Problem 2.25 (Engel)A. Show that these wave functions are orthogonal over the range x: Ø -¶ to ¶, eH-1ê2L a x2 and H2 ax2- 1L eH-1ê2L a x2.Assume a is a real constant greater than zero.SolutionStrategy.The functions are real, so we need to show that Ÿ-¶¶HeH-1ê2L a x2L HH2 ax2- 1L eH-1ê2L a x2L „ x = 0.IH−1ê2L α x2M IH2 α x2− 1L H−1ê2L α x2M−x2αH−1 + 2x2αL‡−∞∞% xIfARe@αD > 0, 0,IntegrateA−x2αH−1 + 2x2αL, 8x, −∞, ∞<, Assumptions → Re@αD ≤ 0EESimplify@%, Re@αD> 0D0The integral vanishes, so the functions are orthogonal.B. Show that these wave functions are orthogonal over the range r: 0 Ø ¶ and q: 0 Ø p and f: 0 Ø 2p,H2 - rêa0L eH-rê2 a0L and Hrêa0L eH-rê2 a0Lcos q.(Omission? It seems likely that a0 is a constant in these wave functions. Remember, the infinitesimal volume elementfor spherical polar coordinates is r2sin q dr dq df.)SolutionStrategy.The functions are real, so we need to show thatŸ02 p Ÿ0p Ÿ0¶HH2 - rê a0L eH-rê2 a0LLHHrêa0L eH-rê2 a0L cos qL r2sin q dr dq df=0.As always, we start with the innermost integral and work our way outwards...ikjj2 −ra0y{zz H−r2 a0L ikjjra0y{zz H−r2 a0L Cos@θD−ra0r Cos@θDH2 −ra0La0·0∞ikjjjjjjjj−ra0r Cos@θDI2 −ra0Ma0y{zzzzzzzz r2 Sin@θD  rIfARe@a0D > 0, −6 Sin@2 θD a03,IntegrateA−−ra0r3Cos@θD Sin@θDHr − 2a0La02,8r, 0, ∞<, Assumptions → Re@a0D ≤ 0EESimplify@%, Re@a0D> 0D−6 Sin@2 θD a03‡0πH−6 Sin@2 θDa03L Sin@θD θ0‡02 πH0L  φ0Of course, it was unnecessary to do the final integral with respect to f because the previous integral had alreadyvanished. The functions are