transmission.
1. Nuclear models like the liquid drop model and shell model describe aspects of nuclear structure and behavior. The liquid drop model treats the nucleus like a liquid drop while the shell model treats nucleons as moving independently in nuclear orbits.
2. The shell model explains nuclear magic numbers and properties like spin and parity. Magic numbers correspond to nuclear stability when the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, etc. The shell model accounts for magic numbers in terms of closed nuclear shells.
3. While insightful, nuclear models have limitations and do not fully describe all nuclear phenomena. The liquid drop model cannot explain magic numbers while the shell model fails to explain the stability of certain
This document extends the Kronig-Penney model for modeling electron interactions with atomic lattices in semiconductors to arbitrary periodic potentials using numerical matrix mechanics. It introduces the Kronig-Penney model and represents periodic potentials using matrix representations in different basis sets. Numerical solutions are obtained and band structures are compared for different periodic potentials, including the Kronig-Penney, harmonic oscillator, and inverted harmonic oscillator potentials. Effective mass is also calculated from the band structure curvature.
Density of States (DOS) in Nanotechnology by Manu ShreshthaManu Shreshtha
1. The document discusses density of states (DOS), which describes the number of accessible quantum states at each energy level in a system. It explains how electrons populate energy bands based on DOS and the Fermi distribution function.
2. Calculation of DOS for a semiconductor is shown, and applications like quantization in low-dimensional structures and photonic crystals are described. Impurity bands formed by dopants are also discussed.
3. In summary, the document provides an overview of density of states, how it is calculated, and its applications in areas like quantization effects and photonic crystals.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
This document summarizes Louis de Broglie's hypothesis of wave-particle duality and its applications. It discusses de Broglie's proposal that particles have wave-like properties with a wavelength given by Planck's constant divided by momentum. The photoelectric effect and Compton effect provide evidence of wave and particle behavior of light and electrons. Wave-particle duality is exploited in technologies like electron microscopy and neutron diffraction to examine structures smaller than visible light wavelengths. While useful, wave-particle duality does not fully explain quantum phenomena like the Heisenberg uncertainty principle.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
The document discusses phase and group velocity of waves. It defines phase velocity as the velocity at which the phase of any single frequency component travels, represented by the crests of a wave. Group velocity is defined as the velocity at which the envelope or outline of a wave packet travels through space. The document demonstrates through equations and diagrams that for wave packets formed from superimposed waves, the phase velocity can be greater than the group velocity.
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
The integral & fractional quantum hall effectSUDIPTO DAS
Introductory idea of integral & fractional quantum hall effect and by imposing the idea of composite fermions showing the existence of fractional charge.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
This document is a project report submitted by Priyanka Verma and Smriti Singh for their Bachelor of Science degree in physics. It discusses elementary particles, including their characteristics, classification, conservation laws, and examples like electrons, positrons, protons, neutrons, pions, and kaons. The report includes certificates of completion from their college principal and physics professors.
The Zeeman effect is the splitting of a spectral line into multiple spectral lines when in the presence of a magnetic field. It was first observed in 1896 by Dutch physicist Pieter Zeeman when he placed a sodium flame between magnetic poles and observed the broadening of spectral lines. Zeeman's discovery earned him the 1902 Nobel Prize in Physics. The pattern and amount of splitting provides information about the strength and presence of the magnetic field.
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
The Davisson-Germer experiment demonstrated the wave-like properties of electrons by firing an electron beam at a nickel crystal target and observing the diffraction pattern of elastically scattered electrons. Electrons were accelerated towards the crystal in a vacuum chamber and the intensity of diffraction was measured at various angles using a movable electron detector. The results supported Bragg's law for diffraction from crystal lattices and de Broglie's hypothesis that particles act as waves, establishing quantum mechanics. The clearest diffraction was observed at an angle of 50° when the electrons' kinetic energy was 54 eV.
This document discusses superconductors and their properties. It begins by defining superconductivity as the phenomenon where certain materials conduct electricity without resistance when cooled below a critical temperature. It then discusses key properties of superconductors including zero electrical resistance, the effects of impurities and pressure, isotope effects, magnetic field effects, critical current density, and the Meissner effect. It categorizes superconductors as either type 1 or type 2 and provides examples of each. Finally, it outlines several applications of superconductors such as magnetic levitation trains, SQUID devices, and RF/microwave filters.
This document compares and contrasts linear and nonlinear optics. In linear optics, light propagates through a medium without changing frequency, while in nonlinear optics the medium's response depends on light intensity. Nonlinear optics involves effects where the induced polarization in a medium does not linearly depend on the electric field of the light. This allows frequency conversion via processes like second harmonic generation and sum frequency generation. Materials can exhibit a nonlinear refractive index, leading to self-focusing or defocusing of high intensity light beams. Nonlinear optical effects enable applications like frequency conversion, optical limiting, and all-optical signal processing.
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
This document defines and classifies different types of magnetic materials. It discusses ferromagnetic, paramagnetic, and diamagnetic materials, and how their properties including permeability and susceptibility differ. It also defines magnetically soft and hard materials, providing examples and characteristics of each. Finally, it outlines some applications of these magnetic materials, such as their use in recording devices, magnetic levitation, electromagnets, and permanent magnets.
The document provides an overview of quantum entanglement, including:
- Entangled particles cannot be described independently and must be described as a whole system. Measurements of one particle seem to instantaneously influence the other, even when separated by large distances.
- Early pioneers like Einstein, Schrodinger, and Podolsky struggled to understand entanglement and viewed it as evidence that quantum mechanics was incomplete.
- A 2015 experiment at Delft University was the first to close all loopholes in verifying Bell's theorem and violations of local realism, providing strong evidence that entanglement involves truly non-local correlations.
- Potential applications of entanglement include quantum cryptography, where entangled particles allow secure communication without a
Este documento presenta un resumen de los cuidados de enfermería en la administración de medicamentos. Fue escrito por cinco estudiantes de enfermería de la Universidad Técnica de Ambato para su clase de Farmacología y cubre temas como la administración segura de medicamentos y la prevención de errores. La bibliografía citada es un libro electrónico sobre este tema.
Este documento presenta los componentes de la sangre como parte de un proyecto escolar de estudiantes de enfermería de la Universidad Técnica de Ambato en Ecuador. El documento fue preparado por cinco estudiantes para su tutor, el Dr. Gustavo Moreno, como parte de sus estudios en el tercer semestre de la carrera de enfermería.
1. Nuclear models like the liquid drop model and shell model describe aspects of nuclear structure and behavior. The liquid drop model treats the nucleus like a liquid drop while the shell model treats nucleons as moving independently in nuclear orbits.
2. The shell model explains nuclear magic numbers and properties like spin and parity. Magic numbers correspond to nuclear stability when the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, etc. The shell model accounts for magic numbers in terms of closed nuclear shells.
3. While insightful, nuclear models have limitations and do not fully describe all nuclear phenomena. The liquid drop model cannot explain magic numbers while the shell model fails to explain the stability of certain
This document extends the Kronig-Penney model for modeling electron interactions with atomic lattices in semiconductors to arbitrary periodic potentials using numerical matrix mechanics. It introduces the Kronig-Penney model and represents periodic potentials using matrix representations in different basis sets. Numerical solutions are obtained and band structures are compared for different periodic potentials, including the Kronig-Penney, harmonic oscillator, and inverted harmonic oscillator potentials. Effective mass is also calculated from the band structure curvature.
Density of States (DOS) in Nanotechnology by Manu ShreshthaManu Shreshtha
1. The document discusses density of states (DOS), which describes the number of accessible quantum states at each energy level in a system. It explains how electrons populate energy bands based on DOS and the Fermi distribution function.
2. Calculation of DOS for a semiconductor is shown, and applications like quantization in low-dimensional structures and photonic crystals are described. Impurity bands formed by dopants are also discussed.
3. In summary, the document provides an overview of density of states, how it is calculated, and its applications in areas like quantization effects and photonic crystals.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
This document summarizes Louis de Broglie's hypothesis of wave-particle duality and its applications. It discusses de Broglie's proposal that particles have wave-like properties with a wavelength given by Planck's constant divided by momentum. The photoelectric effect and Compton effect provide evidence of wave and particle behavior of light and electrons. Wave-particle duality is exploited in technologies like electron microscopy and neutron diffraction to examine structures smaller than visible light wavelengths. While useful, wave-particle duality does not fully explain quantum phenomena like the Heisenberg uncertainty principle.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
The document discusses phase and group velocity of waves. It defines phase velocity as the velocity at which the phase of any single frequency component travels, represented by the crests of a wave. Group velocity is defined as the velocity at which the envelope or outline of a wave packet travels through space. The document demonstrates through equations and diagrams that for wave packets formed from superimposed waves, the phase velocity can be greater than the group velocity.
