M.Sc. QUANTUM MECHANICS Ilnmuacin.in/studentnotice/ddelnmu/2020/M.SC PHY. Quantum... · 2020. 4....

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QUANTUM MECHANICS I

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Dr. Shambhu Prasad-Co-ordinator, DDE, LNMU, Darbhanga

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CONTENTS

Chapters Page No.

1. Origins of Quantum Physics 1

2. Dirac Notation and Hilbert Space 51

3. Postulates of Quantum Mechanics 102

4. One-Dimensional Problems 128

5. Angular Momentum 165

6. Three-Dimensional Problems 191

7. Time Independent Perturbation Theory 225

Origins of Quantum Physics

Self-Instructional Material 1

CHAPTER – 1

ORIGINS OF QUANTUM PHYSICSSTRUCTURE

1.1 Learning Objectives 1.2 Introduction 1.3 Particle Aspect of Radiation 1.4 Wave Aspect of Particles 1.5 Particles verses Waves 1.6 Indeterministic Nature of the Microphysical World 1.7 Atomic Transitions and Spectroscopy 1.8 Quantization Rules 1.9 Wave Packets 1.10 Concluding Remarks 1.11 Summary 1.12 Review Questions 1.13 Further Readings

1.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:

zz In this chapter are going to review the main physical ideas and experimental facts that defied classical physics and led to the birth of quantum mechanics.

zz The introduction of quantum mechanics was prompted by the failure of classical physics in explaining a number of microphysical phenomena that were observed at the end of the nineteenth and early twentieth centuries.

1.2 INTRODUCTION

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Self-Instructional Material2

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1.3 PARTICLE ASPECT OF RADIATION

BlackBody Radiation

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Wien’s Law

Rayleigh-JeansLaw

Planck’s Law

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Wien’s Law

Rayleigh-JeansLaw

Planck’s Law

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Wien’s Law

Rayleigh-JeansLaw

Planck’s Law

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Wien’s Law

Rayleigh-JeansLaw

Planck’s Law

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1.4 WAVE ASPECT OF PARTICLESde BRoglie's hyPothesis: matteR Waves

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Davission-Germer Experiment

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1.5 PARTICLES VERSES WAVES

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PRinciPle of lineaR suPeRPosition

1.6 INDETERMINISTIC NATURE OF THE MICROPHYSICALWORLD

heisenBeRg's unceRtainty PRinciPle

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PRoBaBilistic inteRPRetation

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1.7 ATOMIC TRANSITIONS AND SPECTROSCOPY

RutheRfoRd PlanetaRy model of the atom

BohR model of the hydRogen atom

eneRgy levels of the hydRogen atom

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Spectroscopy of the Hydrogen Atom

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1.8 QUANTIZATION RULES

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1.9 WAVE PACKETS

localized Wave Packets

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Wave Packets and the unceRtainty Relations

motion of Wave Packets

Propagation of a Wave Packet without Distortion

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Propagation of a Wave Packet with Distortion

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1.10 CONCLUDING REMARKS

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1.12 REVIEW QUESTIONS

1.13 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.

zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.

zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.

1. What is Particle Aspect of Radiation and Wave Aspect of Particles?2. Explain Particles verses Waves.3. Describe the Indeterministic Nature of the Microphysical World.4. What is Atomic Transitions and Spectroscopy?5. Explain the Quantization Rules.6. What Wave Packets and Concluding Remarks?

1.11 SUMMARYBohr introduced in 1913 his model of the hydrogen atom. In this work, he argued that atoms can be found only in discrete states of energy and that the interaction of atoms with radiation, i.e., the emission or absorption of radiation by atoms, takes place only in discrete amounts of hv because it results from transitions of the atom between its various discrete energy states. This work provided a satisfactory explanation to several outstanding problems such as atomic stability and atomic spectroscopy. Inspired by Planck’s quantization of waves and by Bohr’s model of the hydrogen atom, Heisenberg founded his theory on the notion that the only allowed values of energy exchange between microphysical systems are those that are discrete: quanta. Expressing dynamical quantities such as energy, position, momentum and angular momentum in terms of matrices, he obtained an eigenvalue problem that describes the dynamics of microscopic systems; the diagonalization of the Hamiltonian matrix yields the energy spectrum and the state vectors of the system. Matrix mechanics was very successful in accounting for the discrete quanta of light emitted and absorbed by atoms.

