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Werner Heisenberg contributed to science at a point when the old quantum physics was discovering a field littered with more and more stumbling blocks. He decided that quantum physics had to be re-thought from the ground up. In so doing he excised several items that were grounded in classical physics and its modeling of the macro world. Heisenberg determined to base his quantum mechanics "exclusively upon relationships between quantities that in principle are observable."〔B.L.Van der Waerden, ''Sources of Quantum Mechanics'', p. 261〕 By so doing he constructed an entryway to matrix mechanics. He observed that one could not then use any statements about the such things as "the position and period of revolution of the electron."〔B.L.Van der Waerden, ''Sources of Quantum Mechanics'', p. 261〕 Rather, to make true progress in understanding the radiation of the simplest case, the radiation of excited hydrogen atoms, one had measurements only of the frequencies and the intensities of the hydrogen bright-line spectrum to work with. In classical physics, the intensity of each frequency of light produced in a radiating system is equal to the square of the amplitude of the radiation at that frequency, so attention next fell on amplitudes. The classical equations that Heisenberg hoped to use to form quantum theoretical equations would first yield the amplitudes, and in classical physics one could compute the intensities simply by squaring the amplitudes. But Heisenberg saw that "the simplest and most natural assumption would be" 〔B.L.Van der Waerden, ''Sources of Quantum Mechanics'', p. 275f〕 to follow the lead provided by recent work in computing light dispersion done by Kramers. 〔H. A. Kramers, ''Nature'' 113 (1924) 673. 〕 The work he had done assisting Kramers in the previous year〔See paper 3 in B.L.Van der Waerden, ''Sources of Quantum Mechanics'.〕 now gave him an important clue about how to model what happened to excited hydrogen gas when it radiated light and what happened when incoming radiation of one frequency excited atoms in a dispersive medium and then the energy delivered by the incoming light was re-radiated — sometimes at the original frequency but often at two lower frequencies the sum of which equalled the original frequency. According to their model, an electron that had been driven to a higher energy state by accepting the energy of an incoming photon might return in one step to its equilibrium position, re-radiating a photon of the same frequency, or it might return in more than one step, radiating one photon for each step in its return to its equilibrium state. Because of the way factors cancel out in deriving the new equation based on these considerations, the result turns out to be relatively simple. ==Development of full quantum mechanical theory== To make a long and rather complicated story short, Werner Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight. By means of an intense series of mathematical analogies that some physicists have termed "magical," Heisenberg wrote out an equation that is the quantum mechanical analogue for the classical computation of intensities. Remember that the one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.〔Heisenberg's paper of 1925 is translated in B. L. Van der Waerden's ''Sources of Quantum Mechanics,'' where it appears as chapter 12.〕 Heisenberg's groundbreaking paper of 1925 neither uses nor even mentions matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)"〔Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details," p. 2〕 of hydrogen radiation. After Heisenberg wrote his ground breaking paper, he turned it over to one of his senior colleagues for any needed corrections and went on a well-deserved vacation. Max Born puzzled over the equations and the non-commuting equations that Heisenberg had found troublesome and disturbing. After several days he realized that these equations amounted to directions for writing out matrices. Matrices were a bit off the beaten track, even for mathematicians of that time, but how to do math with them was already clearly established. He and a few colleagues took up the task of working everything out in matrix form before Heisenberg returned from his time off, and within a few months the new quantum mechanics in matrix form formed the basis for another paper. When quantities such as position and momentum are mentioned in the context of Heisenberg's matrix mechanics, it is essential to keep in mind that a statement such as pq ≠ qp does not refer to a single value of p and a single value q but to a matrix (grid of values arranged in a defined way) of values of position and a matrix of values of momentum. So multiplying p times q or q times p is really talking about the matrix multiplication of the two matrices. When two matrices are multiplied, the answer is a third matrix. Max Born saw that when the matrices that represent pq and qp were calculated they would not be equal. Heisenberg had already seen the same thing in terms of his original way of formulating things, and Heisenberg may have guessed what was almost immediately obvious to Born — that the difference between the answer matrices for pq and for qp would always involve two factors that came out of Heisenberg's original math: Planck's constant h and i, which is the square root of negative one. So the very idea of what Heisenberg preferred to call the "indeterminacy principle" (usually known as the uncertainty principle) was lurking in Heisenberg's original equations. Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical — the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.〔Thomas F. Jordan, ''Quantum Mechanics in Simple Matrix Form'', p. 149〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Heisenberg's entryway to matrix mechanics」の詳細全文を読む スポンサード リンク
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