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
The integral & fractional quantum hall effectSUDIPTO DAS
Introductory idea of integral & fractional quantum hall effect and by imposing the idea of composite fermions showing the existence of fractional charge.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
This document is a project report submitted by Priyanka Verma and Smriti Singh for their Bachelor of Science degree in physics. It discusses elementary particles, including their characteristics, classification, conservation laws, and examples like electrons, positrons, protons, neutrons, pions, and kaons. The report includes certificates of completion from their college principal and physics professors.
The Zeeman effect is the splitting of a spectral line into multiple spectral lines when in the presence of a magnetic field. It was first observed in 1896 by Dutch physicist Pieter Zeeman when he placed a sodium flame between magnetic poles and observed the broadening of spectral lines. Zeeman's discovery earned him the 1902 Nobel Prize in Physics. The pattern and amount of splitting provides information about the strength and presence of the magnetic field.
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
The Davisson-Germer experiment demonstrated the wave-like properties of electrons by firing an electron beam at a nickel crystal target and observing the diffraction pattern of elastically scattered electrons. Electrons were accelerated towards the crystal in a vacuum chamber and the intensity of diffraction was measured at various angles using a movable electron detector. The results supported Bragg's law for diffraction from crystal lattices and de Broglie's hypothesis that particles act as waves, establishing quantum mechanics. The clearest diffraction was observed at an angle of 50° when the electrons' kinetic energy was 54 eV.
This document discusses superconductors and their properties. It begins by defining superconductivity as the phenomenon where certain materials conduct electricity without resistance when cooled below a critical temperature. It then discusses key properties of superconductors including zero electrical resistance, the effects of impurities and pressure, isotope effects, magnetic field effects, critical current density, and the Meissner effect. It categorizes superconductors as either type 1 or type 2 and provides examples of each. Finally, it outlines several applications of superconductors such as magnetic levitation trains, SQUID devices, and RF/microwave filters.
This document compares and contrasts linear and nonlinear optics. In linear optics, light propagates through a medium without changing frequency, while in nonlinear optics the medium's response depends on light intensity. Nonlinear optics involves effects where the induced polarization in a medium does not linearly depend on the electric field of the light. This allows frequency conversion via processes like second harmonic generation and sum frequency generation. Materials can exhibit a nonlinear refractive index, leading to self-focusing or defocusing of high intensity light beams. Nonlinear optical effects enable applications like frequency conversion, optical limiting, and all-optical signal processing.
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
This document defines and classifies different types of magnetic materials. It discusses ferromagnetic, paramagnetic, and diamagnetic materials, and how their properties including permeability and susceptibility differ. It also defines magnetically soft and hard materials, providing examples and characteristics of each. Finally, it outlines some applications of these magnetic materials, such as their use in recording devices, magnetic levitation, electromagnets, and permanent magnets.
The document provides an overview of quantum entanglement, including:
- Entangled particles cannot be described independently and must be described as a whole system. Measurements of one particle seem to instantaneously influence the other, even when separated by large distances.
- Early pioneers like Einstein, Schrodinger, and Podolsky struggled to understand entanglement and viewed it as evidence that quantum mechanics was incomplete.
- A 2015 experiment at Delft University was the first to close all loopholes in verifying Bell's theorem and violations of local realism, providing strong evidence that entanglement involves truly non-local correlations.
- Potential applications of entanglement include quantum cryptography, where entangled particles allow secure communication without a
Este documento presenta un resumen de los cuidados de enfermería en la administración de medicamentos. Fue escrito por cinco estudiantes de enfermería de la Universidad Técnica de Ambato para su clase de Farmacología y cubre temas como la administración segura de medicamentos y la prevención de errores. La bibliografía citada es un libro electrónico sobre este tema.
Este documento presenta los componentes de la sangre como parte de un proyecto escolar de estudiantes de enfermería de la Universidad Técnica de Ambato en Ecuador. El documento fue preparado por cinco estudiantes para su tutor, el Dr. Gustavo Moreno, como parte de sus estudios en el tercer semestre de la carrera de enfermería.
A empresa de tecnologia anunciou um novo smartphone com câmera aprimorada, maior tela e bateria de longa duração. O dispositivo também possui processador mais rápido e armazenamento expansível. O novo modelo será lançado em outubro por um preço inicial de US$799.
El documento presenta un resumen sobre la historia de la farmacología realizado por 5 estudiantes de enfermería de la Facultad de Ciencias de la Salud de la Universidad Técnica de Ambato en Ecuador. El resumen fue realizado bajo la supervisión del Dr. Gustavo Moreno y cubre el tema de la evolución histórica de la farmacología como disciplina científica.
La Unión Europea ha acordado un embargo petrolero contra Rusia en respuesta a la invasión de Ucrania. El embargo prohibirá la mayoría de las importaciones de petróleo ruso a la UE a partir de finales de año. Algunos países aún dependen en gran medida del petróleo ruso y buscan exenciones al embargo.
This document contains a quantum mechanics formula sheet from fiziks, an institute that provides coaching for physics exams. It covers various topics in quantum mechanics over 12 sections, including wave-particle duality, mathematical tools, the Schrodinger equation, angular momentum, perturbation theory, and identical particles. It also provides contact information for fiziks' head office and branch office in New Delhi, India.
La Unión Europea ha acordado un embargo petrolero contra Rusia en respuesta a la invasión de Ucrania. El embargo prohibirá las importaciones marítimas de petróleo ruso a la UE y pondrá fin a las entregas a través de oleoductos dentro de seis meses. Esta medida forma parte de un sexto paquete de sanciones de la UE destinadas a aumentar la presión económica sobre Moscú y privar al Kremlin de fondos para financiar su guerra.
Dokumen tersebut merupakan berita acara pengembalian lembar jawaban ujian nasional oleh Madrasah Aliyah Negeri 3 Bima kepada pihak terkait setelah selesainya ujian nasional tahun pelajaran 2015/2016, yang mencakup penyerahan lembar jawaban, penerimaan dengan tanggung jawab, serta lampiran daftar lembar jawaban yang diterima.
Este documento presenta el portafolio estudiantil de Joselyn Jiménez, estudiante de enfermería de la Universidad Técnica de Ambato. Incluye la visión y misión de la universidad, facultad y carrera, el perfil profesional de un enfermero, el sílabo de la asignatura de fisiología humana y diferentes evaluaciones y actividades realizadas a lo largo del semestre.
Axion Dark Matter Experiment Detailed Design Nbsp And OperationsSandra Long
The document discusses the Axion Dark Matter eXperiment (ADMX) and details its recent technological advances that have improved its sensitivity to detecting dark matter axions. Key advances include implementing state-of-the-art quantum amplifiers like a Microstrip SQUID Amplifier (Run 1A) and Josephson Parametric Amplifier (Run 1B) as well as a dilution refrigerator to reduce cavity temperature. These allowed ADMX to set the most stringent limits to date on axion dark matter in the 2.66-3.1 μeV mass range with DFSZ coupling sensitivity. The document also describes the ADMX detector setup and cavity design used to search for axion-photon conversion signals.
Ab-initio real-time spectroscopy: application to non-linear opticsClaudio Attaccalite
This document discusses ab-initio real-time spectroscopy and its application to non-linear optics. It begins with an overview of non-linear optics and the polarization response. It then discusses using time-dependent density functional theory to calculate nonlinear optical properties in real-time by solving the time-dependent Schrodinger equation under an external electric field. Examples are given of calculating second and third harmonic generation in materials. The document also discusses approaches to address challenges like treating bulk polarization and including many-body effects.
1. The document discusses key concepts in quantum physics including Planck's quantum theory, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's time-independent wave equation.
2. It provides details on experiments that verified the wave-like properties of matter including electron diffraction experiments by Davisson and Germer.
3. The document derives expressions for the energy levels of particles confined in one-dimensional potential wells and boxes in terms of Planck's constant and other variables.
Introduction to computation material science.
The presentation source can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/11/CompMatScience.odp
The document provides information about atomic structure including:
- It covers topics like Bohr model, quantum numbers, wave-particle duality, shapes of atomic orbitals, electronic configurations and periodic trends.
- It also includes practice exercises and answers related to atomic structure.
- The document is from ETOOS ACADEMY and provides contact information and syllabus for their atomic structure course.
This document discusses nonlinear optics and the dynamical Berry phase. It introduces nonlinear optics and summarizes early experiments. It then discusses how the Berry phase is related to nonlinear optical effects like second harmonic generation (SHG). Computational methods are presented for calculating SHG and other nonlinear optical properties from first principles using time-dependent density functional theory and the dynamical Berry phase. Examples of applying these methods to study SHG in semiconductors are provided.
This document provides an overview of the topics to be covered in a Physics 357/457 course on elementary particle physics. The course will examine the elementary particles like quarks and leptons, the forces that govern their interactions, and theoretical models used to understand particle physics, including the Standard Model. It outlines the course structure, evaluation methods, and provides a list of reference materials for further reading on topics covered during the semester.
1) The document provides information about a physical chemistry course on bonding taught by Professor Naresh Patwari, including recommended textbooks, websites with course materials, and what topics will be covered in the course like quantum mechanics, atomic structure, and chemical bonding.
