Applications of Bohr's correspondence principle

Journal Article · Sat Jul 01 00:00:00 EDT 1989 · Am. J. Phys.; (United States)

DOI:https://doi.org/10.1119/1.16142· OSTI ID:6387338

Crawford, F S

The Bohr correspondence-principle (cp) formula /ital dE///ital dn/=/h bar/..omega.. is presented (..omega.. is the classical angular frequency) and its predicted energy levels /ital E//sub /ital n// are compared to those given by the stationary state solutions of the Schr/umlt o/dinger equation, first for several examples in one dimension (1D), including the ''quantum bouncer,'' and then for several examples in three dimensions (3D), including the hydrogen atom and the isotropic harmonic oscillator. For the 3-D cases, the cp predictions based on classical circular orbits are compared with the ''circlelike'' Schr/umlt o/dinger solutions (those with the lowest energy eigenvalue for a given /ital l/) and the cp predictions based on classical ''needle'' orbits (having zero angular momentum) with the Schr/umlt o/dinger /ital l/=0 solutions. For the H atom and the isotropic oscillator, the cp prediction does not depend on the classical orbit chosen because of a ''degeneracy'': the fact that for these systems ..omega.. is independent of the orbit. As a more stringent test of the cp, analogous nondegenerate systems /ital V/=/minus//ital k///ital r//sup 3/2/ in place of the H-atom potential /ital V/=/minus//ital e//sup 2///ital r/ and /ital V/=/ital kr//sup 4/ in place of the oscillator potential /ital V/=(1/2)/ital m/..omega../sup 2//ital r2/ are therefore considered. Aninteresting anomaly that occurs for the harmonic oscillator and itsnondegenerate analog /ital V/=/ital kr//sup 4/ is encountered (but not for the H atomnor its nondegenerate analog /ital V/=/minus//ital k///ital r//sup 3/2/), wherein half of thestates predicted by application of the cp to the needle orbits are ''spurious''in that there are no corresponding Schr/umlt o/dinger /ital l/=0 states.

• Research Organization: Lawrence Berkeley Laboratory and Physics Department, University of California, Berkeley, California 94720(US)
• OSTI ID: 6387338
• Journal Information: Am. J. Phys.; (United States), Journal Name: Am. J. Phys.; (United States) Vol. 57:7; ISSN AJPIA
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ATOMS
BOHR THEORY
DIFFERENTIAL EQUATIONS
ELECTRONIC EQUIPMENT
ELEMENTS
ENERGY LEVELS
EQUATIONS
EQUIPMENT
HARMONIC OSCILLATORS
HYDROGEN
MECHANICS
NONMETALS
ONE-DIMENSIONAL CALCULATIONS
OSCILLATORS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM MECHANICS
SCHROEDINGER EQUATION
THREE-DIMENSIONAL CALCULATIONS
WAVE EQUATIONS