Introduction To Wave Mechanics

introduction to wave mechanics: At the turn of the 20th century, physics was starting to look rather mature and polished. At about this time Albert Michelson wrote, The more important fundamental laws and facts of physical science have all been discovered, and these are so rmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there are apparent exceptions to most of these laws, and this is particularly true when the observations are pushed to a limit, i.e., whenever the circumstances of experiment
are such that extreme cases can be examined. Such examination almost surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the apparent exceptions." [1]

However, there were already some indications that big changes were on the horizon. One was the problem that the version of ether theory in existence at the time could not easily be reconciled with the results of the experiment that Michelson had carried out with Morley. Another was the so-called ultraviolet catastrophe that came about with Lord Rayleigh's 1900 version of the Rayleigh-Jeans law of blackbody radiation.

THE PLANCK-EINSTEIN EQUATION On December 14th 1900, Max Planck presented a derivation of the blackbody radiation law that was based on the assumption that electromagnetic radiation could only be emitted in particle-like packets with a xed ratio of energy to frequency. This assumption can be written
E = h
where  is the frequency and h is a universal constant. This equation came to be known as Planck's equation, though Planck did not initially consider it to be a real physical law. In a letter to a colleague, he wrote

To summarize, all what happened can be described as simply an act of desperation... This was a purely formal assumption and I really did not give it much thought except that, no matter what the cost, I must bring
about a positive result." [2]

After all, light had been considered decisively wave-like in nature since Thomas Young's double-slit experiment in 1803. However, in 1905, Albert Einstein explained the photoelectric e ect by suggesting that this equation was physically true. Einstein also discovered that Planck's assumption provided a solution to the ultraviolet catastrophe, which is something Planck hadn't been aware of. Then in 1923, Arthur Compton performed an experiment in which X-rays were scattered o electrons. His results demonstrated that light must consist of particlelike objects with energy proportional to frequency, thus con rming Einstein's suggestion.

the wave-particle duality concept: Although the wave-particle duality is one of the conceptual cornerstones of quantum mechanics, the waveor- particle dillemma limited to light only is at least 250 years older than the quantum branch of physics. The question of the nature of light has been an important scientific issue since the 17th century, the same time when modern optics was born. One can easily discern three different stages in the evolution of this
problem, and to each of these stages we can attach names of several famous physicists who contributed to our understanding of light. Their discussions and different explanations demonstrate how baffling the nature of light has seemed from the beginning, and how rich is the current of thoughts and ideas that it has stimulated. Only in the last century, thanks to the quantum theory, did the duality problem unexpectedly expand to embrace matter as well.
The first modern scientific inquiries into the realm of optics date back to ca. 1650. Isaac Newton stated that light was composed of particles emitted in all directions from a source, and it was this corpuscular view that became dominant in the 1700s. The second of the aforementioned stages started in the early 19th century when Thomas Young conducted his famous slit experiment which unambiguously proved that light rays interfered just like water waves. Shortly afterwards Augustin-Jean Fresnel presented the wave theory of light grounded firmly within a mathematical framework, and in the 1870s James Clerk Maxwell explained light as propagation of electromagnetic waves. However, in 1905 Albert Einstein, motivated by Max Planck’s scheme of energy quantization, put forward an idea that that light itself propagated in space and interacted with matter as discrete particles (light quanta). Twenty years later Louis de Broglie advanced a hypothesis that all matter manifests a wavelike nature, even if under many experimental circumstances it also behaves as if it were consisted of particles. Quantum mechanics employed this conceptual breakthrough in order to united the wave and the particle views: Light and matter show both wavelike and particle like properties, although not at the same time.
In the following sections we will examine the development of the wave-particle duality concept in more detail, but still rather succinctly. We omit or relegate to later chapters the more detailed quantitative treatments of the presented phenomena, because right now our goal is only to look upon the historical evolution
of the concept. Hence we can better appreciate the colossal amount of scientific research hidden behind and beneath it.