2) Key concepts from quantum mechanics that will be discussed include the particle-wave duality of light and matter demonstrated by experiments, Planck's hypothesis and the photoelectric effect, the de Broglie hypothesis and diffraction of electrons, and the Heisenberg uncertainty principle.
3) Historical models of the atom will also be examined, like the Rutherford model, Bohr's model, and how Schrodinger's wave equation improved our understanding of
This document discusses band theory and several models used to describe electron behavior in solids, including the free electron model, nearly free electron model, and tight binding model. It provides an overview of each model, including their assumptions and how they describe properties like electron energy and band gaps. The free electron model treats electrons as independent particles but fails to explain material properties. The nearly free electron model incorporates a periodic potential and allows electron wavefunctions and energy bands to be described. The tight binding model uses a superposition of atomic orbitals to approximate electron wavefunctions in solids where potential is strong.
What's So Interesting About AMO Phyiscs?Chad Orzel
A talk given at the 2011 meeting of the Division of Atomic, Molecular, and Optical Physics (DAMOP) of the American Physical Society, summarizing recent and exciting results in AMO physics being presented at the meeting.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
The document is a study guide for the topic of Periodic Table & Periodicity. It includes sections on theory, exercises and answers. The theory section covers concepts like the modern periodic law, periodic trends in atomic properties, classification of elements into blocks, and periodic properties. It provides detailed explanations of topics like atomic and ionic radii, ionization energy, electron affinity, oxidation states and more. There are multiple exercises provided after the theory section along with an answer key.
Ultracold atoms in superlattices as quantum simulators for a spin ordering mo...Alexander Decker
This document discusses using ultracold fermionic atoms in optical lattices to simulate spin ordering models. It begins by describing how atoms can be trapped in optical lattices using laser light. It then proposes how a spin ordering Hamiltonian could be used to achieve superexchange interaction in a double well system. Finally, it suggests going beyond double wells to study resonating valence bond states in a kagome lattice, which could provide insights into phenomena like high-temperature superconductivity.
This document summarizes a research project that involves building a toy model of particle collisions using C++ and ROOT. The model simulates collisions by sampling probability distributions measured in real collisions. It generates particles and assigns them properties like momentum and angle. It also models physical processes like jet production and elliptic flow. The goal is to study how properties of particles like jets are affected by a quark-gluon plasma and vice versa. The model allows tuning parameters to learn about collision interactions and switch physics processes on or off.
Black hole entropy leads to the non-local grid dimensions theory Eran Sinbar
Based on Prof. Bekenstein and Prof. Hawking, the black hole maximal entropy , the maximum amount of information that a black hole can absorb, beyond its event horizon is proportional to the area of its event horizon divided by quantized area units, in the scale of Planck area (the square of Planck length).[1]
This quantization in entropy and information in the quantized units of Planck area leads us to the assumption that space is not “smooth” but rather divided into quantized units (“space cells”). Although the Bekenstein-Hawking entropy equation describes a specific case regarding the quantization of the 2D event horizon, this idea can be generalized to the standard 3 dimension (3D) flat space, outside and far away from the black hole’s event horizon. In this general case we assume that these quantized units of space are 3D quantized space “cells” in the scale of Planck length in each of its 3 dimensions.
If this is truly the case and the universe fabric of space is quantized to local 3D space cells in the magnitude size of Planck length scale in each dimension, than we assume that there must be extra non-local space dimensions situated in the non-local bordering’s of these 3D space cells since there must be something dividing space into these quantized space cells.
Our assumption is that these bordering’s are extra non local dimensions which we named as the “GRID” (or grid) extra dimensions, since they look like a non-local 3D grid bordering of the local 3D space cells. These non-local grid dimensions are responsible for all unexplained non-local phenomena’s like the well-known non-local entanglement or in the phrase of Albert Einstein “spooky action at a distance” [2].So by proving that space-time is quantized we prove the existence of the non-local grid dimension that divides space-time to these quantized 3D Planck scale cells.
X-RAY MEASUREMENTS OF THE PARTICLE ACCELERATION PROPERTIES AT INWARD SHOCKS I...Sérgio Sacani
We present new evidence that the bright non-thermal X-ray emission features in the interior of the Cassiopeia A
supernova remnant (SNR) are caused by inward moving shocks based on Chandra and NuSTAR observations. Several
bright inward-moving filaments were identified using monitoring data taken by Chandra in 2000–2014. These inwardmoving shock locations are nearly coincident with hard X-ray (15–40 keV) hot spots seen by NuSTAR. From proper
motion measurements, the transverse velocities were estimated to be in the range ∼2,100–3,800 km s−1
for a distance of
3.4 kpc. The shock velocities in the frame of the expanding ejecta reach values of ∼5,100–8,700 km s−1
, slightly higher
than the typical speed of the forward shock. Additionally, we find flux variations (both increasing and decreasing) on
timescales of a few years in some of the inward-moving shock filaments. The rapid variability timescales are consistent
with an amplified magnetic field of B ∼ 0.5–1 mG. The high speed and low photon cut-off energy of the inward-moving
shocks are shown to imply a particle diffusion coefficient that departs from the Bohm regime (k0 = D0/D0,Bohm ∼ 3–8)
for the few simple physical configurations we consider in this study. The maximum electron energy at these shocks is
estimated to be ∼8–11 TeV, smaller than the values of ∼15–34 TeV inferred for the forward shock. Cassiopeia A is
dynamically too young for its reverse shock to appear to be moving inward in the observer frame. We propose instead
that the inward-moving shocks are a consequence of the forward shock encountering a density jump of & 5–8 in the
surrounding material.
M82 X-2 is the first pulsating ultraluminous X-ray source discovered. The luminosity of these extreme pulsars, if
isotropic, implies an extreme mass transfer rate. An alternative is to assume a much lower mass transfer rate, but
with an apparent luminosity boosted by geometrical beaming. Only an independent measurement of the mass
transfer rate can help discriminate between these two scenarios. In this paper, we follow the orbit of the neutron star
for 7 yr, measure the decay of the orbit (P P orb orb 8 10 yr 6 1 · » - - - ), and argue that this orbital decay is driven by
extreme mass transfer of more than 150 times the mass transfer limit set by the Eddington luminosity. If this is true,
the mass available to the accretor is more than enough to justify its luminosity, with no need for beaming. This also
strongly favors models where the accretor is a highly magnetized neutron star.
Quantum Mechanics_ 500 Problems with Solutions ( PDFDrive ).pdfBEATRIZJAIMESGARCIA
The document provides an overview of the key developments in quantum theory that led to the emergence of quantum mechanics. It discusses Planck's quantum hypothesis, Einstein's explanation of the photoelectric effect and Compton effect, Bohr's theory of the hydrogen atom, and the Wilson-Sommerfeld quantization rule. Various concepts are defined, such as Planck's constant, photons, Compton wavelength, Bohr radius, Rydberg constant, and spectral series of the hydrogen atom. Example problems are provided to illustrate applications of these foundational ideas in quantum theory.
Alessandra Buonanno gave a lecture on the analytical and numerical relativity approaches used to model gravitational waveforms from inspiraling binary systems. She discussed how post-Newtonian theory, effective one body theory, and numerical relativity are used to approximately and exactly solve Einstein's field equations. She emphasized the crucial synergy between analytical and numerical relativity approaches to develop accurate gravitational waveform models like EOBNR and Phenom that have been used to infer astrophysics from LIGO/Virgo detections.
This document numerically analyzes the wave function of atoms under the combined effects of an optical lattice trapping potential and a harmonic oscillator potential, as used in Bose-Einstein condensation experiments. It employs the Crank-Nicolson scheme to solve the Gross-Pitaevskii equation. The results show that the wave function distribution responds to parameters like the trapping frequencies ratio, optical lattice intensity, chemical potential, and energy. Careful adjustment of the time step and grid spacing is needed to satisfy conservation of norms and energy as required by the physical system. Distributions of the overlapping potentials for different q-factors are presented.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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Answers about how you can do more with Walmart!"
