By Kulish P.P. (ed.)

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Wave equation for a particle in a potential energy field The interactions of a non-relativistic particle can usually be described in terms of a potential energy field. For example, an electron in a hydrogen atom can be thought of as moving in the potential energy field V (r) Àe2 a4pE0 r of a nucleus. In classical mechanics, this field implies that an electron at a distance r from the nucleus experiences an attractive force of magnitude e2 a4pE0 r2 . In 30 The SchroÈdinger equation Chap. 2 quantum mechanics, it implies that the wave equation for the electron is not the simple free-particle wave equation given by Eq.

An electron in a circular orbit about a proton can be described by classical mechanics if its angular momentum L is very much greater than "h. e. if r bb a0 4pE0 " h2 X e2 me 6. Assume that an electron is located somewhere within a region of atomic size. Estimate the minimum uncertainty in its momentum. By assuming that this uncertainty is comparable with its average momentum, estimate the average kinetic energy of the electron. 7. Assume that a charmed quark of mass 1X5 GeVac2 is confined to a volume with linear dimension of the order of 1 fm.

Moreover, these undulations move, 22 The SchroÈdinger equation Chap. 2 like a Mexican wave, in the direction of increasing x with velocity 3ak; for example, the maximum of C(x, t) corresponding to kx À 3t 0 occurs at the position x 3tak, and the minimum corresponding to kx À 3t p occurs at the position x la2 3tak; in both cases the position moves with velocity 3ak. The function sin (kx À 3t), like cos (kx À 3t), also represents a sinusoidal travelling wave with wave number k and angular frequency 3.