retical explanation of this behavior it was necessary to assume that the energy

Retical explanation of this behavior it was necessary

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retical explanation of this behavior it was necessary to assume that the energy of a mechanical system cannot assume any sort of value, but only certain discrete values whose mathe- matical expressions were always dependent upon Planck's constant h. Moreover, this conception was essential for the theory of the atom (Bohr's theory). For the transitions of these states into one another,--with or without emission or absorption of radiation,--no causal laws could be given, but only statistical ones; and, a similar conclusion holds for the radioactive decomposition of atoms, which decomposition was carefully investigated about the same time. For more than two decades physicists tried vainly to find a uniform inter- pretation of this " quantum character " of systems and phenomena. Such an attempt was successful about ten years ago, through the agency of two entirely different theoretical methods of attack. We owe one of these to Heisenberg and Dirac, and the other to de Broglie and Schr6dinger. The mathematical equivalence of the two methods was soon recog- nized by Schr6dinger. I shall try here to sketch the line of thought of de Broglie and Schr6dinger, which lies closer to the physicist's method of thinking, and shall accompany the description with certain general considerations. The question is first: How can one assign a discrete suc- cession of energy value H, to a system specified in the sense of classical mechanics (the energy function is a given function of the co6rdinates q~ and the corresponding momenta p,)? Planck's constant h relates the frequency H,/h to the energy values H,. It is therefore sufficient to give to the system a succession of discrete frequency values. This reminds us of the fact that in acoustics, a series of discrete frequency values is co6rdinated to a linear partial differential equation (if
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March, I936.] PHYSICS AND REALITY. 373 boundary values are given) namely the sinusoidal periodic solutions. In corresponding manner, Schr6dinger set himself the task of co6rdinating a partial differential equation for a scalar function ~b to the given energy function ~(qr, Pr), where the q~ and the time t are independent variables. In this he succeeded (for a complex function $) in such a manner that the theoretical values of the energy Ho, as required by the statistical theory, actually resulted in a satisfactory manner from the periodic solution of the equation. To be sure, it did not happen to be possible to associate a definite movement, in the sense of mechanics of material points, with a definite solution ¢(qr, t) of the Schr6dinger equation. This means that the ~ function does not deter- mine, at any rate exactly, the story of the qr as functions of the time t. According to Born, however, an interpretation of the physical meaning of the ~ functions was shown to be possible in the following manner: Sf (the square of the absolute value of the complex function ~) is the probability density at the point under consideration in the configuration-space of the qr, at the time t. It is therefore possible to characterize the
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