Walmart Business+ and Spark Good for Nonprofits.pdf
Quantum formula sheet
1. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
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Head office
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Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
QUANTUM MECHANICS FORMULA SHEET
Contents
1. Wave Particle Duality
1.1 De Broglie Wavelength
1.2 Heisenberg’s Uncertainty Principle
1.3 Group velocity and Phase velocity
1.4 Experimental evidence of wave particle duality
1.4.1 Wave nature of particle (Davisson-German experiment)
1.4.2 Particle nature of wave (Compton and Photoelectric Effect)
2. Mathematical Tools for Quantum Mechanics
2.1 Dimension and Basis of a Vector Space
2.2 Operators
2.3 Postulates of Quantum Mechanics
2.4 Commutator
2.5 Eigen value problem in Quantum Mechanics
2.6 Time evaluation of the expectation of A
2.7 Uncertainty relation related to operator
2.8 Change in basis in quantum mechanics
2.9 Expectation value and uncertainty principle
3. Schrödinger wave equation and Potential problems
3.1 Schrödinger wave equation
3.2 Property of bound state
3.3 Current density
3.4 The free particle in one dimension
3.5 The Step Potential
3.7 Potential Barrier
3.7.1 Tunnel Effect
2. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
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Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
3.8 The Infinite Square Well Potential
3.7.1 Symmetric Potential
3.9 Finite Square Well Potential
3.10 One dimensional Harmonic Oscillator
4. Angular Momentum Problem
4.1 Angular Momentum
4.1.1 Eigen Values and Eigen Function
4.1.2 Ladder Operator
4.2 Spin Angular Momentum
4.2.1 Stern Gerlach experiment
4.2.2 Spin Algebra
4.2.3 Pauli Spin Matrices
4.3 Total Angular Momentum
5. Two Dimensional Problems in Quantum Mechanics
5.1 Free Particle
5.2 Square Well Potential
5.3 Harmonic oscillator
6. Three Dimensional Problems in Quantum Mechanics
6.1 Free Particle
6.2 Particle in Rectangular Box
6.2.1 Particle in Cubical Box
6.3 Harmonic Oscillator
6.3.1 An Anistropic Oscillator
6.3.2 The Isotropic Oscillator
6.4 Potential in Spherical Coordinate (Central Potential)
6.4.1 Hydrogen Atom Problem
3. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 3
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
7. Perturbation Theory
7.1 Time Independent Perturbation Theory
7.1.1 Non-degenerate Theory
7.1.2 Degenerate Theory
7.2 Time Dependent Perturbation Theory
8. Variational Method
9. The Wentzel-Kramer-Brillouin (WKB) method
9.1 The WKB Method
9.1.1 Quantization of the Energy Level of Bound state
9.1.2 Transmission probability from WKB
10. Identical Particles
10.1 Exchange Operator
10.2 Particle with Integral Spins
10.3 Particle with Half-integral Spins
11. Scattering in Quantum Mechanics
11.1 Born Approximation
11.2 Partial Wave Analysis
12. Relativistic Quantum Mechanics
12.1 Klein Gordon equation
12.2 Dirac Equation
4. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 4
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
1.Wave Particle Duality
1.1 De Broglie Wavelengths
The wavelength of the wave associated with a particle is given by the de Broglie relation
mv
h
p
h
where h is Plank’s constant
For relativistic case, the mass becomes
2
2
0
1
c
v
m
m
where m0 is rest mass and v is
velocity of body.
1.2 Heisenberg’s Uncertainty Principle
“It is impossible to determine two canonical variables simultaneously for microscopic
particle”. If q and qp are two canonical variable then
2
qpq
where ∆q is the error in measurement of q and ∆pq is error in measurement of pq and h is
Planck’s constant ( / 2 )h .
Important uncertainty relations
2
xPx (x is position and xp is momentum in x direction )
2
tE ( E is energy and t is time).
2
L (L is angular momentum, θ is angle measured)
5. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 5
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
1.3 Group Velocity and Phase Velocity
According to de Broglie, matter waves are associated with every moving body. These
matter waves moves in a group of different waves having slightly different wavelengths.
The formation of group is due to superposition of individual wave.
Let If tx,1 and tx,2 are two waves of slightly different wavelength and frequency.
tdxdkkAtkxA sin,sin 21
21 tkx
tddk
A
sin
22
cos2
The velocity of individual wave is known as Phase
velocity which is given as
k
vp
. The velocity of
amplitude is given by group velocity vg i.e.
dk
d
vg
The relationship between group and phase velocity is given by
d
dv
vv
dk
dv
kv
dk
d
v
p
pg
p
pg ;
Due to superposition of different wave of slightly different wavelength resultant wave
moves like a wave packet with velocity equal to group velocity.
1.4 Experimental evidence of wave particle duality
1.4.1 Wave nature of particle (Davisson-German experiment)
Electron strikes the crystals surface at an
angle . The detector, symmetrically located
from the source measure the number of
electrons scattered at angle θ where θ is the
angle between incident and scattered electron
beam.
The Maxima condition is given by
p
h
dnor
dn
where
2
cos2
sin2
S D
gv
phv
t
6. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 6
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
1.4.2 Particle nature of wave (Compton and Photoelectric Effect)
Compton Effect
The Compton Effect is the result of scattering of a photon by an electron. Energy and
momentum are conserved in such an event and as a result the scattered photon has less
energy (longer wavelength) then the incident photon.
If λ is incoming wavelength and λ' is scattered
wavelength and is the angle made by
scattered wave to the incident wave then
cos1'
cm
h
o
where
cm
h
o
known as c which is Compton wavelength ( c = 2.426 x 10-12
m) and mo is
rest mass of electron.
Photoelectric effect
When a metal is irradiated with light, electron may get emitted. Kinetic energy k of
electron leaving when irradiated with a light of frequency o , where o is threshold
frequency. Kinetic energy is given by
max 0k h h
Stopping potential sV is potential required to stop electron which contain maximum
kinetic energy maxk .
0seV h h , which is known as Einstein equation
photonincident
photonscattered
ElectronTarget ElectronScattered
7. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 7
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
2. Mathematical Tools for Quantum Mechanics
2.1 Dimension and Basis of a Vector Space
A set of N vectors N ,........, 21 is said to be linearly independent if and only if the
solution of equation
N
i
iia
1
0 is 1 2 Na = a = ..... a =0
N dimensional vector space can be represent as
N
i
iia
1
0 where i = 1, 2, 3 … are
linearly independent function or vector.
Scalar Product: Scalar product of two functions is represented as , , which is
defined as dx*
. If the integral diverges scalar product is not defined.
Square Integrable: If the integration or scalar product
2
, dx is finite then the
integration is known as square integrable.
Dirac Notation: Any state vector can be represented as which is termed as ket
and conjugate of i.e. * is represented by which is termed as bra.
The scalar product of and ψ in Dirac Notation is represented by (bra-ket). The
value of is given by integral rdtrtr 3*
,, in three dimensions.
Properties of kets, bras and brakets:
*
**
aa
Orthogonality relation: If and are two ket and the value of bracket 0
then , is orthogonal.
Orthonormality relation: If and are two ket and the value of bracket 0
and 1 1 then and are orthonormal.
Schwarz inequality:
2
8. fiziks
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Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 8
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
2.2 Operators
An operator A is mathematical rule that when applied to a ket transforms it into
another ket i.e.
A
Different type of operator
Identity operator I
Parity operator rr
For even parity rr , for odd parity rr
Momentum operator xP i
x
Energy operator H i
t
Laplacian operator 2
2
2
2
2
2
2
zyx
Position operator rxrX
Linear operator
For 2211 aa if an operator Aˆ applied on results in 1 1 2 2a A a A
then operator Aˆ is said to be linear operator.
9. fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com
Email: fiziks.physics@gmail.com 9
Head office
fiziks, H.No. 23, G.F, Jia Sarai,
Near IIT, Hauz Khas, New Delhi-16
Phone: 011-26865455/+91-9871145498
Branch office
Anand Institute of Mathematics,
28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
2.3 Postulates of Quantum Mechanics
Postulate 1: State of system
The state of any physical system is specified at each time t by a state vector t which
contains all the information about the system. The state vector is also referred as wave
function. The wave function must be: Single valued, Continuous, Differentiable, Square
integrable (i.e. wave function have to converse at infinity).
Postulate 2: To every physically measurable quantity called as observable dynamical
variable. For every observable there corresponds a linear Hermitian operatorAˆ . The
Eigen vector of Aˆ let say n form complete basis. Completeness relation is given
by
1
n n
n
I
Eigen value: The only possible result of measurement of a physical quantity na is one of
the Eigen values of the corresponding observable.
Postulate 3: (Probabilistic outcome): When the physical quantity A is measured on a
system in the normalized state the probability P(an) of obtaining the Eigen value an of
corresponding observable A is
2
1
ng
i
n
i
n
a
P a
where gn is degeneracy of state and
nu is the Normalised Eigen vector of Aˆ associated with Eigen value an.
Postulate 4: Immediately after measurement.
If the measurement of physical quantity A on the system in the state gives the result
an (an is Eigen value associated with Eigen vector na ), Then the state of the system
immediately after the measurement is the normalized projection
n
n
P
P
where Pn is
projection operator defined by n n .
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fiziks, H.No. 23, G.F, Jia Sarai,
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28-B/6, Jia Sarai, Near IIT
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Projection operator Pˆ : An operator Pˆ is said to be a projector, or projection operator, if
it is Hermitian and equal to its own square i.e. PP ˆˆ 2
The projection operator is represented by n n
n
Postulate 5: The time evolution of the state vector t is governed by Schrodinger
equation: ttHt
dt
d
i , where H(t) is the observable associated with total
energy of system and popularly known as Hamiltonian of system. Some other operator
related to quantum mechanics:
2.4 Commutator
If A and B are two operator then their commutator is defined as A,B AB-BA
Properties of commutators
† † †
, , ; , , ,
, , , ; , ,
, , , , , 0 (Popularly known as Jacobi identity).
