Coordinate changes in Hilbert Space

Coordinate changes in Hilbert Space PhL 11.13.10 1. Here is our first example. We can represent the same point by x,y,z or r,θ,φ. In ket notation, this must mean that | x,y,z> = α |r,θ,φ> where there might be some α which relates to the respective normalizations. The usual normalizations however are these <x',y',z'|x,y,z> = δ(x'-x) δ(y'-y) δ(z'-z) = δ3(r'-r) <r',θ,φ'|r,θ,φ> = δ(r'-r)/r2δ(cosθ-cosθ')δ(φ-φ') = δ3(r'-r) = δ(r'-r)δ(θ-θ')δ(φ-φ')/[ r2sinθ] where r2sinθ is the Jacobian thing we sometimes call v v = = det(Tab) = det( ) = J(x,x') = the Jacobian For an orthogonal system, det(g') = Q12Q22Q32 so v = Q1Q2Q3. For sphericals we have 1,2,3 = r,θ,φ Q1 = Qr = 1 Q2 = Qθ = r Q3 = Qφ = rsinθ so we get v = r2sinθ. So, given these normalizations, we can just say | x,y,z> = |r,θ,φ> where it is understood that on the left, x,y,z are given by x = rsinθsinφ y = rsinθcosφ z = rcosθ 2. Consider: <r1,θ1,φ1| x,y,z> = <r1,θ1,φ1| r,θ,φ> = δ(r1-r)δ(cosθ1-cosθ)δ(φ1- φ)/r2 But we know that r = cosθ = z/ tanφ = x/y Thus we can say <r1,θ1,φ1| x,y,z> = δ(r1-)δ(cosθ1- z/)δ(φ1- tan-2(x/y)) / This is a matrix element you don't see very often. 3. Here is our second example. We can represent the same point in spherical coordinates r,θ,φ or in conical coordinates r,μ,ν. We write | r,μ,ν > = α |r,θ,φ> For this system we find in our conical notes that 1,2,3 = r,θ,φ Q1 = Qr = 1 Q2 = Qθ = r Q3 = Qφ = rsinθ hrr2 = 1 Q1 = 1 r, η, ζ = 1,2,3 hηη2= hζζ2= [μ(η)2-ν(ζ)2] r2 Q2 = Q3 = r Note: This document got aborted, but I will park this piece in sphericals.