Continuous character of the radiation emitted by a glowing solid object constituted one of the major unsolved problems during the second half of the nineteenth century. All attempts to explain this phenomenon by means of the available theories of classical physics (statistical thermodynamics and classical electromagnetic theory) ended up in miserable failure. This problem consisted in essence of specifying the proper theory of thermodynamics that describes how energy gets exchanged between radiation and matter. All attempts to explain this phenomenon by means of the available theories of classical physics (statistical thermodynamics and classical electromagnetic theory) ended up in miserable failure. This problem consisted in essence of specifying the proper theory of thermodynamics that describes how energy gets exchanged between radiation and matter.

Inspired by Planck’s quantization of electromagnetic radiation, Einstein succeeded in 1905 in giving a theoretical explanation for the dependence of photoelectric emission on the frequency of the incident radiation. He assumed that light is made of corpuscles each carrying an energy hv, called photons.

Dirac Notation and Hilbert Space

CHAPTER – 2

DIRAC NOTATION AND HILBERT SPACE

STRUCTURE 2.1 Learning Objectives 2.2 Introduction 2.3 The Hilbert Space and Wave Functions 2.4 Dirac Notation 2.5 Operators 2.6 Representation in Discrete Bases 2.7 Representation in Continuous Bases 2.8 Matrix and Wave Mechanics 2.9 Concluding Remarks 2.10 Summary 2.11 Review Questions 2.12 Further Readings

zz Understanding Hilbert Space and Wave Functions & Dimension and Basis of a Vector Space.

zz To determine the Physical meaning of the scalar product.

2.2 INTRODUCTION

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2.3 THE HILBERT SPACE AND WAVE FUNCTIONSthe lineaR vectoR sPace

the hilBeRt sPace

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dimension and Basis of a vectoR sPace

H H H H

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H H H H

sQuaRe-integRaBle functions: Wave functions

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2.4 DIRAC NOTATION

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2.5 OPERATORS

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2.6 REPRESENTATION IN DISCRETE BASES

Matrix Representation of Kets and Bras

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Matrix Representation of Kets and Bras

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matRix RePResentation of seveRal otheR Quantities

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Properties of a Matrix A

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matRix RePResentation of the eigenvalue PRoBlem

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2.7 REPRESENTATION IN CONTINUOUS BASES

geneRal tReatment

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Position RePResentation

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momentum RePResentation

connecting the Position and momentum RePResentations

Momentum Operator in the Position Representation

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Position Operator in the Momentum Representation

Momentum Operator in the Position Representation

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2.8 MATRIX AND WAVE MECHANICS

matRix mechanics

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2.10 SUMMARYQuantum theory not only works, but works extremely well, and this represents its experimental justification. It has a very penetrating qualitative as well as quantitative prediction power; this prediction power has been verified by a rich collection of experiments. So the accurate prediction power of quantum theory gives irrefutable evidence to the validity of the postulates upon which the theory is built.

An observable is a dynamical variable that can be measured; the dynamical variables encountered most in classical mechanics are the position, linear momentum, angular momentum, and energy. How do we mathematically represent these and other variables in quantum mechanics? In quantum mechanics, however, the measurement process perturbs the system significantly. While carrying out measurements on classical systems, this perturbation does exist, but it is small enough that it can be neglected. In atomic and subatomic systems, however, the act of measurement induces nonnegligible or significant disturbances. Finally, we may state that quantum mechanics is the mechanics applicable to objects for which measurements necessarily interfere with the state of the system. Quantum mechanically, we cannot ignore the effects of the measuring equipment on the system, for they are important. In general, certain measurements cannot be performed without major disturbances to other properties of the quantum system.

Similarly, another measurement of B will yield bn and will leave the system in the same joint eigenstate of A and B. Thus, if two observables A and B are compatible, and if the system is initially in an eigenstate of one of their operators, their measurements not only yield precise values (eigenvalues) but they will not depend on the order in which the measurements were performed. In this case, A and B are said to be simultaneously measurable. So compatible observables can be measured simultaneously with arbitrary accuracy; noncompatible observables cannot.

If the degeneracy persists, we may introduce a fourth operator D that commutes with the previous three and then look for their joint eigenstates which form a complete set. Continuing in this way, we will ultimately exhaust all the operators (that is, there are no more independent operators) which commute with each other. When that happens, we have then obtained a complete set of commuting operators (CSCO). Only then will the state of the system be specified unambiguously, for the joint eigenstates of the CSCO are determined uniquely and will form a complete set (recall that a complete set of eigenvectors of an operator is called a basis). he invariance principles relevant to our study are the time translation invariance and the space translation invariance. We may recall from classical physics that whenever a system is invariant under space translations, its total momentum is conserved; and whenever it is invariant under rotations, its total angular momentum is also conserved.