C C
C C B C
C C C
, 0f
If X is position and xP is conjugate momentum then
1
,n n
xX P nX i
and 1
, n n
x xX P nP i
If b is scalar and A is any operator then , 0b
If [A, B] = 0 then it is said that A and B commutes to each other ie AB BA .
If two Hermition operators A andB , commute ie , 0 and if A has non
degenerate Eigen value, then each Eigen vector of Aˆ is also an Eigen vector ofB .
We can also construct the common orthonormal basis that will be joint Eigen state of
A and B .
The anti commutator is defined as ,
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2.5 Eigen value problem in Quantum Mechanics
Eigen value problem in quantum mechanics is defined as
n n na
where an is Eigen value and n is Eigen vector.
In quantum mechanics operator associated with observable is Hermitian, so its Eigen
values are real and Eigen vector corresponding to different Eigen values are
orthogonal.
The Eigen state (Eigen vector) of Hamilton operator defines a complete set of
mutually orthonormal basis state. This basis will be unique if Eigen value are non
degenerate and not unique if there are degeneracy.
Completeness relation: the orthonormnal realtion and completeness relation is given by
I
n
nnmnmn
1
,
where I is unity operator.
2.6 Time evaluation of the expectation of A (Ehrenfest theorem)
1 A
A,H
d
A
dt i t
where ,A H is commutation between operator A and
Hamiltonian H operator .Time evaluation of expectation of A gives rise to Ehrenfest
theorem .
1d
R P
dt m
, ,
d
P V R t
dt
where R is position, P is momentum and ,V R t is potential operator.
2.7 Uncertainty relation related to operator
If Aˆ and Bˆ are two operator related to observable A and B then
Bˆ,Aˆ
2
1
BˆAˆ
where
2
2
AˆAˆAˆ and
2
2
BˆBˆAˆ .
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2.8 Change in basis in quantum mechanics
If k are wave function is position representation and k are wave function in
momentum representation, one can change position to momentum basis via Fourier
transformation.
1
2
1
2
ikx
ikx
x k e dk
k x e dx
2.9 Expectation value and uncertainty principle
The expectation value A of A in direction of is given by
A
A
or
A n na P where nP is probability to getting Eigen value an in state .
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3. Schrödinger Wave Equation and Potential Problems
3.1 Schrödinger Wave Equation
Hamiltonian of the system is given by
2
2
P
H V
m
Time dependent Schrödinger wave equation is given by
t
iH
Time independent Schrödinger wave equation is given by EH
where H is Hamiltonian of system.
It is given that total energy E and potential energy of system is V.
3.2 Property of bound state
Bound state
If E > V and there is two classical turning point in which particle is classically trapped
then system is called bound state.
Property of Bound state
The energy Eigen value is discrete and in one dimensional system it is non degenerate.
The wave function xn of one dimensional bound state system has n nodes if n = 0
corresponds to ground state and (n – 1) node if n = 1 corresponds to ground state.
Unbound states
If E > V and either there is one classical turning point or no turning point the energy
value is continuous. If there is one turning point the energy eigen value is non-
degenerate. If there is no turning point the energy eigen value is degenerate. The particle
is said to be unbounded.
If E < V then particle is said to be unbounded and wave function will decay at ± ∞. There
is finite probability to find particle in the region.
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3.3 Current density
If wave function in one Dimension is x then current density is given by
xxim
Jx
*
*
2
which satisfies the continually equation 0
J
t
Where *
in general J v
2
J v where v is velocity of particle?
If Ji, Jr, Jt are incident, reflected and transmitted current density then reflection coefficient
R and transmission coefficient T is given by
r
i
J
R
J
and t
i
J
T
J
3.4 The free particle in one dimension
Hψ = Eψ xE
dx
d
m
2
22
2
ikxikx
AeeAx
Energy eigen value E
m
k
2
22
where 2
2
mE
k the eigen values are doubly degenerate
3.5 The Step Potential
The potential step is defined as
0
00
xV
x
xV
o
Case I: E > Vo
0
0
22
11
2
1
xBeAe
xBeAe
xikxik
xikxik
Hence, a particle is coming from left so D = 0.
R = reflection coefficient =
incident
reflected
J
J
=
2
1 2
1 2
k k
R
k k
xo
oV
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T = transmitted coefficient =
incident
dtransmitte
J
J
=
1 2
2
1 2
4k k
T
k k
where
2221
22
oVEm
k
mE
k
Case II: E < Vo
21
2
011
mE
kxBeAe xikxik
I
22
2
02
EVm
kxce oxk
II
1r
t
J
R
J
and 0t
i
J
T
J
For case even there is Transmission coefficient is zero there is finite probability to find
the particle in x > 0 .
3.7 Potential Barrier
Potential barrier is shown in figure.
Potential barrier is given by
ax
axV
x
xV
0
0
00
0
Case I: E > Vo
0
0
0
1
22
11
33
22
11
xEexx
axDeCexx
xBeAexx
xik
xikxik
xikxik
Where 21
2
mE
k
22
2
oVEm
k
Transmission coefficient
1
2
1sin
14
1
1
T
a
Energy
o x
oV
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Reflection coefficient
1
2
1sin
14
1
R Where 2
2
,
o
o
mV
a
V
E
Case II: E < Vo
3.7.1 Tunnel Effect
1 1
2 2
1
0
0
0
ik x ik x
I
k x k x
II
ik x
III
Ae Be x
Ce Be x a
Fe x
21
sin 1
4 1
R h
and
21
1 sin 1
4 1
T h
Where
o
o
V
EmV
a ,
2
2
For E << Vo
2
22
1
16
EVma
oo
o
e
V
E
V
E
T
Approximate transmission probability ak
eT 22
where
22
2
EVm
k o
3.8 The Infinite Square Well Potential
The infinite square well potential is defined as as shown in the figure
ax
ax
x
xV 00
0
Since V(x) is infinite in the region 0x and x a so the wave function corresponding
to the particle will be zero.
The particle is confined only within region 0 ≤ x ≤ a.
Time independent wave Schrödinger wave equation is given by
xV
o a x
ao
oV
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E
dx
d
m
2
22
2
sin cosA kx B kx
B = 0 since wave function must be vanished at boundary ie 0x so sinA kx
Energy eigen value for bound state can be find by ka n where 1,2,3...n
The Normalized wave function is for th
n state is given by
2
sinn
n x
x
a a
Which is energy Eigen value correspondence to th
n
2
222
2ma
n
En
where n = 1, 2, 3 .....
othonormality relation is given by
0
sin sin i.e.
2
0
1
2
a
mn
m x n x a
dx
L L
m n
a
m
If x is position operator xP Px is momentum operator and n x is wave function of
particle in nth
state in one dimensional potential box then
2
sinn
n x
x
a a
then
*
2 2
2 * 2
2 2
*
2 2 2 2
2 *
2 2
2
2 2
0
2
n n
n n
x n n
x n n
a
x x x x
a a
x x x x
n
P x i x
x
n
P x x
m x a
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The uncertainty product is given by 22
2
1
12
1
n
nPxx
The wave function and the probability density function of particle of mass m in one
dimensional potential box is given by
3.8.1 Symmetric Potential
The infinite symmetric well potential is given by
22
0
22
a
x
a
a
xor
a
xxV
Schrondinger wave function is given by
)(0
2
cos
2
sin
)(0
2
cos
2
sin
0,
2
2
cossin
22
;
2 2
22
ii
ka
B
ka
A
i
ka
B
ka
Aso
x
a
xat
mE
kwherekxBkxA
a
x
a
E
dx
d
m
Hence parity ( ) commute with Hamiltonian ( )H then parity must conserve
So wave function have to be either symmetric or anti symmetric
2
a
2
a
a
xV
m
En
2
1
22
1
12 42 EEn
13 93 EEn
a
x
a
sin
2
1
a
x
a
2
sin
2
2
a
x
a
3
sin
2
3
2
1
2
2
2
3
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For even parity
cosB kx and Bound state energy is given by 0
2
cos
ka
,....5,3,10
2
n
a
n
k
ka
Wave function for even parity is given as
a
xn
a
cos
2
For odd parity
Ψ (x) = A sin kx and Bound state energy is given by
sin 0
2
ka
, 0
2
ka n
k
a
2,4,6,........n
For odd parity Wave function is given as
a
xn
a
sin
2
The energy eigen value is given by ,......3,2,1
2 2
222
n
ma
n
En
First three wave function is given by
a
x
a
x
a
x
a
x
a
x
a
x
3
cos
2
2
sin
2
cos
2
3
2
1
where
a
2
is normalization constant.
