EP 228: Quantum Mechanics Lec 30: Addition of angular momentum - - PowerPoint PPT Presentation

EP 228: Quantum Mechanics Lec 30: Addition of angular momentum

Spin ½ particle

• Direct product space spanned by position

ket |x> and two-dimensional spin space given by |ms>

• Rotation operator in such a space is UR(θ)
• Here J is sum of orbital and spin

Direct product space

• Ang. Mom:
• Note: [J2 , L2 ] = [J2 , S2 ] =[J2, Jz] = 0
• Two equivalent basis for the direct product

space:

Direct product state of two spin ½ particles

• Ang. Mom:
• Note: [J2 , S1

2 ] = [J2 , S2 2 ] =[J2, Jz] = 0

• Two equivalent basis for the direct product

space:

Angular momentum basis

• System of two particles with angular

momentum J1 and J2 where

• uncoupled basis states are
• Equivalent basis : coupled basis states

involving total angular momentum

Change of basis

• The dimensionality of both basis are equal :

(2j1 +1) (2j2+1)

• The matrix relating them called Clebsch-Gordan matrix
• The matrix elements are called Clebsch-Gordan (CG)

coefficients

• CG coeffts is non-zero when m= m1 + m2

CG coeffts

• Maximum value for mmax is j1 + j2
• For this max m, jmax = j1 + j2
• We put CG coefft for the such a state as 1
• Consider
• We can obtain this in two ways(degeneracy=2)
• The above m value suggest two possible values

for j : (j1 + j2 ) , (j1 + j2 -1). Proceeding this way, we can determine m values such that mmin = -j1 - j2