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PROOFS PROGRAMME
- Features of Proofs Programme
- PROOFS MAIN CONTENT
- All Quantum Mechanics Content
- Quantum Mechanics Lecture Notes == Home Page
- QM-Lecture-Notes-01 Classical Theories Revisited
- QM-Lecture Notes-02 A Quick Review of Vector Spaces
- QM-Lecture Notes-03 Inner Product Spaces
- QM-Lecture Notes-04 Hilbert Spaces
- QM-Lecture Notes-05 Rise of Quantum Mechanics
- QM Lecture Notes-5.1 :: Advent of Quantum Theory
- QM Lecture Notes 5.2 :: Consequences of wave particle duality
- QM-Lecture Notes 5.3 :: Thought experiments for uncertainty principle
- QM Lecture Notes-5.4 :: Wave particle duality and superposition of states
- QM Lecture Notes-5.5 :: Postulates of quantum mechanics
- QM Lecture Notes-5.6 :: New concepts brought in by quantum theory
- QM Lecture Notes-5.7 :: Early successes of quantum theory
- QM Lecture Notes-06 General Structure of Quantum Mechanics
- QM-Lecture Notes-07 Canonical Quantisation
- QM-Lecture Notes-08 Applications of Canonical Commutation Relations
- QM-Lecture Notes-09 Time Evolution in Quantum Mechanics
- QM-Lecture Notes-10 Coordinate and Momentum Representations
- QM-Lecture Notes-11 Schrodinger Equation in Coordinate Representation
- QM-Lecture Notes-12 Free Particle in One Dimension
- QM-Lecture Notes-13 Potential Problems in One Dimension
- \(\S 13.1\) General Considerations on Poetntial Problems
- \(\S 13.2\) Particle in a Box and Square Well
- \(\S13.3\) Dirac Delta Function Potential
- \(\S 13.4\) Harmonic Oscillator Energy Levels
- \(\S 13.5\) Reflection and Transmission from a Potential
- \(\S 13.6\) General Properties of Motion in One Dimension
- QM-Lecture Notes-14 Problems in Higher Dimensions
- QM-Lecture Notes-15 Methods of Exact Solution
- QM-Lecture Notes-16 Spherically Symmetric Potential Problems
- QM-Lecture Notes-17 Addition of Angular Momenta
- QM-Lecture Notes-18 Integral Equation, Born Approximation
- QM-Lecture Notes-19 Partial Wave Analysis
- QM-Lecture-Notes-20 Spin and Identical Particles
- QM-Lecture Notes-21 Rayleigh Ritz Variation Method
- QM-Lecture Notes-22* Wentzel Kramers Brillouin Approximation
- QM-Lecture Notes-23 Time Independent Perturbation Theory
- QM-Lecture Notes-24 Approximation Schemes for Time Dependent Problems
- QM-Lecture Notes-25 Semi Classical Theory of Radiation
- Quantum Mechanics Cross Roads == Home Page
- QM Cross Roads-01 :: Classical Theories Revisited === Home Page
- QM Cross Roads-02 :: A Quick Review of Vector Spaces
- QM Cross Roads-03 :: Inner Product Spaces
- QM Cross Roads-04 :: Infinite Dimensional Vector Spaces
- QM Cross Roads-05 :: Rise of Quantum Mechanics
- QM Cross Roads-06 :: Postulates of Quantum Mechanics
- QM Points to Remember-06
- QM RC Questions Repository-06 :: General Principles of Quantum Mechanics
- QM AA Problems Repository-06 :: General Principles of Quantum Mechanics
- qm/que/06001 --- Average value
- qm/que/06002 Average value
- qm/que/06003 --- Average value
- qm/que/06004
- qm/que/06005
- qm/que/06006
- qm/que/06007
- qm/que/06008
- qm/que/06108
- qm/que/06009
- qm/que/06010
- qm/que/06011
