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Title+Summary	Name	Date
QFT/17- Course Overview Node id: 2270page	kapoor	22-08-26 10:08:04	n
[QUE/QM-20007] Node id: 2682page Find the direction along which the spin projection is $1/2$ for a spin half particle if the spin wave function the particle is given to be \[ \chi = \begin{pmatrix} 3\over 5 \\ -{4\over 5} \end{pmatrix}\]	kapoor	22-08-26 10:08:33	n
2019-CM @ HCU :: Lessons [LNK] Node id: 3444curated_content	kapoor	22-08-26 00:08:40	n
Classical Mechanics [lnk] Node id: 3385page	kapoor	22-08-26 00:08:02	n
Unit-A Vectors, Summation Convention [LNK} Node id: 3396page	kapoor	22-08-26 00:08:17	n
Info-Page -6 :: Lesson Bundle Template [LNK] Node id: 3356page The following is sample of a Lesson Bundle to illustrate the template of a lesson bundle.	kapoor	22-08-26 00:08:01	n
LSN/VS-02 Vectors and Vector Spaces (LNK) Node id: 3594page	kapoor	22-08-26 00:08:39	n
LSN/VS-06 Matrix Representation in a Vector Space (LNK) Node id: 3610page	kapoor	22-08-26 00:08:04	n
Complex-Variables-Home [CV-HOME] [LNK] Node id: 3488collection Complex Variables --- Principles and Problem Sessions SOLUTIONS and ERRATA Chapter-1 Complex Numbers 1.1 Polar Form of Complex Numbers Q1-Q5 1.2 Curves in Complex Plane Q1-Q3 1.2 Curves in Complex Plane Q4-Q7 Chapter-2 :: Elementary Functions and Differentiation 2.1 Exercise De Moivre's Theorem 2.2 Real and Imaginary Parts 2.4 Exercise Solutions of Equations (Q1-Q2) 2.4 Exercise Solutions of Equations (Q3-Q5) \(\S\S\) 2.6 Tutorial Roots of a complex number; and \(\S\S\) 2.7 Quiz : Roots of Unity. Chapter 3 :: Functions with branch cut singularity 3..1 Questions ----------- Branch Point 3..2 Tutorial ---The square root branch cut 3.3 Exercise Branch cut for 3.4 Quiz Discontinuity and branch cut 3.5 Exercise The Logarithmic function 3.6 Discontinuity across the branch cut 3.8 Mixed Bag Multivalued Functions 3.8 Mixed Bag Q[1] 3.8 Mixed Bag Q[2] 3.8 Mixed Bag Q[3] 3.8 Mixed Bag Q[4] 3.8 Mixed Bag Q[5] 3.8 Mixed Bag Q[7] 3.8 Mixed Bag Q[8] 3.8 Mixed Bag Q[11] 3.8 Mixed Bag Q[12] 3.8 Mixed Bag Q[13] Chapter 4 :: Integration in Complex Plane \(\S\S\) 4.1 Questions Range of Parameters In Improper Integrals Q[1] \(\int_{-\infty}^\infty \frac{}dx{(x^4+1)^c} \), Q[2], Q[3] Q[4] \(\\int_0^\infty \frac{x^c}{\sinh x} Q[5] \(\int_0^\infty \frac{e^{cx}}{\cosh^2x}\), Q[7] \(\int_{-\infty}^\infty \frac{x^{2n+1}{(x^2+1)}\); Q[8]; Q[9] Q[10] \(\int_0^\infty\frac{x^c(e^{ax}-1)}{(x^3+2)^c} \) Q[11] \( \int_0^\infty \frac{x^2}{e^x-1}; Q[12} ; Q{13]; Q[14] \) \(\S\S\) 4.2 Tutorial Computing Line Integrals in Complex Plane Q[1] \(\int_{x_1}^{x_2} \exp(\lambda x) dx = \frac{1}{\lambda}\[\exp(\lambda x_2) -\exp(\lambda x_2) \]\) Q[3] \(\int_\Gamma z^2 dz\) \(\S\S\) 4.3 Exercise Evaluation of Line Integrals \(\S\S\) 4.4 Questions Deformation of Contours \(\S\S\) 4.5 Exercise Deformation of Contours \(\S\S\) 4.6 Exercise Cauchy Theorem \(\S\S\) 4.7 Tutorial Shift of Real Integration Parameter by a Complex Number Q[1] Prove that \(\int_0^\infty exp(-px^2) \cos(2bx)\, dx= \frac{1}{2}\sqrt{\frac{\pi}{p}} \exp(-b^2/p), p>0 \) Q[2] Prove that \(\int_0^\infty e^{-px^2} \sin(ax) \sin(bx) = \frac{1}{4} \left\{\exp\Big(\frac{(a-b)^2}{4p}\Big)- \exp\Big( -\frac{-(a+b)^2}{4p}\Big) \right\}\) \(\S\S\) 4.8 Tutorial Scaling of a Real Integration Variable by a Complex Number Q[1] Prove \(\int_0^\infty e^{ipx^2}\, dx = \frac{1}{2} e^{i\pi/4}\sqrt{\frac{\pi}{p}} \) \(\S\S\) 4.9 Exercise Shift and Scaling by a Complex Number Q[1] \(\int_0^\infty e^{-q^2x^2} \sin^2 ax\, dx\) Q[2] \(\int_0^\infty e^{-q^2x^2}{\,\sin(px+c)\,\choose\, \cos (px+c)\,}\, dx \) Q[3] \( \int_0^\infty x e^{-q^2x^2} \sin ax \, dx\) Q[4] \( \int_0^\infty x^2 e^{-q^2x^2} \cos ax\, dx\) Q[5] \(\int_0^\infty x^3 e^(-q^2x^2) \sin ax \, dx \) Q[6] \(\int xe^{-px} \cos (2x^2+px) \, dx\) Q[7] \(\int xe^{-px} \cos (2x^2-px) \, dx\) Q[8] \(\int_0^\infty e^{-px} \big[ \sin(2x^2+px)+\cos(\cos (2x^2 +px) ) \big]\, dx\) Q[9] \( \int_0^\infty e^{-px} \big[ \sin(2x^2-px)-\cos(\cos (2x^2 -px) ) \big]\, dx\) Q[10] \(\int_0^\infty xe^{-\beta x} \sin ax^2 \sin\beta x\, dx\) Q[11] \(\int_0^\infty \cos ax^2 \cos 2bx, dx\) Q[12] \(\int_0^\infty (\cos ax +\sin ax) \sin(bx)^2\, dx\) Q[13] \(\int_0^\infty \sin ax^2 \sin 2bx \sin 2cx\, dx\) Q[14] \(\int_0^\infty x \sin ax^2 \sin2bx \,dx\) Q[15] \(\int_0^\infty \sin ax^2 \cos bx^2\, dx\) Q[16] \(\int_0^\infty (\sin^2 ax^2 -\sin^2bx^2)\, dx\) Q[17] \(\int_0^\infty \big(\sin (a-x^2) + \cos(a-x^2) \big)\, dx\) \(\S\S\) 4.10 Exercise Rotation of Contour \(\S\S\) 4.11 Mixed Bag Chapter 5 :: Cauchy Integral Formula \(\S\S\) 5.2 5.2 Q 1-Q12 \(\S\S\) 5.3 5.3 Q3 5.3 Q4 Q8 5.5 Q1-Q2 \(\S\S\) 5.5 5.5 Q 3-Q4 \(\S\S\) 5.4 :: Taylor Series Representation 5.4 Q[1] 5.4 Q[2] 5.4 Q[3](a)-(b) 5.4 Q[3](c)-(d) 5.4 Q[3](e)-(g) 5.4 Q[4] 5.4 Q[5] \(\S\S\) 5.8 :: Laurent Expansion of \(f(z)\) in powers of \((z-z_0) 5.8 Q[1] \(f(z) = (z^2+3z+5) e^{1/z}, z_0=0.