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 |m s > • Rotation operator in such a space is U R ( θ ) • Here J is sum of orbital and spin

Direct product space • Ang. Mom: • Note: [J 2 , L 2 ] = [J 2 , S 2 ] =[J 2 , J z ] = 0 • Two equivalent basis for the direct product space:

Direct product state of two spin ½ particles • Ang. Mom: • Note: [J 2 , S 1 2 ] = [J 2 , S 2 2 ] =[J 2 , J z ] = 0 • Two equivalent basis for the direct product space:

Angular momentum basis • System of two particles with angular momentum J 1 and J 2 where • uncoupled basis states are • Equivalent basis : coupled basis states involving total angular momentum

Change of basis • The dimensionality of both basis are equal : (2j 1 +1) (2j 2 +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= m 1 + m 2

CG coeffts • Maximum value for m max is j 1 + j 2 • For this max m , j max = j 1 + j 2 • 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 : (j 1 + j 2 ) , (j 1 + j 2 -1). Proceeding this way, we can determine m values such that m min = -j 1 - j 2