Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 17

The use of spinor functions as a basis for SU(2) and the relations for Jˆ0, Jˆ± given in equations (1.172)–(1.174) lead to the consideration of a functional representation of the ∣lm〉 for (ℏ≡1)

Lˆ0=Lˆz=xˆpˆy−yˆpˆx↔−ix∂∂y+iy∂∂x;(1.177)

Lˆ+=Lˆx+iLˆy=yˆpˆz−zˆpˆy+izˆpˆx−ixˆpˆz,(1.178)

∴Lˆ+↔−iy∂∂z+iz∂∂y+z∂∂x−x∂∂z;(1.179)

Lˆ−↔−iy∂∂z+iz∂∂y−z∂∂x+x∂∂z;(1.180)

where the postion representation has been used. Evidently, Lˆ0, Lˆ± in the form given by equations (1.177), (1.179) and (1.180) leave the degree of a polynomial in x, y and z unchanged. Therefore, we consider the space of homogeneous polynomials in x, y and z, i.e.

f(x,y,z)=(ax+by+cz)l,(1.181)

where a, b, and c are complex numbers.

We start with the homogeneous polynomials ϕlm=−l(r⃗), r⃗≔(x,y,z), that satisfy the so-called ‘lowest weight’ conditions:

Lˆ0ϕl,−l(r⃗)=−lϕl,−l(r⃗)(1.182)

and

Lˆ−ϕl,−l(r⃗)=0.(1.183)

Then, consider

Lˆ0(ax+by+cz)l=−ix∂∂y+iy∂∂x(ax+by+cz)l=−ixl(ax+by+cz)l−1b+iyl(ax+by+cz)l−1a=l(ax+by+cz)l−1(−ibx+iay),(1.184)

and the right-hand side fulfils equation (1.182), i.e.

Lˆ0(ax+by+cz)l=−l(ax+by+cz)l,(1.185)

provided a=1,b=−i,c=0. Thus,

ϕl,−l(r⃗)=(x−iy)l.(1.186)

Evidently,

Lˆ−ϕl,−l(r⃗)=−iy∂∂z+iz∂∂y−z∂∂x+x∂∂z(x−iy)l=izl(−i)(x−iy)l−1−zl(x−iy)l−1=0.(1.187)

We can construct the ϕlm(r⃗) using (ℏ≡1)

Lˆ+ϕlm(r⃗)=(l−m)(l+m+1)ϕl,m+1(r⃗).(1.188)

For l = 1: from

ϕ1,−1(r⃗)=x−iy,(1.189)

Lˆ+ϕ1,−1(r⃗)=−iy∂∂z+iz∂∂y+z∂∂x−x∂∂z(x−iy)=iz(−i)+z=2z≔2ϕ1,0(r⃗);(1.190)

∴ϕ1,0(r⃗)=2z.(1.191)

Then,

Lˆ+ϕ1,0(r⃗)=2−iy∂∂z+iz∂∂y+z∂∂x−x∂∂zz=2(−iy−x)=−2(x+iy)≔2ϕ1,1(r⃗);(1.192)

∴ϕ1,1(r⃗)=−(x+iy).(1.193)

For l = 2: from

ϕ2,−2(r⃗)=(x−iy)2,(1.194)

Lˆ+ϕ2,−2(r⃗)=−iy∂∂z+iz∂∂y+z∂∂x−x∂∂z(x−iy)2=iz2(x−iy)(−i)+z2(x−iy)=4z(x−iy)≔2ϕ2,−1(r⃗);(1.195)

∴ϕ2,−1(r⃗)=2z(x−iy).(1.196)

Then,

Lˆ+ϕ2,−1(r⃗)=−iy∂∂z+iz∂∂y+z∂∂x−x∂∂z2z(x−iy)=−iy2(x−iy)+iz2z(−i)+z2z−x2(x−iy)=−2(x−iy)(x+iy)+4z2=−2(x2+y2)+4z2≔6ϕ2,0(r⃗);(1.197)

∴ϕ2,0(r⃗)=23(−x2−y2+2z2).(1.198)

Similarly,

ϕ2,1(r⃗)=−2z(x+iy),(1.199)

ϕ2,2(r⃗)=(x+iy)2.(1.200)

The functions ϕlm(r⃗) are proportional to the spherical harmonics, Ylm(θ,ϕ) (see table 1.1). This follows from the relationship between Cartesian coordinates and spherical polar coordinates:

x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθ,(1.201)

whence

ϕ1,±1=∓(x±iy)=∓rsinθe±iϕ=r8π3Y1,±1(θ,ϕ).(1.202)

Similarly,

ϕ1,0(r⃗)=2rcosθ=r8π3Y1,0(θ,ϕ).(1.203)

ϕ2,±2(r⃗)=r2sin2θe±2iϕ=r232π15Y2,±2(θ,ϕ).(1.204)

ϕ2,±1(r⃗)=∓2r2cosθsinθe±iϕ=r232π15Y2,±1(θ,ϕ).(1.205)

ϕ2,0(r⃗)=r223(3cos2θ−1)=r232π15Y2,0(θ,ϕ).(1.206)

In general,

ϕl,±l(r⃗)=(∓1)l(rsinθcosϕ±irsinθsinϕ)l,(1.207)

i.e.

ϕl,±l(r⃗)=(∓1)lrlsinlθe±ilϕ.(1.208)

The spherical harmonics Yl,m=±l(θ,ϕ) are:

Yl,±l(θ,ϕ)=(2l+1)!!4π(2l)!!e±ilϕsinlθ,(1.209)

where (2l)!!≔(2l)(2l−2)(2l−4)…2or1 and

∫02πdϕ∫0πsinθdθ∣Yl,±l(θ,ϕ)∣2=1.(1.210)

Thus,

ϕl,±l(r⃗)=rl4π(2l)!!(2l+1)!!Yl,±l(θ,ϕ).(1.211)

It then follows from

ϕlm(r⃗)=(l−m)!(2l)!(l+m)!(Lˆ+)l+m(x−iy)l,(1.212)

which is obtained by repeated application of equations (1.186)–(1.188), that a general spherical harmonic is given by

Ylm(θ,ϕ)=12ll!(2l+1)(l−m)!4π(l+m)!1rl(Lˆ+)l+m(x−iy)l.(1.213)

This leads to the general expression for spherical harmonics:

Ylm(θ,ϕ)=12ll!(2l+1)(l−m)!4π(l+m)!eimϕ(−sinθ)mdd(cosθ)l+m(cos2θ−1)l.(1.214)

The spherical harmonics are related to the Legendre polynomials, P_l by:

Pl(cosθ)=4π2l+1Yl,m=0(θ,ϕ).(1.215)

Table 1.1. The spherical harmonics, Ylm(θ,ϕ), m=l,l−1,l−2,…,1,0,−1,…,−l+1,−l, for l=0,1,2, and 3. They are normalized for 0⩽ϕ⩽2π, 0⩽θ⩽π.

l	m	Ylm(θ,ϕ)
0	0	14π
1	0	34πcosθ
1	±1	∓38πe±iϕsinθ
2	0	516π(3cos2θ−1)
2	± 1	∓158πe±iϕcosθsinθ
2	±2	1532πe±2iϕsin2θ
3	0	6316π53cos3θ−cosθ
3	±1	∓2164πe±iϕ(5cos2θ−1)sinθ
3	±2	10532πe±2iϕsin2θcosθ
3	±3	∓3564πe±3iϕsin3θ