Conservation of

Conservation of Σλ_i

Description:

For a Helbert space (wave function subspace) {φ_i} defined by the operator O whose eigenvalues are λ_i, corresponding to wavefunction  φ_i.

Simply from linear algebra, Σλ_i is actually the trace of the operator matrix O, which is of course conserved, no matter what unitary transformation of the {φ_i} is taken to form new set of basis, say {ψ_i}.

To generalize this, Σλ_iⁿ, is also conserved, because Oⁿ also defines the space.

Example:

For a 4f electron multiplet, say, Nd³⁺, ⁴I_9/2, ΣJ_z,iⁿ is conserved and of course Σμ_z,iⁿ is conserved too because they differ only by a factor of gμ_B

Just to give a few numbers,

sqrt(3Σμ_z,i²/5)=

3.6 μ_B(theoretical value)
3.4, 3.5 and 3.8 from measurement on crystal field split ⁴I_9/2 multiplet.