European Journal of Physics Education | 2019
Solving Hydrogen Atom Problem Using Spherical Polar Coordinates: A Qualitative Study
Abstract
Quantum mechanics has completed century since its genesis. Quantum mechanics is taught at various levels-starting from school and colleges to universities. Regression methods are introduced at under graduate and post graduate levels to solve Schrodinger equation for finding solutions of various trivial and non-trivial physical problems. The common problems, which students encounter at UG level areparticle in a box, potential step and barriers, harmonic oscillator and hydrogen atom. It has been observed that students lack clarity in solving and grasping the hydrogen atom problem. Two reasons can be accounted for this. It is perhaps a lengthy derivation and students, many times, are not well acquainted with the requisite knowledge of Spherical Polar Co-ordinate system. In this article, a brief review on the birth of quantum mechanics is presented judiciously discussing the contribution of Schrodinger, before solving the hydrogen atom problem. Readers are first introduced to spherical-polar coordinate system. The radial solutions, radial probability distribution functions, and hydrogen orbital, are plotted using Mathematica software v.12, for the sake of visualization and understanding . INTRODUCTION The spectra of many elements have been known, since 19 century. The emission and absorption of different, but of definite wavelengths (i.e. color) radiations, work like a fingerprint to recognize chemical elements. In other words, the spectra of atoms are signatures of the electronic distribution inside atoms. But the emission and absorption of radiation, in visible spectra, had remained a mystery for a century, until Planck gave the idea of quanta and proposed that light travels in form of quanta. Quantum mechanics was born in the year 1900, when Max Planck derived a formula for black body radiation in order to explain it, for all possible wavelengths (Alain& Villain, 2017). Planck introduced the concept of quantization of energy. He proposed that radiation travels in form of quanta (i.e. a bundle of energy). In 1905, Einstein introduced the European J of Physics Education Volume 10 Issue 4 1309-7202 Singh 2 notion of “Lichtquanten” (i.e. quantum of light). Twenty years later, it was named photon. He realized the importance of the idea of quantization to explain photoelectric effect. In the year 1905, also rendered as miraculous year in the history of science, Einstein published three historical papers; one dealing with special relativity, another with Brownian motion and the third one with photoelectric effect. He used quantum nature of light to explain photoelectric effect. Wave nature of light could already explain optical phenomena as interference, diffraction and polarization. In 1913, the ground breaking discovery of Bohr atomic model further extended the idea of quantization, because Bohr postulated that electrons in atoms can move only in definite orbits, and it can emit or absorb radiation in form of definite quanta only. The experimental observation of hydrogen atom spectra Figure (1) was already reported by contemporary scientists. But, no satisfactory theory existed, which could explain it.These developments led Louis de Broglie to propose the idea of wave-particle duality in 1923, as a part of his doctoral dissertation. He wanted to introduce the idea of “Atom of Light”, but his principal examiner Paul Lanevin consulted Einstein regarding this. Einstein, appreciated the idea of wave-particle duality, but could not agree with the idea of atom of light. Broglie removed the latter part and obtained his Ph.D degree.Broglie re-derived Bohr’s quantization rules (Alain& Villain, 2017). Figure 1.Absorption and emission lines in hydrogen atom spectra (H-α lines are most intense) It had become quite natural for the contemporary physicists to raise question about the wave equation; that could be solved to obtain such solutions. In 1926, Schrodinger Figure (2) solved problems in a series of papers. In a co-parallel manner, Heisenberg (Heisenberg, 1925) Born and Jordan (Born & Jordan, 1925) published their matrix version of quantum mechanics in 1925 (Aschman & Keaney, 1989; Reiter &Yngvason, 2013; Fedak & Jeffrey 2009). Modern quantum mechanics was born, when Schrodinger demonstrated equivalence with their matrix formalism by exactly solving the hydrogen atom problem and explain it’s spectra. European J of Physics Education Volume 10 Issue 4 1309-7202 Singh 3 Figure 2. Left: Erwin Schrodinger (1887-1961) (Courtesy: R. Braunizer 1926); Right: Tomb of Schrodinger and his wife with plaque imprinted with his famous equation (Courtesy: C. Joas 2008) A relativistic version of his equation came within a year. Klein, Gordon and Fock gave relativistic equation for a free particle. The relativistic treatment of hydrogen atom was introduced by P. A. M. Dirac in 1928.Opposite to matrix calculus of Heisenberg, Schrodinger’s approach is quite simple. Schrodinger equation can be derived considering spatial distribution of the amplitude of the wave ψ(x), at a fixed point in time as follows( ) + ( ) = 0; = & = h/ (1) ( ) + ( ) = 0; = 2 ( − ) (2) ( ) + ( ) ( ) = 0 (3) The premises of quantum mechanics have gotten developed tremendously, since its birth in the beginning of 20 century. Almost every walk of science needs it, especially when the problem is rooted at the bottom of the scale. From nanoscience to cosmology, its knowledge has become mandatory in order to fully narrate natures beauty in terms of mathematical formalism. Here, the exact solution of hydrogen atom, using Schrodinger equation is obtained in a greater detail in a pedagogical manner. MATHEMATICAL ANALYSIS Though, hydrogen atom problem has become a century old problem and numerous text books (Ghatak & Loknathan, 2004; Schiff & Bandhyopadhyay, 2017) (Feynman, 2015) and documents European J of Physics Education Volume 10 Issue 4 1309-7202 Singh 4 are available to solve it including online references (Chapter-10 The Hydrogen Atom) we shall start with a fresh cumulative approach. We shall first convert three-dimensional Schrodinger equation from Cartesian coordinate system to spherical coordinate system. Schrödinger Equation in Spherical Polar Coordinates The vector representations of unit vectors r, θ and φ are as shown in Figure (3). \uf028 \uf029 \uf028 \uf029 r rd r dr r r d r d ˆ ˆ ˆ \uf02b \uf03d \uf03d \uf072 (4) \uf0fa \uf0fb \uf0f9 \uf0ea \uf0eb \uf0e9 \uf0b6 \uf0b6 \uf02b \uf0b6 \uf0b6 \uf02b \uf0b6 \uf0b6 \uf02b \uf03d \uf066 \uf066 \uf071 \uf071 d r d r dr r r r r dr . ˆ . ˆ . ˆ ˆ (5)