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- Show that the spherical harmonics Y2,2(θ,φ)= ((15/32π)^1/2)*sin(2θ)*e^∓2iφ and Y3,3(θ,φ)= ((35/64π)^1/2)*sin(3θ)*e^∓3iφ are normalized.The Brachistochrone Problem: Show that if the particle is projected withan initial kinetic energy 1/2 m v02 that the brachistochrone is still a cycloidpassing through the two points with a cusp at a height z above the initialpoint given by v02 = 2gz.(a) Consider the function x2 + y? – 22 f (x, y, z) = arctan z + 1 Determine the domain and range of f. (b) Suppose we have a function z(x, y) defined implicitly near the point (1, 0, 1) via the equa- tion x³y + y°z+ z°x = 0. %3D Find (1,0). dxðy
- (AA) ²( ▲ B) ²≥ ½ (i[ÂÂ])² If [ÂÂ]=iñ, and  and represent Hermitian operators corresponding to observable properties, what is the minimum value that AA AB can have? Report your answer as a decimal number with three significant figures.Use the commutator results, [â, ô] = iħ, [x²,p] = 2iħâ, and [ÂÂ, Ĉ] = Â[Â, Ĉ] + [‚ Ĉ]B, to find the commutators given below. (a) [x, p²] (b) [x³, p] (c) [x², p²](Mathematical method for physics)