Question

Consider a Hilbert space whose members are functions defined on the surface of the unit sphere, with a scalar product of the form [ langle f | g angle = int d Omega f^* g, ] where ( dOmega ) is the element of solid angle. Note that the total solid angle of the sphere is ( 4pi ). We work here with the three functions ( varphi_1 = Cx/r ), ( varphi_2 = Cy/r ), ( varphi_3 = Cz/r ), with ( C ) assigned a value that makes the ( varphi_i ) normalized. (a) Find ( C ), and show that the ( varphi_i ) are also mutually orthogonal. (b) Form the ( 3 imes 3 ) matrices of the angular momentum operators [ L_x = -ileft(y frac{partial}{partial z} - z frac{partial}{partial y} ight), quad L_y = -ileft(z frac{partial}{partial x} - x frac{partial}{partial z} ight), ] [ L_z = -ileft(x frac{partial}{partial y} - y frac{partial}{partial x} ight). ]

          Consider a Hilbert space whose members are functions defined on the surface of the unit sphere, with a scalar product of the form
[
langle f | g 
angle = int d Omega f^* g,
]
where ( dOmega ) is the element of solid angle. Note that the total solid angle of the sphere is ( 4pi ). We work here with the three functions ( varphi_1 = Cx/r ), ( varphi_2 = Cy/r ), ( varphi_3 = Cz/r ), with ( C ) assigned a value that makes the ( varphi_i ) normalized.
(a) Find ( C ), and show that the ( varphi_i ) are also mutually orthogonal.
(b) Form the ( 3 	imes 3 ) matrices of the angular momentum operators
[
L_x = -ileft(y frac{partial}{partial z} - z frac{partial}{partial y}
ight), quad L_y = -ileft(z frac{partial}{partial x} - x frac{partial}{partial z}
ight),
]
[
L_z = -ileft(x frac{partial}{partial y} - y frac{partial}{partial x}
ight).
]

$Consider a Hilbert space whose members are functions defined on the surface of the unit sphere, with a scalar product of the form [ langle f | g angle = int d Omega f^* g, ] where ( dOmega ) is the element of solid angle. Note that the total solid angle of the sphere is ( 4pi ). We work here with the three functions ( varphi1 = Cx/r ), ( varphi2 = Cy/r ), ( varphi3 = Cz/r ), with ( C ) assigned a value that makes the ( varphii ) normalized. (a) Find ( C ), and show that the ( varphii ) are also mutually orthogonal. (b) Form the ( 3 imes 3 ) matrices of the angular momentum operators [ Lx = -ileft(y fracpartialpartial z - z fracpartialpartial y ight), quad Ly = -ileft(z fracpartialpartial x - x fracpartialpartial z ight), ] [ Lz = -ileft(x fracpartialpartial y - y fracpartialpartial x ight). ]$

Added by Zeynep K.

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