216
3 Quantum Mechanics – II
∴ (S p + S n ) · (S p + S n ) = S · S
S p
2
+ S n
2
+ 2 S p · S n = S
2
= 0
1 / 2 (1/2 + 1) + 1 / 2 (1/2 + 1) + 2 S p · S n = 0
Or S p · S n = −3/4. Or σ p · σ n = −3
(ii) For triplet state S = 1
3/4 + 3/4 + 2 S p · S n = 1(1 + 1)
∴ S p · S n = 1/4
But S p = 1 / 2 σ p and S n = 1 / 2 σ n
∴ σ p · σ n = 1
3.80 From the definition of angular momentum
L = r × p, we can write
L =
i j k
x y z
p x p y p z
= i(yp z − zp y ) + j(zp x − xp z )
+ k(xp y − yp x )
= iL x + jL y + kL z
Fig. 3.26 Cartesian and polar
coordinates
L x = yp z − zp y = −i
y
∂
∂z
− z
∂
∂ y
L y = zp x − xp z = −i
z
∂
∂ x
− x
∂
∂z
(1)
L z = xp y − yp x = −i
x
∂
∂ y
− y
∂
∂ x
If θ is the polar angle, ϕ the azimuthal angle and r the radial distance,
(Fig. 3.26). Then