Uncertainty about the azimuthal angle for spherical harmonics

We know that the components of the angular momentum operator (Lx, Ly, Lz) do not commute each other.

Thus one cannot measure these components simultaneously

Hence the component of L can be measured simultaneously if they are measured with L². Since L_z is a function of azimuthal angle only (Lz = -iℏ∂/∂𝜙), l L l and Lz can be measured exactly. However Lx and Ly remain unknown. Thus we can say that

In the case of spherical harmonics, we know that

We can say that equation vi lies in XY plane with complete uncertainty about the azimuthal angle 𝜙 as shown in figure

For the case, m = L, Ym​(θ,ϕ), Lz has the maximum value ℏL, then the component of sq root Lx² + Ly² doesn't vanish but has the corresponding minimum value ℏL. The uncertainty principle prevents Lz to attain the full value ℏL(L+1) in the z direction. Thus we can write here

(ΔLx)²_min (ΔLy)²_min ⩾ - 1/4 < [ Lx, Ly ] >

= 1/4 ℏ² Lz²

= 1/4 ℏ^4 L²

This expression seems to establish a lower bound on the product of the variances of angular momentum components, given certain conditions on the commutator and expectation values of angular momentum operators. It's often used in quantum mechanics to describe the uncertainty principle related to angular momentum measurements.

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