2
22
2ma
2
a
2
a
2
22
2
9
ma
2
22
2
4
ma
2
a
2
a
2
a
2
a
x3
x2
x1
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3.9 Finite Square Well Potential
For the Bound state E < Vo
Again parity will commute to Hamiltonian
So wave function is either symmetric or
Anti symmetric
For even parity
2
22
cos
2
a
xAex
a
x
a
kxCx
a
xAex
x
III
II
x
I
wave function must be continuous and differentiable at boundaries so using boundary
condition at
2
a
one will get
2
tan
ka
k
tan where
2
a
2
ka
For odd parity
2
22
sin
2
2
1
a
xAex
a
x
a
kxDx
a
xAex
x
III
x
using boundary condition one can get
cot where
2
2
EVm o
2
2
mE
k
and 2
2
22
2
amV
n o
which is equation of circle.
The Bound state energy will be found by solving equation
tann for even
2
a
2
a x
oV
xV
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cotn for odd
2
22
2
amV
n o
one can solve it by graphical method.
The intersection point of = tan (solid curve) and
2
2 2
2
2
omV a
(circle) give
Eigen value for even state and intersection point at = - cot (dotted curve) and
2
2 2
2
2
omV a
(circle) give Eigen value for odd state.
The table below shows the number of bound state for various range of 2
0V a where Voa2
is strength of potential.
Voa2
Even eigen function Odd eigen function No. of Bound
states
m2
22
1 0 1
m
aV
m 2
4
2
22
2
0
22
1 1 2
m
aV
m 2
9
2
22
2
0
22
2 1 3
m
aV
m 2
16
2
9 22
2
0
22
2 2 4
n
tann
o
2
2
3 2
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The three bound Eigen function for the square well
3.10 One dimensional Harmonic Oscillator
One dimensional Harmonic oscillator is given by
xxmxV 22
2
1
The schrodinger equation is given by
Exm
dx
d
m
22
2
22
2
1
2
It is given that x
m
and
E
k
2
The wave function of Harmonic oscillator is given by.
2
2/14/1 2
!2
1
eH
n
m
nnn
and energy eigen value is given by
nE = (n+1/2) ; n = 0, 1, 2, 3, ....
The wave function of Harmonic oscillator is shown
0H ( ) = 1 , 1H ( ) = 2 , 2
2H ( ) = 4 -2
It n and m wave function of Harmonic oscillator then
mnnm dxxx
x
2
a
2
a
x2
x
2
a
2
a
x1
x
x3
1n
2
0
E
xV
0n
2n
2
3
1
E
2
5
2
E
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Number operator
The Hamiltonian of Harmonic operator is given by 22
2
2
1
2
Xm
m
P
H x
Consider dimensioned operator as HPX ˆ,ˆ,ˆ where
X
m
X ˆ
ˆ
x xP m P HH ˆ so 22 ˆˆ
2
1ˆ XPH x .
Consider lowering operator xiPXa ˆ
2
1
and raising † 1 ˆ
2
xa X iP
2/1ˆ NH
where †
N a a and 1/ 2H N N is known as number operator
n is eigen function of N with eigen value n.
N n n n and
1
2
H n N n
nnnH
2
1
where n = 0, 1, 2, 3,………
Commutation of a and †
a : †
[a, a ] = 1, †
[a , a] = -1 and [N, a] = -a , † †
[N a ] = a
Action of a and †
a operator on n
11
001
nnna
abutnnna
Expectation value of 22
,,, XX PPXX in stationary states
mnPn
m
X
PX
m
i
aa
Paa
m
X
X
X
X
2
1
,12
2
0,0
2
,
2
22
2
;0,
2
1
XX PXnfornPX
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4. Angular Momentum Problem
Angular momentum in Quantum mechanics is given kLjLiLL zyx
ˆˆˆ
Where xyzzxYyzX YPXPLXPZPLZPYPL ;;
and
x
iPX
,
y
iPY
,
z
ipZ
Commutation relation
,x y zL L i L , ,y z xL L i L , ,z x yL L i L
2 2 2 2
X Y ZL L L L and 2
, 0XL L , 2
, 0YL L , 2
, 0ZL L
4.1.1 Eigen Values and Eigen Function
iL
iL
SiniL
Z
Y
X
cotsincos
cotcos
Eigen function of ZL is
im
e
2
1
.
and Eigen value of ZL m where m = 0, ± 1, ± 2...
L2
operator is given by
2
2
2
22
sin
1
sin
sin
1
L
Eigen value of 2
L is 2
( 1)l l where l = 0, 1, 2, ...l
Eigen function of 2
L is m
lP where m
lP is associated Legendre function
L2
commute with Lz so both can have common set of Eigen function.
, ( )m m im
l lY P e
is common set of Eigen function which is known as spherical
harmonics .
The normalized spherical harmonics are given by
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2 1 !
( , ) 1 cos
4 !
mm m im
l l
l l m
Y P e
l m
l m l
4
1
,0
0 Y
i
eY sin
8
3
,1
1 , 0
1
3
, cos
4
Y
, 1
1
3
, sin
8
i
Y e
,1, 22 m
l
m
l YllYL l = 0, 1, 2, ……. And
,, m
l
m
lZ YmYL m = -l ,(-l +1) ..0,….. (l – 1)( l ) there is 2 1l
Degeneracy of 2
L is 2 1l .
Orthogonality Relation
2
' '
0 0
, , sinm m
l l ll mmd Y Y d
4.1.2 Ladder Operator
Let X YL L iL and X YL L iL
Let us assume ml, is ket associated with 2
L and ZL operator.
22
, 1 ,L l m l l l m 0,1,2,.......l
,, , ...0,...zL l m m l m m l l
Action of L+ and L- on ml, basis
, 1 1 , 1L l m l l m m l m
, 1 1 , 1L l m l l m m l m
Expectation value of XL and YL in direction of ml,
0xL , 0yL
2
2 2 2
1
2
X YL L l l m
2
2
1
2
X YL L l l m
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4.2 Spin Angular Momentum
4.2.1 Stern Gerlach experiment
When silver beam is passed to the inhomogeneous Magnetic field, two sharp trace found
on the screen which provides the experimental evidence of spin.
4.2.2 Spin Algebra
ˆˆ ˆX Y ZS S i S j S k , 2 2 2 2
X Y ZS S S S
2
, 0XS S , 2
, 0YS S
2
, 0ZS S and
,X Y ZS S iS ,Y Z XS S iS ,Z X YS S iS
2 2
, 1 ,s sS s m s s s m , ,z s sS s m m s m where ss m s
X YS S iS and X YS S iS
, 1 1 , 1s s s sS s m s s m m s m
4.2.3 Pauli Spin Matrices
For Spin
1
2
Pauli matrix
1
2
s ,
1 1
,
2 2
sm
Pauli Matrix is defined as
e.anticommutMatrixSpinPauli;0
10
01
0
0
01
10
222
kj
I
i
i
jkkj
zyx
zyx
1 if is an even permutation of , ,
1 if is an odd permutation of , ,
0 if any two indices among , , are equal
jkl
jkl x y z
jkl x y z
j k l
zzyyxx SSS
2
,
2
,
2
, 1 1 , 1s s s sS s m s s m m s m
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10
01
20
0
201
10
2
zyx S
i
i
SS
01
00
00
10
SS
10
01
4
3 2
2
S
For spin ½ the quantum number m takes only two values
2
1
sm and
2
1
. So that two
states are
2
1
,
2
1
and
2
1
,
2
1
, smS
2
1
,
2
1
4
3
2
1
,
2
1 2
2
S ,
2
1
,
2
1
4
3
2
1
,
2
1 2
2
S
2
1
,
2
1
2
1
2
1
,
2
1
zS ,
2
1
,
2
1
2
1
2
1
,
2
1
zS
0
2
1
,
2
1
S ,
2
1
,
2
1
2
1
,
2
1
S
2
1
,
2
1
2
1
,
2
1
S , 0
2
1
,
2
1
S
4.3 Total Angular Momentum
Total angular momentum J = L + S , kJjJiJJ zyx
ˆˆˆ
jmj, is the Eigen ket at J2
and Jz and x yJ J iJ x yJ J iJ
jj mjjjmjJ ,1, 22
, jjjz mjmmjJ ,,
1,11, jjjj mjmmjjmjJ
1,11, jjjj mjmmjjmjJ
J L S l s j l s
jmjSLJ jzzz and l s jm m m
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5. Two Dimensional Problems in Quantum Mechanics
5.1 Free Particle
Hψ = Eψ
E
yxm
2
2
2
22
2
x and y are independent variable. Thus
1
,
2
1
2
yx
ik yik x
n
i k r
x y e e
e
Energy Eigen value 2
2
22
2
22
k
m
kk
m
yx
As total orientation of k
which preserve its magnitude is infinite. So energy of free
particle is infinitely degenerate.
5.2 Square Well Potential
0V x x a and 0 y a
otherewise
2 2 2
2 2
2
H E
m x y
The solution of Schrödinger wave equation is given by Wave function
, 2
4
sin sinx y
yx
n n
n yn x
a aa
Correspondence to energy eigen value
2
2
2
222
,
2 a
n
a
n
m
E yx
nynx
1,2,3...xn and 1,2,3...yn
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Energy of state (nx, ny) Degeneracy
Ground state 2
22
2
2
ma
(1, 1) Non degenerate
First state 2
22
2
5
ma
(1, 2), (2,1) 2
Second state 2
22
2
8
ma
(2, 2) Non-degenerate
5.3 Harmonic oscillator
Two dimensional isotropic Harmonic oscillators is given by
22
2 2 21
2 2 2
yx
pp
H m x y
m m
1x yn n x yE n n where 0,1,2,3...xn 0,1,2,3...yn
1nE n
degeneracy of the nth
state is given by (n + 1) where n = nx + ny.
