By Robert H. Dicke, J. P. Wittke

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3-6. General potential well. ONE-DIMENSIONAL MOTION IN A POTENTIAL WELL 3-4] I I 1 -a v 51 I I al (a) Region 2 Region 2 (b) FIG. 3-7. (a) A potential well. (b) The wave function for a bound state of the well. Note the oscillatory nature of 1/1 in the classically allowed region 1 and the exponential behavior in the classically forbidden region 2. The wave function shown is a computed eigenfunction of the potential well of (a) for the energy E. is larger than E (Region 2, Fig. 3-7) , the coefficient of u is positive : for positive u the function curves upward, or away from the axis, and for negative u it curves downward, also away from the axis.

Consequently, after passing through the slit, the light is not simply a plane wave, but must be represented by a superposition of plane waves traveling in various directions but all having the same frequency as the original wave. Each of the plane waves of this super position represents photons of a particular momentum, as has been seen. In some strange manner, the slit interacts with the incident photons to change their momenta by an amount not completely predictable. spread of momenta. complementary complementarity principle.

35 PROBLEMS 2-1. (a) Compute energy levels for a hydrogen atom by assuming that the electron moves in circular orbits around the nucleus such that the circumfer ence of an orbit is an integral number of de Broglie wavelengths. (b) From the expression for En thus obtained, compute the frequency of radiation emitted when an atom goes from the (n + l)st to the nth state from J = En + l - En . h (c) Show that in the limit of large quantum numbers, this is identical with the classical frequency of an electron of energy En traveling around the nucleus.

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