1. Define the Hilbert Space and Wave Functions Dirac Notation.2. What is Operators & Representation in Discrete Bases?3. Describe the Representation in Continuous Bases4. What is Matrix and Wave Mechanics?5. Explain Concluding Remarks.

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CHAPTER – 3

POSTULATES OF QUANTUM MECHANICS

STRUCTURE 3.1 Learning Objectives 3.2 Introduction 3.3 The Basic Postulates of Quantum Mechanics 3.4 The State of a System 3.5 Observables and Operators 3.6 Measurement in Quantum Mechanics 3.7 Time Evolution of the System’s State 3.8 Symmetries and Conservation Laws 3.9 Connecting Quantum to Classical Mechanics 3.10 Summary 3.11 Review Questions 3.12 Further Readings

zz Understanding the Basic Postulates of Quantum Mechanicszz To determine the Measurement in Quantum Mechanics

3.2 INTRODUCTION

Postulates of Quantum Mechanics

3.3 THE BASIC POSTULATES OF QUANTUM MECHANICS

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3.4 THE STATE OF A SYSTEM

the state of a system

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the suPeRPosition PRinciPle

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3.5 OBSERVABLES AND OPERATORS

3.6 MEASUREMENT IN QUANTUM MECHANICS

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hoW measuRements distuRB systems

exPectation values

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comPlete sets of commuting oPeRatoRs (csco)

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measuRement and the unceRtainty Relations

3.7 TIME EVOLUTION OF THE SYSTEM’S STATE

time evolution oPeRatoR

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stationaRy states: time-indePendent Potentials

schRödingeR eQuation and Wave Packets

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the conseRvation of PRoBaBility

time evolution of exPectation values

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3.8 SYMMETRIES AND CONSERVATION LAWS

infinitesimal unitaRy tRansfoRmations

finite unitaRy tRansfoRmations

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symmetRies and conseRvation laWs

3.9 CONNECTING QUANTUM TO CLASSICAL MECHANICSPoisson BRackets and commutatoRs

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the ehRenfest theoRem

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3.10 SUMMARYDescribe a system in quantum mechanics, we use a mathematical entity (a complex function) belonging to a Hilbert space, the state vector |(t), which contains all the information we need to know about the system and from which all needed physical quantities can be computed. Observable is a dynamical variable that can be measured; the dynamical variables encountered most in classical mechanics are the position, linear momentum, angular momentum, and energy. How do we mathematically represent these and other variables in quantum mechanics? In quantum mechanics, however, the measurement process perturbs the system significantly. While carrying out measurements on classical systems, this perturbation does exist, but it is small enough that it can be neglected. In atomic and subatomic systems, however, the act of measurement induces nonnegligible or significant disturbances. Finally, we may state that quantum mechanics is the mechanics applicable to objects for which measurements necessarily interfere with the state of the system. Quantum mechanically, we cannot ignore the effects of the measuring equipment on the system, for they are important. In general, certain measurements cannot be performed without major disturbances to other properties of the quantum system. Similarly, another measurement of B will yield bn and will leave the system in the same joint eigenstate of A and B. Thus, if two observables A and B are compatible, and if the system is initially in an eigenstate of one of their operators, their measurements not only yield precise values (eigenvalues) but they will not depend on the order in which the measurements were performed. In this case, A and B are said to be simultaneously measurable. So compatible observables can be measured simultaneously with arbitrary accuracy; noncompatible observables cannot. We will show that for each such symmetry there corresponds an observable which is a constant of the motion. The invariance principles relevant to our study are the time translation invariance and the space translation invariance. We may recall from classical physics that whenever a system is invariant under space translations, its total momentum is conserved; and whenever it is invariant under rotations, its total angular momentum is also conserved.

1. Define the Hilbert Space and Wave Functions Dirac Notation.2. What is Operators & Representation in Discrete Bases?3. Describe the Representation in Continuous Bases4. What is Matrix and Wave Mechanics?5. Explain Concluding Remarks.