\) 5.8 Q[2] \(f(z) = \frac{\text{cosh}\, z}{z(z^4+1)}, z_0=0.]\) 5.8 Q[3] \(f(z) = z^2 \sin(1/(1-z)), z_0=1.\) 5.8 Q[4] \(f(z) = \frac{z}{\text{cosh}\, z -1}, z_0=0. \) 5.8 Q[5] \(f(z) = \cot z , z_0=0. \) 5.8 Q[6] \(f(z) = \text{cosech}\, z, z_0=0. \) 5.8 Q[7] \(f(z) = \text{sec } z, z_0=\pi/2.\) 5.8 Q[8] \(f(z)=\frac{z+3}{\sin z^2}, z_0=0.\) 5.8 Q[9]\(f(z)=\frac{z}{\sin^3 z}, z_0=n\pi.\) 5.8 Q[10]\(f(z)= \frac{z}{\sin\pi z^2}, z_0=1.\) 5.8 Q[11]\(f(z)= \frac{1}{(\exp(z)-1)}, z_0=0.\) 5.8 Q[12] \(\frac{1}{(\exp(z)+1)^2)}, z_0=i\pi\) 5.8 Q[13] Prove \(\exp(t(z+1/z))= \sum_{-\infty}^\infty z^n J_n(t).\) 5.8 Q[14]\(f(z) = \sin(\pi/z), z_0=1.\) Chapter 6 :: The Residue Theorem \(\S\S\) 6.10 Integrals Using the Residue at Infinity Some general, common remarks 6.10 Q[1] \(\oint \sqrt{(z-a)(z-b)}\, dz= -(\pi i/4)(a-b)^2\) 6.10 Q[2] \(\oint \sqrt{\frac{z-a}{z-b}\, dz} = -(\pi i/2)(a-b)\) 6.10 Q[3] \(\oint \frac{z}{\sqrt{z-a}{z-b}}\, dz = \pi i (a+b)\) 6.10 Q[5] \(\int_0^1 \frac{x^{1-p}(1-x)^p}{(1+x)^3}\, dx\) 6.10 Q[6] \(\int_0^1\frac{x^{1-p}(1-x)^p}{1+x^2}\, dx\) 6.10 Q[7] \(\int_0^1 \frac{x^{1-p}(1-x)^p}{(1+x^2)^2}\, dx\) 6.10 Q[8] \(\int_0^1 \frac{1}{(1+x)^3(x^2(1-x))^{1/3}}\, dx\) 6.10 Q[9] \(\int \frac{x^{2n}}{(1+x^2)\sqrt{1-x^2}}\, dx\) 6.10 Q[10] \(\int_0^1 \frac{x^{2n}}{(x(1-x^2))^{1/3}}\) 6.10 Q[11] \(\int_0^1 \frac{1}{(1-x^n)^{1/n}}\, dx\) Chapter 7 :: Contour Integration \(\S\S\) 7.1 Tutorial Integrals of Rational Function 7.1 Q[1] 7.1 Q[2] \(\S\S\) 7.2 7,2. Q[1] 7.2 Q[2} \(\S\S\) 7.3 Exercise 7.3 Q[1] \(\int_0^\infty\frac{dx}{(x^2+p^2)^2} \) 7.3 Q[2] \(\int_0^\infty \frac{x\, dx}{x^4+1}\) 7.3 Q[4]\(\int_{-\infty}^\infty \frac{dx}{(x^2+x+1)^2}\) 7.3 Q[5] \(\int_{-\infty}^{\infty} \frac{(x^2+2) dx}{(x^4+10x^2+9)}\) 7.3 Q[6] \(\int_0^\infty \frac{(x-2)dx}{(x^4+10x^2+9)}\) 7.3 Q[7] \(\int_0^\infty \frac{x^2\, dx}{x^6+1}\) 7.3 Q[8] \(\int_0^\infty \frac{dx}{(x^2+p^2)(x^2+q^2)}\) 7.3 Q[10] \(\int_0^\infty \frac{x^2\, dx}{x^{2N+1}+1}\) 7.3 Q[11] \(\int_0^\infty \frac{x^4\, dx}{x^{2N}+1}\) 7.3 Q[12] \(\int_0^\infty \frac{dx}{x^N+1}\) 7.3 Q[13] \(\int_0^\infty \frac{x^{2N}\,dx}{x^{2M}+1}\)7.3 Q[14] \(\int_0^\infty \frac{dx}{(p+qx^2)^N}\) \(\S\S\) 7.4 Exercise ---Integrals of \(\sin x / \cos x\) with rational functions. 