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This condition gives descrete values of the electron momentum p = mv, which in turn leads to descrete energy levels. (b) Later de Broglie found that Bohr’s circular orbits could exactly hold integral numbers of de Broglie wavelength of the electron. As pr = n = 2πr = n nh , 2π h = nλ , p where λ is the de Broglie wavelength, which is associated with the group velocity of matter wave. 1043 In radio astronomy, hydrogen atoms are observed in which, for example, radiative transitions from n = 109 to n = 108 occur.

C) Insert the result of (b) in Poisson’s equation to obtain a nonlinear partial diﬀerential equation for φ. (Princeton) Solution: 2 p (a) For a bound electron, its energy E = 2m − eφ(x) must be lower than that of an electron on the Fermi surface. Thus p2max − eφ(x) = 0 , 2m where pmax = pf , the Fermi momentum. Hence p2f = 2meφ(x) . (b) Consider the electrons as a Fermi gas. The number of electrons ﬁlling all states with momenta 0 to pf is N= V p3f . 3π 2 3 The charge density is then ρ(x) = ep3f eN e = 2 3 = V 3π 3π 2 3 3 [2meφ(x)] 2 .

13 the dashed lines represent absorption transitions, the solid lines, emission transitions. 1045 The “plum pudding” model of the atom proposed by J. J. Thomson in the early days of atomic theory consisted of a sphere of radius a of positive charge of total value Ze. Z is an integer and e is the fundamental unit of charge. The electrons, of charge −e, were considered to be point charges embedded in the positive charge. (a) Find the force acting on an electron as a function of its distance r from the center of the sphere for the element hydrogen.