La Funcion De Onda

‘An Undulatory Theory of the Mechanics of Atoms and Molecules’ and the Development of the Shrödinger Wave Equation Joshua Sidgwick * I Preamble—why Schrödinger?

In an introductory course to quantum physics the central mechanical element is the Schrödinger Wave Equation. Here is its one-dimensional representation:
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The scope and significance of this equation in quantum physics can beconsidered as ‘logically analogous’ to classical physics’ second law of motion, F=ma.2 This is because Ψ(x,t) solutions to this equation encode all needed physical information about the state of the system that Ψ(x,t) represents.3 Arguably it is the most important equation of modern science, as it mathematically models sub-microscopic particles and so appears widely in not just the many fields ofphysics but also organic and inorganic chemistry.4 It is for these reasons that I decided to further investigate the development of this equation.

II Focus, context and justification—selecting An Undulatory Theory of the Mechanics of Atoms and Molecules
In this research assignment I will examine important aspects of Erwin Schrödinger’s contribution to the theory of quantum mechanics. I will do thisby reviewing his paper An Undulatory Theory of the Mechanics of Atoms and Molecules. This paper was published in America in The Physical Review late in 1926. During 1926 Schrodinger gave talks at universities in Munich, Berlin and Zurich, to eminent physicists of the time including Planck, Sommerfield and Wein, discussing his developments in a wave theory of mechanics. This paper is in effect aJosh Sidgwick is in his second year of a Bachelor of Advanced Science degree at the Australian National University. He is a current resident of Bruce Hall. 1 D. Griffiths, Introduction to Quantum Mechanics (2005). Ψ is the Greek letter Psi. 2 Ibid. 3 N. Zettili, Quantum Mechanics: Concepts and Application (2001), 158. 4 J. Mehra, The Historical Development of Quantum Theory, Volume Five, Part One(1987).
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Cross-sections | Volume V 2009

summation of the content of these talks so they could be accessible to a wider audience.5 1926 was also a year of prominent publication for Schrödinger, in which six important papers of his were published in German.6 One of these involved suggesting quantisation as an Eigenvalue problem—crucially important for the use of operators, which maybe used to determine expectation values for observables. Others included dealing with the continuation of micro-mechanical to macro-mechanical systems, which, vitally, determined the previous classical physics as an upper limit of his new quantum theory, thus unifying the two coherently.7 Schrödinger states early on that views in An Undulatory Theory were ‘first published in a series of Germanpapers’.8 As such, many important elements of these previous papers are incorporated into this paper. For these reasons, An Undulatory Theory, is a broad account of the ‘author’s work on a new form of quantum theory’ as Schrödinger terms it in the paper, and so therefore a useful paper to review.

III

Overview of An Undulatory Theory

The first four sections of Schrödinger’s paper can beloosely grouped together in summary. They are concerned with exploring the Hamiltonian and explaining how, from simple modification of a Hamiltonian principle of action, a comparison between mechanical and optical systems can be drawn. De Broglie’s postulates are then described and applied in the context of the Hamiltonian, so that we may consider a particle or ‘one material point’ as a ‘parcel ofwaves’.9 The significance of wavelength to a situation being a macro or micro mechanical problem is also considered. This important idea is dealt with in most introductions to quantum mechanics through using the De Broglie wavelength as a limiting factor on whether a system would be treated quantum mechanically or not.10 Most importantly, in these sections the Schrödinger wave equation is described....