Abstract
The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.
Highlights
Few expressions are more frustrating to the student than “it is easy to prove that” followed by a nontrivial result
Bohm [4] leaves the integral as an exercise, and Born [5] only cites the result to highlight the remarkable coincidence between this and Dirac’s later results. He later solves the equation in an appendix using the same method Sommerfeld [6, 10] employed to solve the integral. This fact does not constitute a severe hindrance in the learning process, since Schrodinger’s equation provides the complete solutions for the hydrogen atom. [12, 13] the hydrogen atom is one of the most important elementary systems that can be solved with quantum mechanics, and we consider relevant that the students know how to work out the details even in the old quantum theory
In the period of time between Max Planck’s proposed quantization of energy in 1900 [14,15] and the development of the first forms of matrix [16] and wave [17] mechanics - beginning in 1925 and 1926, respectively - the physics of the atomic phenomena evolved into a theory known as old quantum mechanics (OQM). [15, 18] Analyses of Planck’s reasoning to derive the quantum of action h are beyond the scope of this article and can be found in the references, [14, 15, 19,20,21] along with translations of Planck’s original works. [22,23,24] Below, we will see how this was a hybrid theory that prescribed a classical approach to the problem before the quantization could be, somehow, introduced
Summary
Few expressions are more frustrating to the student than “it is easy to prove that” followed by a nontrivial result. [12, 13] the hydrogen atom is one of the most important elementary systems that can be solved with quantum mechanics (together with the potential well and the harmonic oscillator), and we consider relevant that the students know how to work out the details even in the old quantum theory We explore these calculations in detail, so the readers can focus more of their time on the physical interpretation of the results. The article is organized as follows: Sec. 2 provides a brief historical introduction to the old quantum theory; Sec. 3 shows how the integral that quantizes the radial component of the action can be computed without previous knowledge of its orbits; Sec. 4 describes the original method Arnold Sommerfeld employed to solve the integral using the properties of an ellipse and a few other tricks.
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