7.5: Eigenvalues of L²
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider the angular wavefunction ψ(θ,ϕ)=L+Yl,m(θ,ϕ). We know that
∮ψ∗(θ,ϕ)ψ(θ,ϕ)dΩ≥0,
because ψ∗ψ≡|ψ|2 is a positive-definite real quantity. Hence, making use of Equations ([e5.48]) and ([e8.14]), we find that
∮(L+Yl,m)∗(L+Yl,m)dΩ=∮Y∗l,m(L+)†(L+Yl,m)dΩ=∮Y∗l,mL−L+Yl,mdΩ≥0.
It follows from Equations ([e8.17]), and ([e8.29])–([e8.31]) that
∮Y∗l,m(L2−L2z−ℏLz)Yl,mdΩ=∮Y∗l,mℏ2[l(l+1)−m(m+1)]Yl,mdΩ=ℏ2[l(l+1)−m(m+1)]∮Y∗l,mYl,mdΩ=ℏ2[l(l+1)−m(m+1)]≥0.
We, thus, obtain the constraint l(l+1)≥m(m+1).
Without loss of generality, we can assume that l≥0. This is reasonable, from a physical standpoint, because l(l+1)ℏ2 is supposed to represent the magnitude squared of something, and should, therefore, only take non-negative values. If l is non-negative then the constraints ([e8.42]) and ([e8.44]) are equivalent to the following constraint:
−l≤m≤l.
We, thus, conclude that the quantum number m can only take a restricted range of integer values.
Now, if m can only take a restricted range of integer values then there must exist a lowest possible value that it can take. Let us call this special value m−, and let Yl,m− be the corresponding eigenstate. Suppose we act on this eigenstate with the lowering operator L−. According to Equation ([e8.32]), this will have the effect of converting the eigenstate into that of a state with a lower value of m. However, no such state exists. A non-existent state is represented in quantum mechanics by the null wavefunction, ψ=0. Thus, we must have L−Yl,m−=0.
L2=L+L−+L2z−ℏLz
or
l(l+1)Yl,m−=m−(m−−1)Yl,m−,
where use has been made of ([e8.29]), ([e8.30]), and ([e8.46]). It follows that
l(l+1)=m−(m−−1).
Assuming that m− is negative, the solution to the previous equation is m−=−l.
We can now formulate the rules that determine the allowed values of the quantum numbers l and m. The quantum number l takes the non-negative integer values 0,1,2,3,⋯. Once l is given, the quantum number m can take any integer value in the range −l,−l+1,⋯0,⋯,l−1,l.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)