Physical Basis of Quantum Mechanics

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

In this chapter, we will discuss the experimental evidence that led to the development of quantum mechanics, as well as some of the basic principles of the theory. In quantum mechanics, systems are described in terms of wave functions, which specify the probability of finding a system in a given state. This is in contrast to classical physics, which describes systems in terms of well-defined positions and velocities of particles. While many classical systems, such as fluids, are described in terms of statistical theories, said theories are only convenient approximations to the underlying deterministic theories that describe the individual particles that make up the system. Quantum mechaincs, on the other hand, is not a statistical approximation to some underlying deterministic theory. It proposes an inherent randomness to the behavior of physical systems, and all of its laws are written in terms of wave functions rather than definite properties of particles.

The concept of a wave function leads to some of the basic ideas of quantum mechanics, such as wave-particle duality and the principle of complementarity. Wave-particle duality is the idea that particles possess a wave-like nature, and vice versa. The principle of complementarity states that it is impossible to observe both particle-like and wave-like behaviors in the same experiment.