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6. Three Dimensional Problems in Quantum Mechanics
6.1 Free Particle
EH
E
zyxm
2
2
2
2
2
22
2
Hence x, y, z are independent variable. Using separation of variable one can find the
zikyikxik
k
zyn
eeezyx 2/3
2
1
,,
3/2 .
2 ik r
e
Energy Eigen value 2
2
222
2
22
k
m
kkk
m
zyx
As total orientation of k which preserve its magnitude is infinite, the energy of free
particle is infinitely degenerate.
6.2 Particle in Rectangular Box
Spinless particle of mass m confined in a rectangular box of sides Lx, Ly, Lz
, , 0 ,xV x y z x L 0 ,yy L 0 ,zz L
= other wise .
The Schrodinger wave equation for three dimensional box is given by
E
zyxm
H
2
2
2
2
2
22
2
Solution of the Schrödinger is given by Eigen function x y zn n n and energy eigen value
is x y zn n nE is given by
z
z
y
y
x
x
zyx
nnn
L
zn
L
yn
L
xn
LLLzyx
sinsinsin
8
2
2
2
2
2
222
2 z
z
y
y
x
x
nnn
L
n
L
n
L
n
m
E zyx
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6.2.1 Particle in Cubical Box
For the simple case of cubic box of side a, the
i.e. Lx = Ly = Lz = a
a
zn
a
yn
a
xn
a
zyx
nnn zyx
sinsinsin
8
3
222
2
22
2
zyxnnn nnn
ma
E zyx
1,2,3...xn 1,2,3...yn 1,2,3...zn
Energy of state (nx, ny, nz) Degeneracy
E of ground state 2
22
2
3
ma
(1, 1, 1) Non degenerate
E of first excited state 2
22
2
6
ma
(2, 1, 1) (1, 2, 1) (1, 1, 2 3
E of 2nd
excited state 2
22
2
9
ma
(2, 2, 1) (2, 1, 2) (1, 2, 2) 3
6.3 Harmonic Oscillator
6.3.1 An Anistropic Oscillator
222222
2
1
2
1
2
1
,, ZmYmXmZYXV ZYX
zzyyxxnnn nnnE zyx
2
1
2
1
2
1
where
0,1,2,3...xn 0,1,2,3...yn 0,1,2,3...xn
6.3.2 The Isotropic Oscillator
x y z
3
2x y zn n n x y zE n n n
where x y zn n n n 0,1,2,3...n
Degeneracy is given by = 21
2
1
nn
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6.4 Potential in Spherical Coordinate (Central Potential)
Hamiltonion in spherical polar co-ordinate
2 2
2
2 2 2 2 2
1 1 1
sin
2 sin sin
r V r E
m r rr r r
2 2
2
2 2
1
2 2
L
r V r
m r rr mr
L2
is operator for orbital angular momentum square.
So ,m
lY are the common Eigen state of and L2
because [H, L2
] = 0 in central force
problem and 2 2
, 1 ( , )m m
l lL Y l l Y
So ,,r can be separated as ,m
lf r Y
2 22
2 2
12
0
2 2
d f r l lf
f r V r E f r
m r rdr mr
To solve these equations
r
ru
rf
So one can get
0
2
1
2 2
2
2
22
uE
mr
ll
rV
dr
ud
m
Where
2
2
2
1
mr
ll
is centrifugal potential and
2
2
2
1
mr
ll
rV
is effective potential
The energy Eigen function in case of central potential is written as
, , , ,m m
l l
u r
r f r Y Y
r
The normalization condition is
2
, , 1r d
2 2
22
0 0 0
, sin 1m
l
u r
r dr Y d d
r
2
0
1u r dr
or
2 2
0
1R r r dr
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6.4.1 Hydrogen Atom Problem
Hydrogen atom is two body central force problem with central potential is given by
2
04
e
V r
r
Time-independent Schrondinger equation on centre of mass reference frame is given by
2 2
2 2
, ,
2 2
R r V r R r E R r
m
where R is position of c.m and r is distance between proton and electron.
R,r R r
The Schronginger equation is given by
h2
2M
1
R
R
2
R
h2
2
1
r
R
2
rV r
E
Separating R and r part
RER
M
R 2
2
2
rrVrr
2
2
2
Total energy R rE = E + E
ER is Energy of centre of mass and Er is Energy of reduce mass µ
3/ 2
1
2
ik R
R e
2 2
2
R
k
E
M
i.eCentre of mass moves with constant momentum so it is free particle.
Solution of radial part
h2
2
d2
u r
dr2
l l 1 h2
2r2
e2
4 0 r
u r E r
For Hydrogen atom em
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The energy eigen value is given by nE =
4
2 2 2
02 4
em e
n
eV
n2
6.13
1,2,3...n
And radius of th
n orbit is given by
2 2
0
2
4
n
n
r
m e
1,2,3...n where me is mass of
electron and e is electronic charge.
n is known as participle quantum number which varies as 1, 2,...n l n For the
given value of n the orbital quantum numbern can have value between 0 and 1n (i.e. l
= 0, 1, 2, 3, ….. n – 1) and for given value of l the Azimuthal quantum number m varies
from – l to l known as magnetic quantum mechanics .
Degeneracy of Hydrogen atom without spin = 2
n and if spin is included the degeneracy
is given by
1
2
0
2 2 1 2
n
n
l
g l n
For hydrogen like atom
2
2
6.13
n
z
En where z is atomic number of Hydrogen like atom.
Normalized wave function for Hydrogen atom i.e. Rnl (r) where n is principle quantum
number and l is orbital quantum.
n l E(eV) Rnl
1 0 2
-13.6z
0
3/ 2
/
0
2 zr az
e
a
2 0 2
- 3.4 z
0
3/ 2
/ 2
0 0
2
2
zr az zr
e
a a
2 0 2
- 3.4 z
0
3/ 2
/ 20
0 0
1
24
zr az zr
e
a a
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The radial wave function for hydrogen atom is Laguerre polynomials and angular part of
the wave function is associated Legendre polynomials .
For the nth
state there is n – l – 1 node
If Rnl is represented by ln, then
2
0
1
3 1
2
nl r nl n l l a
2 2 2 2
0
1
5 1 3 1
2
nl r nl n n l l a
1
2
0
1
nl r nl
n a
2
3 2
0
2
2 1
nl r nl
n l a
rR10
0/arr
rR20
0/arr
21R
0/ar
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7. Perturbation Theory
7.1 Time Independent Perturbation Theory
7.1.1 Non-degenerate Theory
For approximation methods of stationary states
o PH H H
Where H Hamiltonian can be divided into two parts in that oH can be solved exactly
known as unperturbed Hamiltonian and pH is perturbation in the system eigen value
of oH is non degenerate
It is known n
o
nno EH and 1 pH W where 1
Now o n n nH W E where n is eigen function corresponds to eigen value nE
for the Hamiltonian H
Using Taylor expansion
........2210
nnnn EEEE and .........221
nnnn
First order Energy correction 1
nE is given by 1
n n nE W
And energy correction up to order in is given by 1
n n n nE E W
First order Eigen function correction 1
0 0
m n
n m
m n n m
W
E E
And wave function up to order correction in 0 0
m n
n n m
m n n m
W
E E
Second order energy correction
2
2
0 0
m n
n
m n n m
W
E
E E
Energy correction up to order of 2
2
2
0 0 0
m n
n n n
m n n m
W
E E W
E E
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7.1.2 Degenerate Theory
( )n o p n n nH H H E
0
nE is f fold degenerate fEH nnno ,.....3,2,10
To determine the Eigen values to first-order and Eigen state to zeroth order for an f-fold
degenerate level one can proceed as follows
First for each f-fold degenerate level, determine f x f matrix of the perturbation pHˆ
ff
f
f
ppfpf
ppp
ppp
p
HHH
HHH
HHH
H
.........
.
.
.........
.........
ˆ
21
22221
11211
where p n p nH H
then diagonalised pH and find Eigen value and Eigen vector of diagonalized pH which
are nE and n respectively.
10
nnn EEE and
f
nn q
1
7.2 Time Dependent Perturbation Theory
The transition probability corresponding to a transition from an initial unperturbed
state i to perturbed f is obtained as
1
'
0
'fiiw t
if f i
i
P t V t e dt
Where
f i
fi
E E
w
and
f o f i o i
fi
H H
w
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8. Variational Method
Variational method is based on energy optimization and parameter variation on the basis
of choosing trial wave function.
1. On the basis of physical intuition guess the trial wave function. Say
noo ,.....,, 321 where 321 ,, are parameter.
