The Correspondence Principle

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

Thus far, we have been focusing our attention on experiments that defy explanation in terms of classical mechanics and, at the same time, isolate certain aspects of the laws of quantum mechanics. We must not lose sight, however, of the fact that there exists an enormous domain, the domain of the macroscopic physics, for which classical physics works extremely well. There is thus an obvious requirement that quantum mechanics must satisfy - in the appropriate limit, namely the classical limit, it must lead to the same predictions as classical mechanics. Mathematically, this limit is that in which $\hbar$ may be regarded as small. For the electromagnetic field, for example, this means that the number of quanta in the field must be very large. For particles, it means that the de Broglie wavelengths must be very small compared to all relevant length scales. Of course, the statements of quantum mechanics are, as stated earlier, probabilistic in nature, while those of classical mechanics are completely deterministic. Thus, in the classical limit, quantum mechanical probabilities must become practical certainties; fluctuations must become negligible. This principle, that, in the classical limit, the predictions of the laws of quantum mechanics must be in one-to-one correspondence with the predictions of classical mechanics, is called the correspondence principle.

For example, in classical mechanics, physical quantities are functions $A(\mathbf {r} ,\mathbf {p} )\!$ of the position and momentum variables. The correspondence principle leads us to assume that, in quantum mechanics, these quantities are replaced with operators whose functional form in terms of position and momentum, also now operators, is the same. In other words, to a given classical quantity $A\left(\mathbf {r} ,\mathbf {p} \right),$ there corresponds an operator, $A\left({\hat {\mathbf {r} }},{\hat {\mathbf {p} }}\right).$ For instance, the Hamiltonian, $H={\frac {p^{2}}{2m}}+V(\mathbf {r} ),$ becomes

${\hat {H}}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}+V({\hat {\mathbf {r} }})=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\hat {\mathbf {r} }}),$

and the angular momentum, $\mathbf {L} =\mathbf {r} \times \mathbf {p} ,$ becomes

${\hat {L}}={\hat {\mathbf {r} }}\times {\hat {\mathbf {p} }}={\frac {\hbar }{i}}\left({\vec {r}}\times {\vec {\nabla }}\right).$