Publisher Summary

This chapter discusses the orbital momentum operator and spherical harmonics and related functions. Orbital angular momentum operators and spherical harmonics are related. It may be seen from the standpoint of a central force problem. The chapter presents equations that exhibit the basic connections between orbital angular momentum operators and spherical harmonics. It also lists several formulas involving spherical harmonics. The quantum numbers in the coupled representation must be related in some fashion to those in the uncoupled representation. Quantum–mechanical wave functions and their first derivatives must be everywhere continuous, single-valued, and finite. The methods developed for the coupling of two angular momentum operators may be extended to any number of operators but not without additional complications.

1.1 Orbital Angular Momentum

The orbital angular momentum operator L is defined by

=1ℏ(r×p) (1.1-1)

where r is a vector whose components ri are x, y, z (or x1, x2, x3) and

=−iℏ∇ (1.1-2)

is the linear momentum operator; the rectangular components of the gradient operator . Expanding (1.1-1),

x=1ℏ(ypz−zpy)=−i(y∂∂z−z∂∂y)         =i(sinφ∂∂θ+cotθcosφ∂∂φ), (1.1-3a)

y=1ℏ(zpx−xpz)=−i(z∂∂x−x∂∂z)         =i(−cosφ∂∂θ+cotθsinφ∂∂φ), (1.1-3b)

z=1ℏ(xpy−ypx)=−i(x∂∂y−y∂∂x)         =−i∂∂φ. (1.1-3c)

In (1.1-3) the angles θ and φ are the polar and azimuth angles, respectively.

The operators Lx, Ly, and Lz are Hermitian, i.e.,

i †=Li(i=x,y,z), (1.1-4)

and, as functions of the coordinates, Lx, Ly, and Lz are pure imaginary operators.

It will often be convenient to use spherical components of L; these are defined as

+1=−12(Lx+iLy)=−12eiφ(∂∂θ+icotθ∂∂φ),L−1=12(Lx−iLy)=−12e−iφ(∂∂θ−icotθ∂∂p),L0=Lz. (1.1-5)

The inverse relations are

x=−12(L+1−L−1),   Ly=−i2(L+1−L−1),  Lz=Lo. (1.1-6)

In contrast to the rectangular components of L, L+1 and L−1 are not Hermitian since

† +1=−L−1,L† −1=−L+1. (1.1-7)

The components of r and p satisfy certain commutation relations:

ri,pj]=iℏδij, (1.1-8a)

ri,rj]=[pi,pj]=0, (1.1-8b)

ri,p2]=2iℏpi, (1.1-8c)

pi,p2]=0 (1.1-8d)

in which ri, rj = x, y, z; pi pj = px, py, pz, and p2 = px2 + py2 + pz2. The definition of L (1.1-1) together with (1.1-8) imply that

Lx,Ly]=iLz,[Ly,Lz]=iLx,[Lz,Lx]=iLy. (1.1-9)

These may be written in any of the compact forms:

Li,Lj]=iLk(i,j.k cyclic), (1.1-10a)

×L=iL, (1.1-10b)

Li,Lj]=iεijkLk, (1.1-10c)

in which εijk is the antisymmetric unit tensor of rank 3 defined by

ijk={+1,−1,0,i,j,k in cyclic order,i, j, k not in cyclic order,two indices  alike. (1.1-11)

The three statements (1.1-10a)–(1.1-10c) are equivalent in all respects. Additional commutator relations among the components of L, r, and p are

Li,rj]=iεijkrk, (1.1-12a)

Li,pj]=iεijkpk, (1.1-12b)

L0,L±1]=±L±1,[L+1,L−1]=−L0. (1.1-13)

Another important operator is L2, also known as the total orbital angular momentum operator. It may be expressed in various equivalent forms:

2=Lx 2+Ly 2+Lz 2         =−[∂2∂θ2+cotθ∂∂θ+(1+cot2θ)∂2∂φ2]         =−[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ∂2∂φ2]         =−L+1L−1+L0 2−L−1L+1         =∑q(−1)qLqL−q(q=1,0,−1). (1.1-14)

Employing relations (1.1-13) we also have

2=−2L+1L−1+L0(L0−1)=−2L−1L+1+L0(L0+1). (1.1-15)

L2 commutes with all components of L, i.e.,

L2,Lμ]=0 (1.1-16)

where Lμ refers to either rectangular components (Lx, Ly, Lz) or spherical components (L+1, L0, L−1 of L.

1.2 Spherical Harmonics and Related Functions

The spherical harmonics are defined by

lm(θ,φ)=(−1)m+|m|2l+14π(l−|m|)!(l+|m|)!Pl |m|(cosθ)eimφ (1.2-1)

with

=0,1,2,…, (1.2-2a)

=l,l−1,…,−l, (1.2-2b)

and Pl|m|(cos θ) an associated Legendre polynomial. The phase convection in (1.2-1) is not universal; the one adopted here is known as the Condon-Shortley convention. Some of the commonly used spherical harmonics are listed in Table 1.1; among their properties are:

l−m(θ,φ)=(−1)mY lm∗(θ,φ), (1.2-3a)

lm(π−θ,π+φ)=(−1)lYlm(θ,φ). (1.2-3b)