By Drinfeld V.G., Manin Yu.I.
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I (b) The effective radial potential for a hydrogen atom is e2/r - l(Z + 1) /i2/2mr2. Comparing this with V(x) we see that the l/r2 term is formally identical with the 1/x2 term with the angular momentum 1 taking the place of n. The i term of V(x) depends on n = I, while the ! (Coulomb) term in the effective potential for the hydrogen atom is independent of the orbital angular momentum 1. This is the difference between the two potentials. 1019 Consider the following one-dimensional potential wells: V(x) Q -4 b-____P Fig.
Therefore (V) = (E)/2. e. 16). V(x) 26 X Fig. 16 (a) What is the low-lying energy spectrum under the approximation that the barrier is completely impenetrable? (b) Describe qualitatively the effect on the spectrum of the finite pen+ trability of the barrier. (MIT) Solution: (a) For the low-lying energy spectrum, as the barrier is completely impenetrable, the potential is equivalent to two separate halves of a harmonic oscillator potential and the low-lying eigenfunctions must satisfy the condition $J (x) = 0 at x = 0.
This is the meaning of being “far away”. k Fig. 15 shows line 1 representing y = k and curve 2 representing y = ve [l-exp(-2kd)], wh ere ye = mVo/li2. The condition for the equation k = mVo [l - exp (-2kd)]/h2 Basic Principles and One-Dimensional Motions 41 to have a solution is that the slope of curve 2 at the origin is greater than that of line 1: dy = 2mVod/h2 > 1. z k=O Hence if Vod > &, there is one bound state. 1027 The wave function of the ground state of a harmonic oscillator of force constant k and mass m is $0 (z) = (o/r> 114 pz=/2 , a = rrw~/fi, ~0” = k/m.