- qm/que/06012
- qm/que/06013
- qm/que/06014
- qm/que/06015
- qm/que/06016
- qm/que/06025
- qm/que/06017
- qm-que-06019
- qm/que/06020
- qm-que-06021
- qm/que/06022
- qm/que/06022
- qm/que/06023
- qm/que/06024
- QM Cross Roads-07 :: Canonical Quantization
- QM Cross Roads-08 :: Eigenvalues Using Commutation Relations
- QM Cross Roads-09 :: Time Evolution
- QM Cross Roads-10 :: Working With Representation === Home Page
- QM Cross Roads-11 :: Schrodinger Equation in Coordinate Representation
- QM Cross Roads-12 :: Free Particle, Particle in a Box
- QM Cross Roads-13 :: Potential Problems In One Dimension
- QM Cross Roads-14 :: Potential Problems In Two and Three Dimensions
- QM Cross Roads-15 :: Potentials With Special Properties
- QM Cross Roads-16 :: Spherically Symmetric Potential Problems
- QM Cross Roads-17 :: Addition of Angular Momenta
- QM Cross Roads-18 :: Scattering-I Integral Equation and Born Approximation
- QM Cross Roads-19 :: Scattering-II Partial Wave Expansion
- QM Cross Roads-20 :: Spin and Identical Particles
- QM Cross Roads-21 :: Variation Method
- QM Cross Roads-22 :: WKB Approximation
- QM Cross Roads-23 :: Time Independent Perturbation Theory
- QM Cross Roads-24 :: Approximation Method for Time Dependent Problems
- QM Cross Roads-25 :: Semi-classical Theory of Radiation
- QM Special Purpose Units
- Quantum Mechanics - Question Bank
- QM Question Bank-01 Review of Classical Theories
- QM-Question Bank-02 Fundamental Concepts in Vector Spaces
- QM-Question Bank-03 Inner Product Spaces
- QM Question-Bank-04 Hilbert Spaces
- QM-Question Bank-05 Rise of Quantum Mechanics
- QM-Question Bank-06 Postulates of Quantum Mechanics
- QM-Question Bank-07 Canonical Quantisation
- QM-Question Bank-08 Commutation Relations, Ladder Operators
- QM-Question Bank-09 Time Development
- QM-Question Bank-10 Coordinate and Momentum Representations
- QM-Question Bank-11 Schrodinger Equation in Coordinate Representation
- QM-Question Bank-12 Free Particle
- QM-Question Bank-13 Potential Problems in One Dimension
- QM-Question Bank-14 Problems in Higher Dimensions
- QM Question Bank-15 Methods of Exact Solution
- QM-Question Bank-16 Spherically Symmetric Potential Problems
- QM-Question Bank-17 Addition of angular momenta
- QM-Question Bank-18 Quantum Theory of Scattering - Born Approximation
- QM-Question Bank-19 Quantum Theory of Scattering - Partial Wave Analysis
- QM-Question Bank-20 Spin and Identical Particles
- QM-20 Spin and Identical Particles Problem Set-1)
- QM-20 Spin and Identical Particles (Problem Set-2)
- QM-20 Spin and Identical Particles (Problem Set-3)
- QM-20 Spin and Identical Particles (Problem Set-4)
- QM-20 Spin and Identical Particles (Problem Set -5)
- QM-20 Spin and Identical Particles (Problem Set -6)
- QM-20 Spin and Identical Particles (Problem Set -7)
- QM-Question Bank-21 Rayleigh Ritz Variation Method
- QM-Question Bank-22 Wentzel Kramers Brillouin Approximation
- QM-Question Bank-23 Time Independent Perturbation Theory
- Quantum Mechanics - Problem Sets
- QM-Problem-Sets-05 Applications of Uncertainty Principle
- QM Problem Sets-06 Postulates of Quantum Mechanics
