(a) Use the triangular function (7.35) as a variation function for the ground state of the particle in a box. ((( (x) is infinite at x = 1/2 l because of the discontinuity in (( (x) at this point. Therefore, in evaluatingwe run into difficulty in evaluating the integral of ((((. One way around this problem is to first show that

for any function obeying the boundary conditions. Then, using the expression on the right of (8.96), we can calculate the variational integral. Prove (8.96), and then calculate the percent error in the ground-state energy, using this triangular function. There is no parameter in this trial function. [If you are ambitious, try this alternative procedure: (( (x) involves the Heaviside step function (Section 7.7), and therefore ((( (x) involves the Dirac delta function. Use the properties of the delta function to evaluate and find the percent error using this triangular function.]
(b) The variation function ( of the first example in Section 8.1 has discontinuities in (( at x = 0 and x = l, so, strictly speaking, we should use one of the procedures of part (a) of this problem to evaluate Do this and show that the same value of is obtained.

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