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Table of Contents
1. The Schrödinger equation
2. Physical motivation
3. Classical mechanics of a single particle
4. Quantum mechanics of a particle
5. Time-dependent Schrödinger equation
6. Time-independent Schrödinger equation
7. Quantum observable and Hamiltonian
8. Scattering theory
9. Variants and generalizations of Schrödinger equation
Summary
The Schrödinger equation is one of the most fundamental equations in non-relativistic quantum mechanics. It has applications in both physics and mathematics. The document discusses the classical and quantum mechanics of a single particle, the time-dependent and time-independent Schrödinger equations, conservation of energy, wave functions, quantum observables, and scattering theory. It also touches on the spectral theory of Schrödinger operators, scattering amplitudes, and the relationship between potential functions and scattering amplitudes. The document concludes by mentioning generalizations and variants of the Schrödinger equation.