Quantum mechanics physics problem

Question

What is the magnitude of the angular momentum for an l =1 electron? What is the magnitude of the electrons spin angular momentum?

Answer 1

In quantum mechanics there is angular momentum, and there is angular momentum.

Operator of angular momentum is vector, its components are

Lψ = (Lx, Ly, Lz) ψ = (∂/∂φ_x, ∂/∂φ_y, ∂/∂φ_z) ψ,

where Lx = ∂/∂φ_x is rotation arond x-axis.

It so happens that neither pair of Li and Lj commute, so a particle can NEVER have definite angular momentum L.

However operator L²

L² = ∂²/∂φ_x² + ∂²/∂φ_y² + ∂²/∂φ_z²

is scalar, it commutes with any component Li and has eigen values

L(L+1)

Answer:

magnitude of the angular momentum for an l =1 electron is

h√( 1 * (1+1)) = √2 h

magnitude of the electrons spin 1/2 angular momentum is

h√( 1/2 * (1/2+1)) =  Ã¢ÂˆÂš3/2 h