- QM-Problem-Sets-08 Applications of Canonical Commutation Relations
- QM-Probelm Sets-10 Coordinate and Momentum Representations
- QM-Problem Sets-13 Potential Problems in One Dimension (Problem Sets)
- QM-Problem Sets-17 Angular Momentum in Quantum Mechanics
- QM-ProblemSets-18 Quantum Theory of Scattering - Born Approximation
- QM-Question Bank-17 Addition of angular momenta
- QM-Question Bank-21 Rayleigh Ritz Variation Method
- QM-Question Bank-23 Time Independent Perturbationn Theory
- QM-Problem Sets-22 Wentzel Kramers Brillouin Approximation
- Quantum Mechanics Lecture Notes == Home Page
- All Mathematical Physics Content
- Mathematical Physics Lecture Notes
- MM Lecture Notes-01 Topics in Set Theory
- MM Lecture Notes-02 A Quick Review of Matrices
- MM Lecture Notes-03 Vector Spaces
- VS-Lecture Notes-01 Groups, Fields, Vector Spaces, Subspace
- VS-Lecture Notes-02 Linear Independence and Basis
- VS-Lecture Notes-03 Linear Functional
- VS-Lecture Notes-04 New Spaces from Old Spaces
- VS-Lecture Notes-05 Linear Operators
- VS-Lecture Notes-06 Matrix Representation
- VS-Lecture Notes-08 Vector Spaces with Inner Product
- VS-Lecture Notes-09 Orthogonality
- VS-Lecture Notes-10 Operators in Inner Product Spaces
- VS-Lecture Notes -11 Representation in o.n. Basis
- MM Lecture Notes-04 Theory of Finite Groups
- GT-01 Group and Subgroup
- GT-02 Group Multiplication Table, Cosets
- GT-03 Factor and Quotient Group
- GT-04 Group of Permutations
- GT-05 Representations-I
- GT-06 Representations -II
- GT-07 Irreducible Representations
- GT-08 First Orthogonality Theorem
- GT-09 Group Characters
- GT-10 Second Orthogonality Theorem
- MM Lecture Notes-06 Ordinary Differential Equations
- MM Lecture Notes-07 Partial Differential Equations
- PDE Lecture Notes-01 Separation of Variables
- PDE-Lecture Notes-02 Boundary Value Problem in One Dimension
- PDE Lecture Notes-03 Rectangular Boundary
- PDE-Lecture Notes-04 Boundary Value Problem for a Circular Domain
- PDE Lecture Notes-05 Solution Requiring Use of Fourier Transform
- PDE Lecture Notes-06 An Example Requiring Expansion in Legendre Polynomials
- MM Lecture Notes-10 Real Number System
- Mathematical Physics Cross Roads
- MM Cross Roads-01 Basics of Set Theory
- MM Cross Roads-02 A Quick Review of Matrices
- MM Cross Roads-03 Vector Spaces
- MM Cross Roads-04 Group Theory
- MM Cross Roads-05 Complex Variables
- Solutions to Problems from "Complex Variables-- Principles and Problem Sessions"
- \(\S\S\) 7.3 Exercise ----- Integrals of Type \(\int Q(x) \,dx\)
- \(\S\S\) 7.3 Q[1] \(\int_0^\infty\frac{1}{(x^2+p^2)^2} dx \)
- \(\S\S\) 7.3 Q[2] \(\int_0^\infty \frac{x}{x^4+1} dx\)
- \(\S\S\) 7.3 Q[3] \(\int \frac{dx}{x^4+x^2+1}\)
- \(\S\S\) 7.3 Q[4] \(\int_{-\infty}^\infty \frac{dx}{(x^2+x+1)^2}\)
- \(\S\S\) 7.3 Q[5] \int_{-\infty}^{\infty}\(\frac{(x^2+2)}{(x^4+10x^2+9)}\)
- \(\S\S\) 7.3 Q[6] \(\int_0^\infty \frac{(x-2)}{(x^4+10x^2+9)} dx\)
- \(\S\S\) 7.3 Q [7] \(\int_0^\infty \frac{x^2\, dx}{x^6+1}\)
- \(\S\S\) 7.3 Q [8] \(\int_0^\infty \frac{dx}{(x^2+p^2)(x^2+q^2)}\)
- \(\S\S\) 7.3 Q[9] \(\int_0^\infty \frac{x^2}{x^4+1}dx \)
