Contents The quantization of energy 33 Black-body radiation 34 Brief illustration 4.1: Wien's displacement law 34 Example 4.1: Using the Planck distribution 35 Spectroscopy 36 Brief illustration 4.2: The Bohr frequency condition 37 Wave-particle duality 37 The photoelectric effect 37...
[概要]量子力学は, 現代の科学技術文明にとってもっとも重要な理論である. 量子なる概念は, 二十世紀初頭に出現した. これを量子革命と称する. われわれは, 本論で量子論の発展過程を見てゆく. とりわけ本論が主題とするのは, 黒体放射の問題がいかに解き明かされてプランクの放射公式に結実したか, ということである. かかる議論の過程で, われわれは, 量子の二重性がどのようにして出現したのかが明確にされるのを見るであろう. 言い替えれば, 量子の二面性の根源が, 完成された理論のどこに潜んでいるかを見いだすであろう. また, 本論は, 量子の認識論に関して, その哲学的含意についても若干の議論をすることになる. [Abstract] Quantum mechanics is the most important theory for the modern scientific and technological civilization. The notion of quantum had been emerged at the beginning of the 20th century that is called quantum revolution. We will see the process of the development of the quantum theory in this paper. Especially, we will see and discuss how the black-body problem had been solved by the Planck's formula for radiation. This discussion will make a clear understanding to the emergence of the so-called wave-particle duality. That is to say, we will see the origins of the quantum duality that are hid behind of the accomplished theory. And we discuss a few philosophical implications that are about the epistemology of quantum.
Abstract There are two rather different phenomena that furthered the development of quantum theory: blackbody radiation and atomic spectra. In fact quantum theory began in 1900 with Planck’s radiation law and his introduction of the quantum of action h (Episode 7). His derivation of the law was by no means stringent. He assumed the existence of many resonators (atoms or molecules) of different but fixed frequencies ν, which could emit or absorb radiation only in quanta of energy E = hν. Einstein, in his light-quantum hypothesis of 1905, postulated that the energy quantization is a property of the radiation field itself and not due to the material resonators. He found a similarity between a cavity filled with radiation consisting of quanta and a vessel filled with gas consisting of atoms. However, he could not obtain Planck’s law but only Wien’s (Episode 10).
The fundamental fact for quantum physics which has emerged from experimental investigations in the field is a double duality of wave and particle concepts. The wave theory of light, firmly grounded by the wave explanations of interference and diffraction and by Maxwell’s electromagnetic theory of light waves, is now recognized as giving but one aspect of the nature of radiation. The corpusclar aspect of radiation is revealed in the photoelectric effect and in the Compton effect. In these effects Planck’s constant appears as a universal constant which connects the wave and particle modes of description of the same radiation.
We tried to introduce the idea of quantization in Chapter 9 as a completion of a one semester course. However, we skipped over some really interesting events in the history of Science between 1900 and 1913 when Bohr derived the quantized energy of the H atom. First we want to carry out the 1901 Planck derivation of the formula for blackbody radiation [1]. Many texts just show the curve, write e¼ hn, and move on. As a student, this author found that limited explanation very frustrating since energy quantization is a fundamental concept. Even among graduate texts in quantum mechanics, we are aware of only one that does the complete treatment which we will draw upon for the mathematics [2] but supplement with a narrative that we have found helpful to students over the years. Then in 1905, Albert Einstein (1879-1955), one of the most influential scientists of all time, gave an explanation of the photoelectric effect [3] for which he received the Nobel Prize in 1921. Even reducing our list to essential topics, we need to discuss the Davisson-Germer experiment [4]. The photoelectric effect introduces the idea that light waves can act as particles while the Davisson-Germer experiment showed that particles can act like waves and confirmed the De Broglie equation [5]. However, you can be assured that it will not be as difficult as you might have anticipated and if you can absorb the meaning of just these three key experiments you should be able to begin thinking in terms of quantum mechanics! Despite our slow historical development, this is 2010 and we have to get to the twenty-first century somehow!
Most textbooks say that quantum mechanics came into being when Max Planck, in a desperate attempt to construct a theory that explained experimental data, explored the radical idea of energy being radiated or absorbed by all bodies in discrete packets (or quanta). The energy content of a quantum of radiation is proportional to the frequency of the radiation emitted or absorbed, as in , and the proportionality constant bears the name of Planck. Planck himself was rather uncomfortable with the idea of energy packets, and spent a considerable amount of time in later years trying to “restore” the classical picture of continuous energy exchange between bodies and surroundings. However, in 1905 Einstein was able to use the idea of the quantum to explain the mechanism of the photoelectric effect, which cannot be explained in any other way! This established the reality of the quantum and so, it may be said that Einstein, rather than Planck, is the father of quantum mechanics. Within a few years, Neils Bohr constructed his model of the hydrogen atom in which the electron can only occupy certain special orbits around the nucleus, and any jump between these orbits is accompanied by energy being absorbed or emitted in the form of quanta. The nearly exact agreement of Bohr’s predictions based on the model and observed hydrogen atomic spectra was among the first impressive accomplishments of quantum mechanics.