Thad Dankel, Jr. and Jack B. Levy
University of North Carolina Wilmington, 28401
An introduction to the quantum mechanical theory of the hvdroeen atom in textbooks almost always heains with the ~ h h Lheory r of circular orbits. The reasons fo;the continued presentation of this semi-classical approach include: 1) the easy visualization of the Bohr model, 2) the remarkable fact that it gives the correct energy levels (ignoring fine structure), and 3) the qualitative agreement of the radii of the Bohr orbits with the average distances of the electron from the nucleus as calculated from the corresponding wave functions (la). One flaw in the Bohr picture, however, is its prediction of nonzero orbital angular momentum for all atomic states, in contrast to the existence of s-orbitals with zero angular momentum in wave mechanics. Pauling (2) suggested that a linear Bohr theory could overcome this difficulty, but he gave no mathematical details, and his suggestion seems to have been subsequently ignored. Earlier, Sommerfeld had encountered linear orbits as a degenerate case of his theory of elliptical orbits and had rejected them for mathematical reasons (3). However, a close analysis of Sommerfeld's reasoning shows that he assumes the quantum number rn to he positive (44,when in fact it may he either positive or negative (4b). Thus, Sommerfeld's rejection of linear orbits rests on dubious grounds. In this paper, we present a linear Bohr theory of the hydrogen atom and show that the allowed energy levels are exactly those of Bohr. The Semi-Classical Linear Model of the Hydrogen Atom
We assume that the electron is a particle with one degree of freedom and with charge e moving in a Coulomb potential well arising from a charge at the origin (the nucleus) which is equal in magnitude but opposite in sign. We assume that at the origin the particle reverses its direction with no loss of energy. The energy E of the electron satisfies
Linear B0hr-Sommerfeld Electron Paths
where X,, = - e 2 / E is the classical maximum displacement. Evaluating the above integral and solving for E, we find
the energy levels of Bohr. The figure shows X,,, as a function of n, the principal quantum number. Note that X,,, is twice the radius of the corresponding Bohr orbit. The relation between energy and frequency can also he used to quantize the linear model (recall that Planck used such a relation to quantize the harmonic oscillator). In the Bohr theory, the energy-frequency relation is E
1 2
(1)
= --nhu
In the linear model, the time required for one complete period is
This integral can be evaluated, and n = l/t is found to satisfy
nucleus 0
-
electron Path e
where rn is the mass of the electron, and u its velocity. The Wilson-Sommerfeld quantization rule (5) requires
--
--
Presented before the North Carolina Academy of Science in Charlotte, April 1973.
398
/ Journal ot Chemical Education
XmaX (8) 1.1 -
4.2 9.5 17 27 38
52
Maximum distance from the nucleus as a function of the principal quantum number in the semi-classical linear model of the hydrogen atam.
This equation, together with eqn. (I),implies E=*
- me4
-
the Bohr levels again.
pear to be based on the idea that the electron must pass through the nucleus. This model therefore appears to differ in this important respect from other proposals for linear electron paths.
Conclusion
Literature Cited
We have shown how to supplement the usual Bohr theory, giving a semi-classical interpretation of s-orbitals. Linear by assumption, this model does not tax the intuition by demanding that a particle with no angular momentum move in a circular orbit. We believe that the model presented here should aid beginners in visualizing s-orbitals. In conclusion, one feature of this model should be emphasized, viz. that the particle elastically bounces off of the nucleus. Some objections ( l b , 6, 7) to linear orbits ap-
111 . . Eiherz. R. M.. Tundamentals of Modem Phvsie~." , John Wilev & Sons. Inc. New ~ o r < 1 9 6 1 , i h 8 , ibl ~ 3 1 3 . (21 Pauline. Linus. "The Nature of the Chemical Bond." 3rd Ed.. Cornall Univsrnitv PIPS;, I t h s c ~ Now . Ywk, 1960. P35. (31 Sommorfoid, Amold, "Atomic Structure and Spectral Liner," 3rd Ed.. Methuen. London. 1931.p 115. (41 Sommoifold, Arnold. "Wave Mechanics," Dutton, New York. 1930. la1 p 10, lbl p 8. (51 Cuss. IrstsrS..J.CHEM. EDUC.. 19. 371 (19421. (61 Arya, Afsm P.. "Fundsmentals of Modern Physics." Allyn and Bamn, Bmfon. 1911.~281. (71 Jammer. Mar. "The Conceptual Development of Quantum Mechanics." McGrawHill BmkCo.. Now York, 1966. p93.
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