2. Find
nn
nn
n
H
E
,.....,,,.....,,
,.....,,,.....,,
,.....,,
32103210
32103210
3210
3. Find 0,.....,, 321
n
i
oE
Find the value of ,.....,, 321 so that it minimize E0.
4. Substitute the value of ,.....,, 321 in ,.....,, 3210E one get minimum value
of E0 for given trial wave function.
5. One can find the upper level of 1 on the basis that it must be orthogonal to 0 i.e.
1 0 0
Once 1 can be selected the 2, 3, 4 step can be repeated to find energy the first Eigen
state.
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9. The Wentzel-Kramer-Brillouin (WKB) method
WKB method is approximation method popularly derived from semi classical theory
For the case 1
d
dx
where ( )
2 ( ( )
x
m E V x
If potential is given as V(x) then and there is three region
WKB wave function in the first region i.e 1x x
1
1 1
exp ' '
x
I
x
c
P x dx
P x
WKB wave function in region II: i.e 1 2x x x
' "
2 2
exp ' exp ' 'II
x
c ci i
P x dx P x dx
P x P x
WKB wave function in region III: i.e 2x x
2
3 1
exp ' '
x
III
x
c
P x dx
P x
9.1.1 Quantization of the Energy Level of Bound state
Case I: When both the boundary is smooth
1
( ) ( )
2
p x dx n where 0,1,2...n
2
1
1
2 2
2
x
n
x
m E V x dx n
where 0, 1, 2,......n and 1x and 2x are turning
point
Case II: When one the boundary is smooth and other is rigid
2
1
3
( ) ( )
4
x
x
p x dx n where 0,1,2...n
2
1
3
2
4
x
n
x
m E V x dx n
where 0, 1, 2,......n and 1x and 2x are turning
points
xV
I II III
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Case III: When both boundary of potential is rigid
2
1
( ) ( 1)
x
x
p x dx n where 0,1,2...n
2
1
2 1
x
n
x
m E V x dx n where 0, 1, 2,......n and 1x and 2x are turning
points .
9.1.2 Transmission probability from WKB
T is defined as transmission probability through potential barrier V is given by
exp 2T where
2
1
1
2
x
n
x
m E V x dx
and 1x and 2x are turning points.
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10. Identical Particles
Identical particle in classical mechanics are distinguishable but identical particle in
quantum mechanics are Indistinguishable.
Total wave function of particles are either totally symmetric or totally anti-symmetric.
11.1 Exchange Operator
Exchange operator ijP as an operator as that when acting on an N-particle wave
function 1 2 3( .... ... ... )i j N interchanges i and j.
i.e 1 2 3 1 2 3( .... ... ... ) ( .... ... ... )ij i j N j i NP
sign is for symmetric wave function s and sign for anti symmetric wave
function a .
11.2 Particle with Integral Spins
Particle with integral spins or boson has symmetric states.
1 2 1 2 2 1
1
, ,
2
s
For three identical particle:
123213312
132231321
321
,,,,,,
,,,,,,
6
1
s
For boson total wave function(space and spin) is symmetric i.e if space part is symmetric
spin part will also symmetric and if space part is ant symmetric space part will also also
anti symmetric.
11.3 Particle with Half-integral Spins
Particle with half-odd-integral spins or fermions have anti-symmetric.
For two identical particle:
122121 ,,
2
1
a
For three identical particle:
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1 2 3 1 3 2 2 3 1
1 2 3
2 1 3 3 1 2 3 2 1
, , , , , ,1
, , , , , ,6
a
For fermions total wave function(space and spin) is anti symmetric .ie if space part is
symmetric spin part will anti symmetric and if space part is ant symmetric space part will
also symmetric.
11. Scattering in Quantum Mechanics
Incident wave is given by .oik r
inc r Ae
. If particle is scattered with angle θ which is
angle between incident and scattered wave vector ok
and k
Scattered wave is given by
.
,
ik r
sc
e
r Af
r
, where ,f is called scattering amplitude wave function.
is superposition of incident and scattered wave
r
e
feA
rik
rik
o
o
,
differential scattering cross section is given by
2
,
o
d k
f
d k
where is solid angle
For elastic collision
2
,
d
f
d
If potential is given by V and reduce mass of system is µ then
k
ok
2 22 ' 3
2
, ' ' '
4
ikrd
f e V r r d r
d
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11.1 Born Approximation
Born approximation is valid for weak potential V(r)
2
0
2
, ' ' sin ' 'f r V r qr dr
q
Where 2 sin
2
oq k k k
for and for ok k
2
2
4 2
0
4
' ' sin ' '
d
r V r qr dr
d q
11.2 Partial Wave Analysis
Partial wave analysis for elastic scattering
For spherically symmetric potential one can assume that the incident plane wave is in z-
direction hence exp cosinc r ikr . So it can be expressed in term of a superposition of
angular momentum Eigen state, each with definite angular momentum number l
cos
0
2 cosik r ikr l
l l
l
e e i l l J kr p
, where Jl is Bessel’s polynomial function and Pl
is Legendre polynomial.
0
, 2 1 cos
ikr
l
l l
l
e
r i l J kr P f
r
1
2 1 sin cosli
l lf l e P
k
Total cross section is given by
2
2
0 0
4
2 1 sinl l
l l
l
k
Where σl is called the partial cross section corresponding to the scattering of particles in
various angular momentum states and l is phase shift .
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12. Relativistic Quantum Mechanics
12.1 Klein Gordon equation
The non-relativistic equation for the energy of a free particle is
2
2
p
E
m
and
2
2
p
E
m
2
2
p
i
m t
where
p i is the momentum operator ( being the del operator).
The Schrödinger equation suffers from not being relativistic ally covariant, meaning it
does not take into account Einstein's special relativity .It is natural to try to use the
identity from special relativity describing the energy: 2 2 2 4
p c m c E Then, just
inserting the quantum mechanical operators for momentum and energy yields the
equation 2 2 2 2 4
c m c i
t
This, however, is a cumbersome expression to work with because the differential operator
cannot be evaluated while under the square root sign.
which simplifies to
2
2 2 2 2 4
2
c m c
t
Rearranging terms yields
2 2 2
2
2 2 2
1 m c
E
c t
Since all reference to imaginary numbers has been eliminated from this equation, it can
be applied to fields that are real valued as well as those that have complex values.
Using the inverse of the Murkowski metric we ge
2 2
2
0
m c
where
2
( ) 0
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In covariant notation. This is often abbreviated as 2
( ) 0 where
mc
and
2
2
2 2
1
c t
This operator is called the d’Alembert operator . Today this form is interpreted as the
relativistic field for a scalar (i.e. spin -0) particle. Furthermore, any solution to the Dirac
equation (for a spin-one-half particle) is automatically a solution to the Klein–Gordon
equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac
equation.
The Klein–Gordon equation for a free particle and dispersion relation
Klein –Gordon relation for free particle is given by
2
2
2 2
1
E
c t
with the same solution as in the non-relativistic case:
dispersion relation from free wave equation ( , ) exp ( . )r t i k r t which can be
obtained by putting the value of in
2
2
2 2
1
E
c t
equation we will get
dispersion relation which is given by
2 2 2
2
2 2
m c
k
k
.
12.2 Dirac Equation
searches for an alternative relativistic equation starting from the generic form describing
evolution of wave function:
H
t
i ˆ
If one keeps first order derivative of time, then to preserve Lorentz invariance, the space
coordinate derivatives must be of the first order as well. Having all energy-related
operators (E, p, m) of the same first order:
t
iE
ˆ and
z
ip
y
ip
x
ip zyx
ˆ,ˆ,ˆ
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mpppE zyx ˆˆˆˆ
321
By acting with left-and right-hand operators twice, we get
2
1 2 3 1 2 3
ˆ ˆ ˆ ˆ ˆ ˆ ˆx y z x y zE p p p m p p p m
which must be compatible with the Klein-Gordon equation
22222
ˆˆˆˆ mpppE zyx
This implies that
,0 ijji for ji
0 ii
12
i
12
Therefore, parameters α and β cannot be numbers. However, it may and does work if they
are matrices, the lowest order being 4×4. Therefore, ψ must be 4-component vectors.
Popular representations are
0
0
i
i
i
and
10
01
where i are 2 2 Pauli matrices:
10
01
0
0
01
10
321
i
i
The equation is usually written using γµ-matrices, where i i for
The equation is usually written using γµ-matrices, where i i for 1,2,3i and
0 (just multiply the above equation with matrix β and move all terms on one side of
the equation):
0
m
x
i
where
0
0
i
i
i
and
10
01
0
Find solution for particles at rest, i.e. p=0:
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0
4
3
2
1
0
m
t
i
B
A
B
A
m
t
i
10
01
It has two positive energy solutions that correspond to two spin states of spin-½
electrons:
0
1imt
A e and
1
0imt
A e
and two symmetrical negative-energy solutions
0
1imt
B e and
1
0imt
B e