- \(\S\S\) 7.3 Q[10] \(\int_0^\infty \frac{x^2}{x^{2N+1}+1}\, dx\)
- \(\S\S\) 7.3 Q[11] \(\int_0^\infty \frac{x^4}{x^{2N}+1} dx\)
- \(\S\S\) 7.3 Q[12] \( \int_0^\infty \frac{dx}{x^N+1} \)
- \(\S\S\) 7.3 Q[13] \(\int_0^\infty \frac{x^{2N}}{x^{2M}+1} \, dx\)
- \(\S\S\) 7.3 Q[14] \(\int_0^\infty \frac{dx}{(p+qx^2)^N}\)
- \(\S\S\) 7.4 Exercise ------ Integrals of \({\sin x \choose \cos x}\) with Rational Functions
- \(\S\S\) 7.4 Q[1] \(\int_0^\infty \frac{x\sin p x\, dx}{x^2+1} \)
- \(\S\S\) 7.4 Q[2] \(\int \frac{\cos p x\, dx}{(x^2+1)^2}\)
- \(\S\S\) 7.4 Q[3] \(\int_0^\infty \frac{\cos px\, dx}{(x^2+1)(x^2+4)}\)
- \(\S\S\) 7.4 Q[4] \(\int_0^\infty \frac{\sin x \, dx}{(x^2+4)(x^2+9)}\)
- \(\S\S\) 7.4 Q[5] \(\int_0^\infty \frac{x^2\cos px \, dx }{x^4+1}\)
- \(\S\S\) 7.4 Q[6] \( \int_0^\infty \frac{x\sin p x\, dx}{x^4+1}\)
- \(\S\S\) 7.4 Q[7] \(\int_0^\infty \frac{x\cos \alpha x\, dx}{x^2-2x+10} \)
- \(\S\S\) 7.4 Q[9] \( \int_0^\infty \frac{\cos a x \, dx}{(x^2+b^2)^2+c^2}\)
- \(\S\S\) 7.4 Q[10] \(\int_0^\infty \frac{x \sin ax\, dx}{(x^2+b^2)^2 + c^2}\)
- \(\S\S\) 7.4 Q[11] \(\int_0^\infty \frac{\sin^2 ax \cos^2 bx}{\beta^2+x^2} dx\)
- \(\S\S\) 7.4 Q[12] \(\int_0^\infty \frac{x\sin^2 ax \cos^2bx}{\beta^2+x^2} dx\)
- \(\S\S\) 7.6 Integration around a branch cut
- \(\S\S\) 7.6 Q[1] \(\int_0^\infty \frac{x^{\frac{1}{2}}}{(x+4)(x+25)}\)
- \(\S\S\) 7.6 Q[2] \(\int_0^\infty \frac{x^{p-1}}{x^2+2x+2}\)
- \(\S\S\) 7.6 Q[3] \(\int_0^\infty \frac{x^{p-1}}{(x+1)^2}\, dx\)
- \(\S\S\) 7.6 Q[4] \(\int_0^\infty \frac{x^p}{x^2+a^2}\, dx\)
- \(\S\S\) 7.6 Q[5] \(\int_0^\infty \frac{x^p}{(x^2+a^2)^2}\, dx\)
- \(\S\S\) 7.6 Q[7] \( \int_0^\infty \frac{x^p}{x^6+1}\, dx \)
- \(\S\S\) 7.6 Q[8] \( \int_0^\infty \frac{x^{1/2}}{(x^2+1)(x^2+4)}\, dx\)
- \(\S\S\) 7.6 Q[9] \(\int_0^\infty \frac{x^{p-1}}{1+x^q}\)
- \(\S\S\) 7.6 Q[11] \(\int_1^\infty \frac{(x-1)^{p-1}}{x^2}\,dx\)
- \(\S\S\) 7.6 Q[12] \(\int_1^\infty\frac{(x-1)^{p-1}}{x^3}\, dx\)
- \(\S\S\) 7.6 Q[13] \(\int_0^\infty \frac{ x^{p-1} dx}{x^3+b^3}\)
- \(\S\S\) 7.6 Q[14] \(\int_0^\infty \frac{x^p}{x^2+2x\cos\alpha +1} \)
- \(\S\S\) 7.7 Integral of Type \(\int \log x Q(x) \,dx\)
- \(\S\S\) 7.7 Q[1] \( \int_0^\infty \frac{\log x}{x^2+1}\,dx\)
- \(\S\S\) 7.7 Q[2] \(\int_0^1 \frac{\log x}{x^3+1}\)
- \(\S\S\) 7.7 Q[3] \(\int_0^\infty \frac{\log^2x}{x^2+1}\, dx\)
- \(\S\S\) 7.7 Q[4]-[6] \(\int_0^\infty \frac{\log^n x}{x^2+1}\, dx, n=4,6,8 \)
- \(\S\S\) 7.7 Q[7] \(\int_0^\infty \frac{\log(px) }{x^2+q^2}\, dx\)
- \(\S\S\) 7.7 Q[8] \(\int_0^\infty \frac{x^2\log x}{(x^2+1)^2} \, dx\)
- \(\S\S\) 7.7 Q[9] \(\int\frac{\log x}{(x^2+1)(1+q^2x^2)}\, dx\)
- \(\S\S\) 7.7 Q[11] \(\int_0^\infty \frac{\log x\, dx}{x^2+2ax\cos\alpha + a^2}\)
- \(\S\S\) 7.7 Q[12] \(\int_0^\infty \frac{x^2 \log x\, dx }{(a^2+b^2x^2)(1+x^2)}\)
- \(\S\S\) 7.9 Exercise ----- Exponential and Hyperbolic Functions
- \(\S\S\) 7.9 Q[1] \(\int_{-\infty}^\infty \frac{x\exp(-x)}{1+\exp(-4x)}\, dx\)
- \(\S\S\) 7.9 Q[3] \(\int_{-\infty}^\infty \frac{x \exp(p x)}{1+ \exp(x)}\, dx\)
- \(S\S\) 7.9 Q[4] \(\int_{-\infty}^\infty x^2 \frac{\exp(px)}{1+ \exp(x)}\, dx\)
