We present an introduction to quantum mechanics based on the formal correspondence between the atomic properties of quantum jumps and the classical harmonics of the electron’s periodic motion. By adding a simple quantum condition to the classical Fourier analysis, we readily find the energies of the stationary states, calculate the transition probabilities between the states, and construct the line spectrum of the emitted light. We provide examples to illustrate the asymptotic, and sometimes exact, agreement between the classical-quantum results (Fourier harmonics) and the exact quantum results (Heisenberg harmonics).
REFERENCES
1.
N. Bohr, “On the quantum theory of line-spectra,” in Niels Bohr Collected Works, Vol. 3, edited by L. Rosenfeld and J. Rud Nielsen (North-Holland, New York, 1981), pp. 67–184. Reprinted in Ref. 6, pp. 95–137.
2.
F. S.
Crawford
, “Applications of Bohr’s correspondence principle
,” Am. J. Phys.
57
, 621
–628
(1989
).3.
J.
Sivardière
and E.
Belorizky
, “Old quantum mechanics and perturbation theory
,” Am. J. Phys.
53
, 653
–655
(1985
).4.
J.
Nag
, V. J.
Menon
, and S. N.
Mukherjee
, “Ehrenfest theorem and the classical trajectory of quantum motion
,” Am. J. Phys.
55
, 802
–804
(1987
).5.
G. Q.
Hassoun
and D. H.
Kobe
, “Synthesis of the Planck and Bohr formulations of the correspondence principle
,” Am. J. Phys.
57
, 658
–662
(1989
).6.
Sources of Quantum Mechanics, edited by B. L. Van Der Waerden (Dover, New York, 1968).
7.
M. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983), pp. 307–310.
8.
The absolute intensity depends on the number of atoms in the initial state n because the total energy radiated per unit time is The population of excited states, as described by depends on the experimental conditions, for example, spark tube, temperature equilibrium, etc.
9.
A. P. French and E. F. Taylor, An Introduction to Quantum Physics (W. W. Norton, New York, 1978), pp. 15–16.
10.
W.
Heisenberg
, “Quantum-theoretical re-interpretation of kinematic and mechanical relations
,” Z. Phys.
33
, 879
–893
(1925
), reprinted in Ref. 6, pp. 261–276.11.
K. Krane, Modern Physics (Wiley, New York, 1996), pp. 196–198.
12.
R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, New York, 1985), pp. 117–118.
13.
J. H. Van Vleck, Quantum Principles and Line Spectra (Bulletin of the National Research Council, Washington, DC, 1926), p. 123.
14.
A. E. Ruark and H. C. Urey, Atoms, Molecules and Quanta (McGraw-Hill, New York, 1930), p. 173.
15.
S. Tomonaga, Quantum Mechanics (North-Holland, Amsterdam, 1962), p. 149.
16.
T-Y. Wu, Quantum Mechanics (World Scientific, Singapore, 1986), p. 63.
17.
C. J.
Latimer
, “Teaching Bohr theory
,” Phys. Educ.
18
, 86
–90
(1983
).18.
D.
Kleppner
, M. G.
Littman
, and M. L.
Zimmerman
, “Highly excited atoms.
” Sci. Am.
244
, 130
–149
(1981
).19.
T. F. Gallagher, Rydberg Atoms (Cambridge University Press, New York, 1994).
20.
M. I.
Syrkin
, “Semiclassical approach to Rydberg-atom intercombination transitions in collisions with electrons
,” Phys. Rev. A
50
, 2284
–2291
(1993
).21.
For the hydrogen atom, the spectroscopic frequency is and the fundamental frequency of revolution of the electron in an elliptical orbit is where is the Rydberg constant. See Ref. 14, pp. 122–125. As a function of n, the orbital frequencies for elliptical and circular orbits are identical. However, because circular motion possesses no overtones, the allowed transitions obey the selection rule while for elliptical orbits all transitions are possible.
22.
H. Kramers, Intensities of Spectra Lines (A. F. Host & Son, Kobenhavn, 1919).
23.
D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1964), p. 41.
24.
D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1999), p. 462.
25.
V. Rojansky, Introductory Quantum Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1938), pp. 164–165, pp. 341–349.
26.
D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1995), p. 35.
27.
Reference 26, p. 314.
28.
Reference 26, p. 26.
29.
Reference 26, pp. 222–225.
30.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), p. 499.
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