Introduction to Quantum Mechanics : With a Focus on Physics and Operator Theory

Acknowledgments
Introduction
Conventions
A Short Summary of the Quantization Process
Constraints on Quantization
Contents
List of Symbols
1 Quantization of a Free Particle in N-Dimensional Space
	1.1 The Operators Corresponding to the Measurement of the Components of Position
	1.2 Partial Integration and an Auxiliary Sequence
	1.3 The Operators Corresponding to the Measurement …
	1.4 The Hamilton Operator Governing Free Motion in double struck upper R Superscript nmathbbRn
	1.5 Time Evolution Generated by the Free Hamilton Operator
		1.5.1 Large Time Asymptotics of the Evolution
		1.5.2 Time Evolution and Causality
		1.5.3 Free Propagation of Gaussians
	1.6 Perturbations of the Free Hamilton Operator in less than or slanted equals 3leqslant3 Space Dimensions
2 Commutators, Symmetries and Invariances
	2.1 Commuting Operators
	2.2 Closed Invariant Subspaces of Observables
	2.3 Insert: Decomposition of Spectra of DSLO\'s
	2.4 Quantization of Angular Momentum
		2.4.1 A Change of Representation Induced by Introduction of Spherical Coordinates
		2.4.2 Spherical Harmonics
		2.4.3 Analysis of the Reduced Operators
		2.4.4 The Operator Corresponding to the Square of Angular Momentum
		2.4.5 The Commuting of the Operators ModifyingAbove upper L With caret Subscript 33 and ModifyingAbove upper L With caret squared2
	2.5 Symmetry and Invariance
	2.6 An Unitary Representation of Translations in Euclidean Space
		2.6.1 Generators Corresponding to Continuous One-Parameter Subgroups
	2.7 An Unitary Representation of Orthogonal Transformations in Euclidean Space
		2.7.1 An Unitary Representation of upper O left parenthesis n right parenthesisO(n) on upper L Subscript double struck upper C Superscript 2 Baseline left parenthesis double struck upper R Superscript n Baseline right parenthesisL2mathbbC(mathbbRn)
		2.7.2 Generators Corresponding to Rotations About the Coordinate Axes in 33 Space Dimensions
		2.7.3 Symmetries of Perturbations of the Free Hamilton Operator
	2.8 An One-Parameter Group of Symmetries
	2.9 Galilean Invariance
	2.10 Gauge Invariance
3 Simple Quantum Systems in bold 11 Space Dimension
	3.1 Auxiliary Results About Perturbations of the Free Hamilton Operator in 11D
	3.2 Quantum Tunneling
	3.3 Potential Wells and Potential Barriers
		3.3.1 Eigenvalues of ModifyingAbove upper H With caret for upper V 0 less than 0V0 < 0
		3.3.2 Motion in a ``deltaδ-Function\'\' Potential
	3.4 Weyl\'s Limit Point/Limit Circle Criterion
		3.4.1 The Free Hamiltonian in bold 11 Space Dimension, a Simple Application
	3.5 Further Results About the Hamilton Operator of the Harmonic Oscillator
	3.6 Motion in a Repulsive Pöschl–Teller Potential and Resonances
		3.6.1 Calculation of the Resolvent and the Resonances of upper AA
		3.6.2 Calculation of the Functional Calculus for upper AA
4 Motion in a Central Force Field
	4.1 A Change of Representation Induced by Introduction of Spherical Coordinates
	4.2 Reduction of upper U ModifyingAbove upper H With caret Subscript 0 Baseline upper U Superscript negative 1U 0   U-1
	4.3 Analysis of the Reduced Operators
	4.4 Motion in a Coulomb Field
		4.4.1 An Application of Weyl\'s Criterion
		4.4.2 Extensions of the Purely Radial Operator ModifyingAbove upper H With caret Subscript 0000
		4.4.3 Eigenvalues of ModifyingAbove upper H With caret Subscript 00 e Superscript asterisk00e* and ModifyingAbove upper H With caret Subscript script l script m Superscript asteriskmathcallmathcalm* for left parenthesis script l comma script m right parenthesis element of(mathcall,mathcalm) in script upper I minus left brace left parenthesis 0 comma 0 right parenthesis right bracemathcalI   {(0,0)}
		4.4.4 Corresponding Normalized Eigenfunctions
		4.4.5 Continuous Spectrum of ModifyingAbove upper H With caret Subscript 00 e Superscript asterisk00e* and ModifyingAbove upper H With caret Subscript script l script m Superscript asteriskmathcallmathcalm* for left parenthesis script l comma script m right parenthesis element of(mathcall,mathcalm) in script upper I minus left brace left parenthesis 0 comma 0 right parenthesis right bracemathcalI   {(0,0)}
		4.4.6 The Spectrum of ModifyingAbove upper H With caret Subscript 00 e Superscript asterisk00e* and ModifyingAbove upper H With caret Subscript script l script m Superscript asteriskmathcallmathcalm* for left parenthesis script l comma script m right parenthesis element of(mathcall,mathcalm) in script upper I minus left brace left parenthesis 0 comma 0 right parenthesis right bracemathcalI   {(0,0)}
5 Motion in an Axially-Symmetric Force Field
	5.1 A Change of Representation Induced by Introduction of Cylindrical Coordinates
	5.2 Reduction of ModifyingAbove upper H With caret Subscript 00
	5.3 Motion of a Charged Particle in a Homogeneous Magnetic Field
		5.3.1 Construction of a Hilbert Basis of bold upper L Subscript double struck upper C Superscript bold 2 Baseline bold left parenthesis bold 0 bold comma normal infinity bold right parenthesisL2mathbbC(0,infty)
		5.3.2 Synthesis and Properties of the Hamilton Operator ModifyingAbove upper H With caret of the System
		5.3.3 Calculation of the Time Evolution Generated by ModifyingAbove upper H With caret
6 The Path Integral Approach to Quantum Mechanics
	6.1 A Connection Between the Solutions of the Schrödinger and the Heat Equation
7 Conclusion
References
Index