Quantum Mechanics-1

CHAPTER – 4

ONE-DIMENSIONAL PROBLEMS

STRUCTURE 4.1 Learning Objectives 4.2 Introduction 4.3 Properties of One-Dimensional Motion 4.4 The Free Particle: Continuous States 4.5 The Potential Step 4.6 The Potential Barrier and Well 4.7 The Infinite Square Well Potential 4.8 The Finite Square Well Potential 4.9 The Harmonic Oscillator 4.10 The Harmonic Oscillator 4.11 Summary 4.12 Review Questions 4.13 Further Readings

zz Understanding the Free Particle: Continuous Stateszz To determine the Finite Square Well Potential

4.2 INTRODUCTION

One-Dimensional Problems

4.3 PROPERTIES OF ONE-DIMENSIONAL MOTION

discRete sPectRum (Bound states)

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continuous sPectRum (unBound states)

mixed sPectRum

symmetRic Potentials and PaRity

4.4 THE FREE PARTICLE: CONTINUOUS STATES

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4.5 THE POTENTIAL STEP

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4.6 THE POTENTIAL BARRIER AND WELL

the case e >v0

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the case e >v0: tunneling

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the tunneling effect

4.7 THE INFINITE SQUARE WELL POTENTIAL

the asymmetRic sQuaRe Well

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the symmetRic Potential Well

4.8 THE FINITE SQUARE WELL POTENTIAL

the scatteRing solutions (e >v0)

the Bound state solutions (0 < e < v0)

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4.9 THE HARMONIC OSCILLATOR

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4.10 THE HARMONIC OSCILLATOR

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4.11 SUMMARYThe wave packet solution cures and avoids all the subtleties raised above. First, the momentum, the position and the energy of the particle are no longer known exactly; only probabilistic outcomes are possible. Second, as shown in Chapter 1, the wave packet (4.12) and the particle travel with the same speed vg= p/m, called the group speed or the speed of the whole packet. Third, the wave packet (4.12) is normalizable. We are now going to show that the quantum mechanical

Quantum Mechanics-1

1. What is Properties of One-Dimensional Motion?2. Explain the Free Particle: Continuous States.3. Describe the Potential Step 4. What is the Potential Barrier and Well?5. What is the Infinite Square Well Potential & Finite Square Well Potential.6. Describe the Harmonic Oscillator.

predictions differ sharply from their classical counterparts, for the wave function is not zero beyond the barrier. Classically, we would expect total reflection: every particle that arrives at the barrier (x) will be reflected back; no particle can penetrate the barrier, where it would have a negative kinetic energy.

Quantum mechanical effect which is due to the wave aspect of microscopic objects; it is known as the tunneling effect: quantum mechanical objects can tunnel through classically impenetrable barriers. This barrier penetration effect has important applications in various branches of modern physics ranging from particle and nuclear physics to semiconductor devices. For instance, radioactive decays and charge transport in electronic devices are typical examples of the tunneling effect.

This is in sharp contrast to classical mechanics, where the lowest possible energy is equal to the minimum value of the potential energy, with zero kinetic energy. In quantum mechanics, however, the lowest state does not minimize the potential alone, but applies to the sum of the kinetic and potential energies, and this leads to a finite ground state or zero-point energy. This concept has far-reaching physical consequences in the realm of the microscopic world. For instance, without the zero-point motion, atoms would not be stable, for the electrons would fall into the nuclei. Also, it is the zero-point energy which prevents helium from freezing at very low temperatures.

Harmonic oscillator is one of those few problems that are important to all branches of physics. It provides a useful model for a variety of vibrational phenomena that are encountered, for instance, in classical mechanics, electrodynamics, statistical mechanics, solid state, atomic, nuclear, and particle physics.

Angular Momentum

CHAPTER – 5

ANGULAR MOMENTUMSTRUCTURE

5.1 Learning Objectives 5.2 Introduction 5.3 Orbital Angular Momentum 5.4 General Formalism of Angular Momentum 5.5 Matrix Representation of Angular Momentum 5.6 Geometrical Representation of Angular Momentum 5.7 Spin Angular Momentum 5.8 Eigenfunctions of Orbital Angular Momentum 5.9 Summary 5.10 Review Questions 5.11 Further Readings

zz Understanding Geometrical Representation of Angular Momentumzz To determine the general Formalism of Angular Momentum

5.2 INTRODUCTIONAfter treating one-dimensional problems in Chapter 4, we now should deal with three-dimensional problems. However, the study of three-dimensional systems such as atoms cannot be undertaken unless we first cover the formalism of angular momentum.

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5.3 ORBITAL ANGULAR MOMENTUM

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5.4 GENERAL FORMALISM OF ANGULAR MOMENTUM

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5.5 MATRIX REPRESENTATION OF ANGULAR MOMENTUM

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5.8 EIGENFUNCTIONS OF ORBITAL ANGULAR MOMENTUM

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eigenfunctions and eigenvalues of

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eigenfunctions of

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PRoPeRties of the sPheRical haRmonics

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5.9 SUMMARYAdditionally, angular momentum plays a critical role in the description of molecular rotations, the motion of electrons in atoms, and the motion of nucleons in nuclei. The quantum theory of angular momentum is thus a prerequisite for studying molecular, atomic, and nuclear systems. There are many ways to represent the angular momentum operators and their eigenstates. In this section we are going to discuss the matrix representation of angular momentum where eigenkets and operators will be represented by column vectors and square matrices, respectively. This is achieved by expanding states and operators in a discrete basis. We will see later how to represent the orbital angular momentum in the position representation.

