7.1: Orbital Angular Momentum
( \newcommand{\kernel}{\mathrm{null}\,}\)
From classical physics we know that the orbital angular momentum of a particle is given by the cross product of its position and momentum
L=r×p or Li=ϵijkrjpk,
where we used Einstein’s summation convention for the indices. In quantum mechanics, we can find the operator for orbital angular momentum by promoting the position and momentum observables to operators. The resulting orbital angular momentum operator turns out to be rather complicated, due to a combination of the cross product and the fact that position and momentum do not commute. As a result, the components of orbital momentum do not commute with each other. When we use [rj,pk]=iℏδjk, the commutation relation for the components of L becomes
[Li,Lj]=iℏϵijkLk.
A set of relations like this is called an algebra, and the algebra here is called closed since we can take the commutator of any two elements Li and Lj, and express it in terms of another element Lk. Another (simpler) closed algebra is [x,px]=iℏI and [x,I]=[px,I]=0.
Since the components of angular momentum do not commute, we cannot find simultaneous eigenstates for Lx,Ly, and Lz. We will choose one of them, traditionally denoted by Lz, and construct its eigenstates. It turns out that there is another operator, a function of all Lis, that commutes with any component Lj, namely L2=L2x+L2y+L2z. This operator is unique, in that there is no other operator that differs from L2 in a nontrivial way and still commutes with all Lis. We can now construct simultaneous eigenvectors for Lz and L2.
Since we are looking for simultaneous eigenvectors for the square of the angular momentum and the z-component, we expect that the eigenvectors will be determined by two quantum numbers, l, and m. First, and without any prior knowledge, we can formally write down the eigenvalue equation for Lz as
Lz|l,m⟩=mℏ|l,m⟩,
where m is some real number, and we included ℏ to fit the dimensions of angular momentum. We will now proceed with the derivation of the eigenvalue equation for L2, and determine the possible values for l and m.
From the definition of L2, we have L2−L2z=L2x+L2y, and
⟨l,m|L2−L2z|l,m⟩=⟨l,m|L2x+L2y|l,m⟩≥0
The spectrum of Lz is therefore bounded by
l≤m≤l
for some value of l. We derive the eigenvalues of L2 given these restrictions. First, we define the ladder operators
L±=Lx±iLy with L−=L†+.
The commutation relations with Lz and L2 are
[Lz,L±]=±ℏL±,[L+,L−]=2ℏLz,[L±,L2]=0.
Next, we calculate Lz(L+|l,m⟩):
Lz(L+|l,m⟩)=(L+Lz+[Lz,L+])|l,m⟩=mℏL+|l,m⟩+ℏL+|l,m⟩=(m+1)ℏL+|l,m⟩.
Therefore L+|l,m⟩∝|l,m+1⟩. By similar reasoning we find that L−|l,m⟩∝|l,m−1⟩. Since we already determined that −l≤m≤l, we must also require that
L+|l,l⟩=0 and L−|l,−l⟩=0.
Counting the states between −l and +l in steps of one, we find that there are 2l+1 different eigenstates for Lz. Since 2l+1 is a positive integer, l must be a half-integer (l=0,12,1,32,2,…). Later we will restrict this further to l=0,1,2,…
The next step towards finding the eigenvalues of L2 is to calculate the following identity:
L−L+=(Lx−iLy)(Lx+iLy)=L2x+L2y+i[Lx,Ly]=L2−L2z−ℏLz
We can then evaluate
L−L+|l,l⟩=0⇒(L2−L2z−ℏLz)|l,l⟩=L2|l,l⟩−(l2+l)ℏ2|l,l⟩=0
It is left as an exercise (see exercise 1b) to show that
L2|l,m⟩=l(l+1)ℏ2|l,m⟩.
We now have derived the eigenvalues for Lz and L2.
One aspect of our algebraic treatment of angular momentum we still have to determine is the matrix elements of the ladder operators. We again use the relation between L±, and Lz and L2:
⟨l,m|L−L+|l,m⟩=l∑j=−l⟨l,m|L−|l,j⟩⟨l,j|L+|l,m⟩.
Both sides can be rewritten as
⟨l,m|L2−L2z−ℏLz|l,m⟩=⟨l,m|L−|l,m+1⟩⟨l,m+1|L+|l,m⟩,
where on the right-hand-side we used that only the m+1-term survives. This leads to
[l(l+1)−m(m+1)]ℏ2=|⟨l,m+1|L+|l,m⟩|2.
The ladder operators then act as
L+|l,m⟩=ℏ√l(l+1)−m(m+1)|l,m+1⟩,
and
L−|l,m⟩=ℏ√l(l+1)−m(m−1)|l,m−1⟩.
We have seen that the angular momentum L is quantized, and that this gives rise to a discrete state space parameterized by the quantum numbers l and m. However, we still have to restrict the values of l further, as mentioned above. We cannot do this using only the algebraic approach (i.e., using the commutation relations for Li), and we have to consider the spatial properties of angular momentum. To this end, we write Li as
Li=−iℏϵijk(xj∂∂xk),
which follows directly from the promotion of r and p in Eq. (7.1) to quantum mechanical operators. In spherical coordinates,
r=√x2+y2+z2,ϕ=arctan(yx),θ=arctan(√x2+y2z),
the angular momentum operators can be written as
Lx=−iℏ(−sinϕ∂∂θ−cotθcosϕ∂∂ϕ)Ly=−iℏ(cosϕ∂∂θ−cotθsinϕ∂∂ϕ)Lz=−iℏ∂∂ϕ,L2=−ℏ2[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ∂2∂ϕ2].
The eigenvalue equation for Lz then becomes
Lzψ(r,θ,ϕ)=−iℏ∂∂ϕψ(r,θ,ϕ)=mℏψ(r,θ,ϕ)
We can solve this differential equation to find that
ψ(r,θ,ϕ)=ζ(r,θ)eimϕ.
A spatial rotation over 2π must return the wave function to its original value, because ψ(r,θ,ϕ) must have a unique value at each point in space. This leads to ψ(r,θ,ϕ+2π)=ψ(r,θ,ϕ) and
eim(ϕ+2π)=eimϕ, or e2πim=1
This means that m is an integer, which in turn means that l must be an integer also.