- \(\S\S\) 7.9 Q[5] \( \int _0^\infty x \frac{[1-\exp(-x) ]\exp(-x)}{1+ \exp(-3x)} \,dx\)
- \(\S\S\) 7.9 Q[6] \(\int_0^\infty \frac{x\{\exp(-qx) + \exp[(q-p)x]\}}{1- \exp(-px)}\, dx \)
- \(\S\S\) 7.9 Q[7]-[10] \(\int_0^\infty \frac{x^n}{\cosh \pi x} \, dx, n=2,4,6,8\)
- \(\S\S\) 7.9 Q[11] \(\int_0^\infty \frac{x \sinh ax}{\cosh bx}\, dx\)
- \(\S\S\) 7.9 Q[12] \( \int_0^\infty \frac{\cosh ax}{\cosh bx}\, dx \)
- \(\S\S\) 7.9 Q[13] \(\int_0^\infty \frac{x^2 \cosh ax}{\cosh bx}\, dx \)
- \(\S\S\) 7.9 Q[14] \(\int_0^\infty \frac{dx}{\cosh ax + \cos t} \)
- \(\S\S\) 7.11 Exercise Integrals Requiring Indented Contours
- $ \S\S 7.11$ Q[1] $ \int_0^\infty \frac{(1-\cos a x)}{x^2}dx, \quad a >0 $
- \(\S\S 7.11\) Q[2] \(\int_0^\infty \frac{(\sin x - x\cos x)}{x^3} dx \)
- \(\S\S\) 7.11 Q[3] \( \int_0^\infty \frac{\sin^3x}{x^3} dx \)
- \(\S\S\) 7.11 Q[4] \( \int_0^\infty \frac{x^3-\sin^3x}{x^5} dx\)
- \(\S\S\) 7.11 Q[5] \( \int_0^\infty \frac{\sin (p x) \sin (q x)}{x^2} dx\)
- \(\S\S\) 7.11 Q[6] \( \int_0^\infty \frac{\sin^2 px \cos 2qx}{x^2} dx \)
- \(\S\S\) 7.11 Q[7] \(\int_0^\infty \frac{\cos px - \cos qx}{x^2} dx\)
- \(\S\S\) 7.11 Q[8] \( \int_0^\infty \frac{\sin \pi x}{x(1-x^2)} dx\)
- \(\S\S\) 7.11 Q[9] \( \int_0^\infty \frac{1}{1-x^4} dx \)
- \(\S\S\) 7.11 Q[10] \(\int_0^\infty \frac{x^{p-1}}{1-x^3} dx\)
- \(\S\S\) 7.3 Exercise ----- Integrals of Type \(\int Q(x) \,dx\)
- Solutions to Problems from "Complex Variables-- Principles and Problem Sessions"
- MM Cross Roads-05 ComplexVariables
- MM Cross Roads -06 Ordinary Differential Equations
- MM Cross Roads-07 Partial Differential Equations
- MM Cross Roads-08 Fourier Series Fourier Transform
- MM Cross Roads-09 Green Functions
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- CM Cross Roads-01 Review of Newtonian Mechanics
- CM Problem Repository -01 Periodic Motion In One Dimension
- Qld cm/que/01/001; Periodic Motion
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- CM Problem Repository 02 Variable Mass Problems
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- Random Mix-01 Boundary Condition on Solutions of Schrodinger Equation
- Random Mix-02 Improper Riemann Integrals
- Random Mix-03 What all information does state of a quantum system give?
- Random Mix-04 Coordinate Representation
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- Random Mix-07 Chio's Rule For Determinants
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- DYK-01 Pauli's Reaction to Discovery of Matrix Mechanics
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- DYK-04 Who Revived the Klein Gordon Equation?
- DYK-05 Change in Sign of Fermion Wave Function under 2$\pi$ Rotation
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- Part-I Rise of Quantum Mechanics (2017)
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- Part-III Lecture Notes Point Particle in a Potential Well
- Part-IV Lecture Notes Spherically Symmetric Potentials in 3 Dimensions
- Part-V Lecture Notes General Structure of Quantum Mechanics
- Part-VI Lecture Notes Matrix Mechanics
- Part-VII Identical Particles and Spin
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