It is possible to take the finite well wave functions further than (5-21) without approxim...

It is possible to take the finite well wave functions further than (5-21) without approximation, eliminating all but one normalization constant C. First, use the continuity/smoothness conditions to eliminate A, B, and G in favor of C in (5-21). Then make the change of variables z = x - L/2 and use the trigonometric relations

sin(a + b) = sin(a) cos(b) + cos(a) sin(b) and

cos(a+b) = cos(a) cos(b) - sin(a) sin(b) on the

functions in region I, -L/2 <z< +L/2, The change of variables shifts the problem so that it is symmetric about Z = 0, which requires that the probability density be symmetric and thus that ψ(z) be either an odd or even function of z. By comparing the region II and region III functions, argue that this in turn demands that (α/k) sin (kL) + cos (kL) must be either +1 (even) or — 1 (odd). Next, show that these conditions can be expressed, respectively, as α/k = tan(kL/2) and α/k = -cot(kL/2). Finally, plug these separately back into the region I solutions and show that

Note that C is now a standard multiplicative normalization constant. Setting the integral of |ψ(z)|2 over all space to 1 would give it in terms of k and a, but because we can't solve (5-22) exactly for k (or E), neither can we obtain an exact value for C