Original Idea of Schrödinger Equation
In 1926, it had been well established (by Rutherford) that atoms are comprised of a dense, positively-charged nucleus which is surrounded by a region of negative charge. It had been established (by Thompson) that the negatively-charged region was comprised of particles called electrons. In the period between the discovery of these basic atomic attributes and quantum theory, little was known as to why the electrons in atoms are not instantly attracted into the nucleus. Afterall, there does exist an electrostatic potential between the nucleus and the electrons. What kept them from falling inward, thereby causing the atom to collapse? An Austrian physicist named Erwin Schrödinger confronted the problem, utilizing a new approach which was emerging at the time. He used the idea that matter in motion possesses wave-like attributes. This idea, which may seem unusual to those first exposed to it, was not based on pure speculation. In fact, much work was being produced (largely by Planck and Einstein) at this point which showed that light exists as both particle and wave, a concept which originated from experiments concerning the diffraction of light around barriers and blackbody radiation (the ultraviolet catastrophe). A French physicist named Louis Victor de Broglie had applied this particle-wave concept to matter and had devised an expression which relates the momentum of an object with the wavelength of its 'matterwave'. Not only did Schrödinger utilize the idea of waves in his new theory, but also the idea of quantization. A Danish physicist named Niels Bohr had attempted to work with this idea after he realized that the emission spectrum of hydrogen only possessed certain bands of color. These bands indicated to him that the electrons of hydrogen were making certain energy transitions within the atom, meaning the energy which an electron could possess was restricted to a discrete set of values. As it turns out, standing waves do in fact display a sort of quantization in that there can only exist integer amounts of antinodes and nodes along the wave. Schrödinger's model for electron behavior is usually referred to as the wave-mechanical model. This model states that all the possible positions which an electron can occupy, can be represented by a wave. It does so by ascribing a specific function to the electron. Since the function is based on wave properties, it is named a wave function. From this wave function, all information which one wishes to ascertain about the electron can be extracted. Wave functions will vary depending on the situation. The only conditions placed on them is that they must be continuous, single valued and square-integrable. The Schrödinger Equation is the mathematical relation between an electron's wavefunction and total mechanical energy. The Schrödinger Equation is a very generalized equation, making it versatile in terms of its application to various types of electron behavior.