Show that Rp = p x D and Lp = p x + D where D = d/dx, are raising and lowering operators

Chapter 12, Problem 29

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Show that Rp = p x D and Lp = p x + D where D = d/dx, are raising and lowering operators for Bessel functions, that is, show that RpJp(x) = Jp+1(x) and LpJp(x) = Jp1(x). Hint: Use equations (15.5). Note that these operators depend on p as well as x, so they are not as simple as the Hermite function raising and lowering operators (22.7) and (22.8). If you want to operate, say, on Jp+1, you must change p in R or L to p+ 1, etc. Making this adjustment, show that the equations LRJp = Jp and RLJp = Jp both give Bessels equation.

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