By S R de Groot; L G Suttorp

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360◦ thus corresponds to 2π. 1 Schr¨ odinger If the electron is a standing wave, why should it be confined to a circle? After the insight that particles can behave like waves, which de Broglie gained ten years after Bohr postulated the quantization of angular momentum, it took less than three years for the mature (“new”) quantum theory to be formulated, not once but twice in different mathematical attire, by Werner Heisenberg in 1925 and by Erwin Schr¨ odinger in 1926. If we imagine the electron as a standing wave in three dimensions, we have almost all it takes to arrive at the equation that is at the heart of the new theory.

27) −∞ Believe it or not, a significant fraction of the literature in theoretical physics concerns variations and elaborations of this integral. √ One such variation can be obtained by substituting a x for x: +∞ dx e−ax 2 /2 = −∞ 2π . 28) Another variation can be obtained by treating both sides of this equation as functions of a and differentiating them with respect to a. The result is +∞ dx e−ax 2 2π . 28. Prove the last two equations. One method that sometimes helps evaluating an integral is known as integration by parts.

18) 2 This “uncertainty relation,” as it is generally called in the English speaking world, was first derived by Werner Heisenberg, also in 1926. It is essential to our understanding of why the laws of quantum mechanics have the form that they do. Bohr, as you will remember, postulated the quantization of angular momentum in an effort to explain the stability of atoms. An atom “occupies” hugely more space than its nucleus (which is tiny by comparison) or any one of its electrons (which do not appear to “occupy” any space at all).

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