7.4 Q[1] 7.4 Q[2] \(\int \frac{\cos p x\, dx}{(x^2+1)^2}\) 7.4 Q[3] 7.4 Q[4] 7.4 Q[5] 7.4 Q[6] 7.4 Q[7] 7.4 Q[9] 7.4 Q[10] 7.4 Q[11] 7.4 Q[12] \(\S\S\) 7.5 Tutorial Integration Around a Branch Cut \(\S\S\) 7.6 Integration around a branch cut 7.6 Q[1] 7.6 Q[2] 7.6 Q[3] 7.6 Q[4] 7.6 Q[5] 7.6 Q[7] 7.6 Q[8] 7.6 Q[9] 7.6 Q[11] 7.6 Q[12] 7.6 Q[13] 7.6 Q[14] \(\S\S\) 7.7 Integral of Type \(\int 7.7 Q[1] 7.7 Q[2] 7.7 Q[3] 7.7 Q[4]-[6] 7.7 Q[7] 7.7 Q[8] 7.7 Q[9] 7.7 Q[11] 7.7 Q[12] \(\S\S\) 7.8 Hyperbolic Functions \(\S\S\)7.9 Exercise ----- Exponential and Hyperbolic Functions 7.9 Q[1] 7.9 Q[3] 7.9 Q[4] 7.9 Q[5] 7.9 Q[6] 7.9 Q[7]-[10] 7.9 Q[11] 9 Q[12] 7.9 Q[13] 7.9 Q[14] \(\S\S\) 7.10 Tutorial Principal Value Integrals 7.10 Q[1] 7.10 Q[2] \(\S\S\) 7.11 Exercise Integrals Requiring Indented Contours 7.11 Q[1] 7.11 Q[2] 7.11 Q[3] 7.11 Q[4] 7.11 Q[5] 7.11 Q[6] 7.11 Q[7] 7.11 Q[8] 7.11 Q[9] 7.11 Q[10] \(\S\S\) 7.12 Series Summation and Expansion 7.12 Q[1] 7,12 Q[5] \(\S\S\) 7.13 What you see is not what you get 7.13 Q[1] 7.13 Q[2] 7.13 Q[3] 7.13 Q[4] 7.13 Q[5] 7.13 Q[6] 7.13 Q[7] 7.13 Q[8] 7.13 Q[9] 7.13 Q[10] 7.13 Q [11] 7.13 Q[12] \(\S\S\) 7.14 Integrals from Statistical Mechanics 7.14 Q[1] 7.14 Q[2] 7.14 Q[3] 7.14 Q[4] 7.14 Q[5] \(\S\S\) 7.15 Alternate Routes Improper Integrals 7.15 Q[1] 7.15 Q[2] 7.15 Q[4] \(\S\S\)7.16 Killing Two Birds with One Stone 7.16 Q1] 7.16 Q[2] 7.16 Q[3] 71.6 Q[4] \(\S\S\) 7.17 7.17 Q[1] 7.17 Q[6] 7.17 Q[7] 7.17 Q[8] \(\S\S\) 7.18 Mixed Bag : Improper Integrals Errata for Complex Variables ----Principles and Problem Sessions Errata--Part-I :: Chapters 1-2 Errata -- Part-I:: Chapter 4 Integration in Complex Plane Errata -- Part-I :: Chapter 5 Cauchy's Integral Formula Errata -- Part-I :: Chapter 6 Residue Theorem Errata -- Part-II :: \(\S\S\) 1.9Mixed Bag Errata -- Part-II ::\(\S\S\) 3.6Exercise Discontinuity across the branch cut	kapoor	22-08-26 00:08:04	n
PSET/CM-01 Periodic Motion Node id: 3701page	kapoor	22-08-26 00:08:15	y
LEC/QM-Mod08 Applications of Canonical Commutation Relations (LNK) Node id: 1047page Harmonic Oscillator Energy Levels Angular Momentum Eigenvalues and Eigenvectors	kapoor	22-08-26 00:08:07	n
LEC/QM-Mod01 Classical Theories Revisited (LNK) Node id: 1784page 1. An Overview of Classical Concepts Introduction;Thought experiments with waves;Thought experiments with bullets;Waves versus particles. 2. Structure and Formulations of Physical Theories Structure of physical theories; Formulations of classical mechanics;Thermodynamics and statistical mechanics. 3. Inadequacy of Classical Theories Click on a section heading to see details.	kapoor	22-08-26 00:08:06	n
