216 3 Quantum Mechanics – II
∴(Sp+Sn)·(Sp+Sn)=S·S
Sp^2 +Sn^2 +2Sp·Sn=S^2 = 0(^1) / 2 (1/ 2 +1)+ (^1) / 2 (1/ 2 +1)+2Sp·Sn= 0
Or Sp·Sn=− 3 /4. Orσp·σn=− 3
(ii) For triplet state S= 1
3 / 4 + 3 / 4 +2Sp·Sn=1(1+1)
∴Sp·Sn= 1 / 4
But Sp=^1 / 2 σpand Sn=^1 / 2 σn
∴σp·σn= 1
3.80 From the definition of angular momentum
L=r×p, we can write
L=
∣
∣∣
∣
∣
∣
ijk
xyz
pxpypz∣
∣
∣
∣
∣∣
=i(ypz−zpy)+j(zpx−xpz)
+k(xpy−ypx)
=iLx+jLy+kLzFig. 3.26Cartesian and polar
coordinates
Lx=ypz−zpy=−i(
y∂
∂z−z∂
∂y)
Ly=zpx−xpz=−i(
z∂
∂x−x∂
∂z)
(1)
Lz=xpy−ypx=−i(
x∂
∂y−y∂
∂x)
Ifθ is the polar angle,φthe azimuthal angle andr the radial distance,
(Fig. 3.26). Then