By Daniel T. Gillespie

During this self-contained and systematic improvement, the writer provides a transparent and concise account of formal quantum mechanics. by means of rigorously simplifying the idea and mostly ignoring its extra tricky functions, he conveys a significant standpoint of the quantum thought with out wasting rigour of therapy.

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A* denote the eigenfunctions of H, and expand P(Z) = c G&l (x) . n 56 Problems and Solutions on Electromagnetism As n there is at least an eigenfunction $i (x) satisfying the inequality Hence there exists at least one bound state in V(x). Method 2: Let the wave function be @(CT) = (b/~)l/~ exp (4x2/2), where b is an undetermined parameter. c+f db (V)- L)“’ b +CO J--oo x2V(x) e-b22 dx =g+; (V)-(z2V)=0, giving Substitution in the expression for (H) yields 2(xW) - ;I (V) i? = (H) = 2 (XW) - ; 1- Basic Principles and One-Dimensional Motions 57 As V(z) 5 0 for all 2, V(fm) = 0, and V is not everywhere zero, we have (V) < 0, (x2 V) < 0 and hence I?

B) Using (a) show that the expectation value of z, (x), as a function of time can be written as A cos wt + B sin wt, where A and B are constants. (c) Using (a) show explicitly that the time average of the potential energy satisfies (V) = a(E) for a general $(z, t). Note the equality ~x4~=~~d,+1+&5~-i. J(x, t) ) as fi does not depend on time explicitly, we get 7+5(x, t) = emiHtlh $(x, 0). We can expand $J(x, 0) in terms of & (CC): Icl(T 0) = c on4n (2) 7 n where % = (471 (x1 I $ (27 0)) 7 Basic Principles and One-Dimensional Motions 47 and & (x) are the eigenfunctions of H&(z) = En&(Z), w i t h E,, = n + 5 fw.

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.

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