\(\S\) 6.4 Compatible Observables (LNK) Node id: 1442page General Structure of Quantum Mechanics Postulates of Quantum Mechanics Applying Postulates to Compute Probability and Average Values Compatible Observables Use navigation links at the bottom of the page	kapoor	22-08-26 00:08:34	n
\(\S\) 6.2 Postulates of Quantum Mechanics (LNK) Node id: 1440page General Structure of Quantum Mechanics Postulates of Quantum Mechanics Applying Postulates to Compute Probability and Average Values Compatible Observables	kapoor	22-08-26 00:08:13	n
\(\S 7.1\) Canonical Quantization (LNK) Node id: 1792page \(\S 7.1\)Canonical Commutation Relation \(\S 7.2\) General Form of Uncertainty Relation	kapoor	22-08-25 23:08:26	n
\(\S 9.2\) Time Evolution of Quantum Systems (LNK) Node id: 1387page \(\S 9.1\) Time Evolution of Quantum Systems-I \(\S 9.2\) Time Evolution of Quantum Systems-II \(\S 9.3\) Perturbation Expansion of S Matrix in Interaction Picture	kapoor	22-08-25 23:08:16	n
\(\S 9.3\) Perturbation Expansion of S-Matrix in Interaction Picture (LNK) Node id: 1388page \(\S 9.1\) Time Evolution of Quantum Systems-I \(\S 9.2\) Time Evolution of Quantum Systems-II \(\S 9.3\) Perturbation Expanstion of S Matrix in Interaction Picture	kapoor	22-08-25 23:08:46	n
LEC/QM-Mod09 Time Evolution in Quantum Mechanics (LNK) Node id: 1385page \(\S 9.1\) Time Evolution of Quantum Systems-I \(\S 9.2\) Time Evolution of Quantum Systems-II \(\S 9.3\) Perturbation Expanstion of S Matrix in Interaction Picture	kapoor	22-08-25 23:08:11	n
\(\S 13.5\) Reflection and Transmission from a Potential (LNK) Node id: 861page General Considerations on Solutions of Schrodinger Equation Particle in a Box, Square Well Dirac Delta Function Potential Harmonic Oscillator Reflection and Transmission from a Potential General Properties of Motion in One Dimension	kapoor	22-08-25 23:08:06	n
\(\S 13.4\) Harmonic Oscillator Energy Levels (LNK) Node id: 847page General Considerations on Solutions of Schrodinger Equation Particle in a Box, Square Well Dirac Delta Function Potential Harmonic Oscillator Energy Levels Reflection and Transmission from a Potential General Properties of Motion in One Dimension	kapoor	22-08-25 23:08:11	n