The Principle of Complementarity

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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.
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
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
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
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
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

The idea of wave-particle duality has no classical counterpart. In classical physics, a given entity must be exclusively one or the other. But this has come at the expense of great conceptual difficulty. We must somehow accommodate the classically irreconcilable wave and particle concepts. This accommodation involves what is known as the principle of complementarity, first enunciated by Bohr. The wave-particle duality is just one of many examples of complementarity. The idea is the following. Objects in nature are neither particles nor waves; a given experiment or measurement which emphasizes one of these properties necessarily does so at the expense of the other. An experiment properly designed to isolate the particle properties, such as Compton scattering or the observation of cloud chamber tracks, provides no information about the wave aspects. Conversely, an experiment properly designed to isolate the wave properties, for example the diffraction, provides no information about its particle aspects. The conflict is thus resolved in the sense that irreconcilable aspects are not simultaneously observable in principle. Other examples of complementary aspects are the position and linear momentum of a particle, the energy of a given state and the length of time for which the state exists, the angular orientation of a system and its angular momentum, and so on. The quantum mechanical description of the properties of a physical system is expressed in terms of pairs of mutually complementary variables or properties. Increasing precision in the determination of one such variable necessarily implies decreasing precision in the determination of the other.

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