1. What Orbital Angular Momentum? 2. Describe General Formalism of Angular Momentum, 3. Explain Matrix Representation of Angular Momentum. 4. What is Geometrical Representation of Angular Momentum? 5. Define Spin Angular Momentum. 6. What is Eigenfunctions of Orbital Angular Momentum?

Three-Dimensional Problems

CHAPTER – 6

THREE-DIMENSIONAL PROBLEMS

STRUCTURE 6.1 Learning Objectives 6.2 Introduction 6.3 3D Problems in Cartesian Coordinates 6.4 3D Problems in Spherical Coordinates 6.5 Concluding Remarks 6.6 Summary 6.7 Review Questions 6.8 Further Readings

zz Understanding the Problems in Spherical Coordinates & zz To determine the Concluding Remarks

6.2 INTRODUCTION

6.3 3D PROBLEMS IN CARTESIAN COORDINATES

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geneRal tReatment: sePaRation of vaRiaBles

the fRee PaRticle

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the Box Potential

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the haRmonic oscillatoR

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6.4 3D PROBLEMS IN SPHERICAL COORDINATEScentRal Potential: geneRal tReatment

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the fRee PaRticle in sPheRical cooRdinates

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the sPheRical sQuaRe Well Potential

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the isotRoPic haRmonic oscillatoR

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the hydRogen atom

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effect of magnetic fields on centRal Potentials

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6.6 SUMMARYThus, the zero-point energy for a particle in a three-dimensional box is three times that in a one-dimensional box. The factor 3 can be viewed as originating from the fact that we are confining the particle symmetrically in all three dimensions. Degeneracy occurs only when there is a symmetry in the problem. For the present case of a particle in a cubic box, there is a great deal of symmetry, since all three dimensions are equivalent. Note that for the rectangular box, there

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1. What is 3D Problems in Cartesian Coordinates>2. Describe the 3D Problems in Spherical Coordinates.3. What is the Concluding Remarks?

is no degeneracy since the three dimensions are not equivalent. Moreover, degeneracy did not exist when we treated one-dimensional problems in Chapter 4, for they give rise to only one quantum number. Since the energy depends on the sum of nx, ny, nz, any set of quantum numbers having the same sum will represent states of equal energy.

Hydrogen

Time Independent Perturbation Theory

CHAPTER – 7

TIME INDEPENDENT PERTURBATION THEORY

STRUCTURE 7.1 Learning Objectives 7.2 Introduction 7.3 Time-Independent Perturbation Theory 7.4 The Variational Method 7.5 Summary 7.6 Review Questions 7.7 Further Readings

zz Understanding the Degenerate Perturbation Theory.zz To determine Nondegenerate Perturbation Theory.

7.2 INTRODUCTION

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7.3 TIME-INDEPENDENT PERTURBATION THEORY

nondegeneRate PeRtuRBation theoRy

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degeneRate PeRtuRBation theoRy

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fine stRuctuRe and the anomalous zeeman effect

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7.5 SUMMARYPerturbation theory is based on the assumption that the problem we wish to solve is, in some sense, only slightly different from a problem that can be solved exactly. In the case where The WKB method is useful for finding the energy eigenvalues and wave functions of systems for which the classical limit is valid. Unlike perturbation theory, the variational and WKB methods do not require the existence of a closely related Hamiltonian that can be solved exactly.

Hyperfine corrections are included, they would split each of the fine structure levels into a series of hyperfine levels. For instance, when the hyperfine coupling is taken into account in the ground state of hydrogen, it would split the 1S1/2 level into two hyperfine levels separated by an energy of 5.89 × 10–6 eV. This corresponds, when the atom makes a spontaneous transition from the higher hyperfine level to the lower one, to a radiation of 1.42 × 109 Hz frequency and 21 cm wavelength. We should note that most of the information we possess about interstellar hydrogen clouds had its origin in the radioastronomy study of this 21 cm line.

Quantum Mechanics-1

1. Explain the Time-Independent Perturbation Theory.2. What is the The Variational Method?3. Describe the Fine Structure and the Anomalous Zeeman Effect.

M.Sc. QUANTUM MECHANICS Ilnmuacin.in/studentnotice/ddelnmu/2020/M.SC PHY. Quantum... · 2020. 4....

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