The Correspondence Principle

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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
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Free Particle in Spherical Coordinates
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Hydrogen Atom
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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

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).

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