Consider a particle in a spherically symmetric potential wit

Consider a particle in a spherically symmetric potential with wavefunction: Assume the radial wave function R(r) is normalized by itself. If you measured the orbital angular momentum L^2, what values will you get, and with what probability for each? What about the z-component L_z of the orbital angular momentum? Same for the spin angular momentum and its z-component. If you measured the total angular momentum J^2 (where J = L + S), what values will you get. mid with what probability for each? Same for J_z. If you measured the position of the particle, what is the probability density for finding it at r, theta, phi? If you measured both the z-component of the spin and the distance from the origin, what is the probability density for finding the particle at radius r with spin up?

Solution

psi = R(r) ( 1/sqrt3 Y₂⁰ X₊  + sqrt(2/3) Y₁¹ X_- )

a) orbital angular momentum L² and its eigen value is l(l+1) (h/2pi)²

< L²> =  psi^* L² psi

= R(r)* (1/sqrt3 Y₂⁰ X₊  + sqrt(2/3) Y₁¹ X_- ) L²   R(r) ( 1/sqrt3 Y₂⁰ X₊ + sqrt(2/3) Y₁¹ X_- )

= 1/3 x 2 (2+1) H² + 2/3 x 1 (1+1) H²   since H = h/2pi

   = (2 + 4/3 ) H²

<L²> = 7/3 H²

probability P₊ = psi^* X₊

= R(r)* (1/sqrt3 Y₂⁰ X₊ + sqrt(2/3) Y₁¹ X_-)  x R(r) sqrt(1/3) Y₂⁰ X₊

P₊ = 1/3

  P_- = psi^* X_-

= R(r)* (1/sqrt3 Y₂⁰ X₊ + sqrt(2/3) Y₁¹ X_- ) x R(r) sqrt(2/3) Y₁¹ X_-

P_- = 2/3

b) orbital angular momentum L_z

<L_z> = mH

= R(r)* ( 1/sqrt3 Y₂⁰ X₊ + sqrt(2/3) Y₁¹ X_- ) L_z  R(r) ( 1/sqrt3 Y₂⁰ X₊ + sqrt(2/3) Y₁¹ X_-)

= 1/3 x 0 + 2/3 x 1 H

<L_z> = 2/3 H

c) spin angular momentum

< S²> = s(s+1) H²

s = s₊ + s_- = +1 + -1

s = 0

<S²> = 0