Quantum Mechanics II angular Momentum II : Addition of Angular Momentum Clebsch-Gordan Coefficients

Quantum Mechanics - II Angular Momentum - II : Addition of Angular Momentum - Clebsch-Gordan
Coefficients

Dipan Kumar Ghosh

UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai 400098
September 1, 2019

1. Introduction

Consider a spin-half particle. The state space of the particle is spanned by position kets
{| →x⟩} and two dimensional spin space spanned by | ↑⟩ and | ↓⟩. If the spin-orbit coupling is weak, the Hilbert space is the product of the position space and the spin space, and is spanned by
| →x, ±⟩ =| →x⟩⊗ | ↑, ↓⟩
The rotation operator is still given by exp −iJ^→ · nˆθ/k , where J^→ is given by
J^→ = L^→ ⊗ I ⊕ I ⊗ S^→
which is usually abbreviated as J^→ = L^→ + S^→. The rotation operator in the product space is given by the product of the operators for the orbital and spin parts :
U_R(nˆ, θ) = exp −iL^→ · nˆθ/k exp −iS^→ · nˆθ/k

The total wavefunction is a a product of the space part and a spin part

ψ_α(→x) = ψ(→x)χ_α



where α =↑ or ↓. The wavefunction is a two component tensor
ψ_↑(→x)
ψ_↓(→x)
We have seen that the orbital angular momentum is described by the operator L² and L_z, while the spin operators by S² and S_z. The set of operators required to describe the composite system is {L², L_z, S², S_z}. Since the state space is a product of independent states, we can write

where
| l, s, m_l, m_s⟩ =| l, m_m⟩⊗ | s, m_s⟩

L² | l, m_l⟩ = l(l + 1)k² | l, m_l⟩ L_z | l, m_l⟩ = m_lk | l, m_l⟩
S² | s, m_s⟩ = s(s + 1)k² | s, m_s⟩ S_z | s, m_s⟩ = m_sk | s, m_s⟩

Instead of considering spin and orbital angular momenta of a single particle, we could con- sider more complex system consisting of two or more particles. We could, for instance, talk about orbital momenta of two spinless particles or angular momenta of more com- plex systems with many particles with different angular momenta. However, the basic formulation remains the same as for adding two angular momenta. Let us consider two angular momenta J₁ and J₂. We define the total angular momenta of the system by