Total Angular Momentum

Total angular momentum:- J = L + S for L -> |l, m_l> S -> |s, m_s> J -> |j, m_j> J^2 = (L + S)^2 J^2 = L^2 + S^2 + 2L * S (L and S commute to each other) Each Comp of spin angular momentum Commutes with each Comp. of Total angular momentum:- J = L + S for L -> |l, m_l> S -> |s, m_s> J -> |j, m_j> J^2 = (L + S)^2 J^2 = L^2 + S^2 + 2L * S (L and S commute to each other) Each Comp of spin angular momentum Commutes with each Comp. of position, linear momentum and orbital angular momentum. ⇒ (\vec{L}\cdot s=\cdot\frac{J^{2}-L^{2}-s^{2}}{d}) [=\frac{\hbar^{2}}{2}[J(J+1)-I(I+1)-\beta(S+1)]] spin is ⇒ ([s_{n},x]=0=[s_{n},Y]=[s_{n},z]=0) [⇒[s_{x},L_{y}]=[s_{n},L_{z}]=0] (\vec{L}\cdot\vec{s}=L_{x}S_{x}+L_{y}s_{y}+L_{z}s_{z}) , [\mathbb{L}.\vec{s}+\vec{s}\cdot\vec{L}=2\vec{L}\cdot\vec{s}^{\prime}] [\vec{s}\cdot\vec{l}=S_{\eta}L_{\pi}+S_{y}L_{y}+S_{z}L_{z}] Prob: Find the following commutator Brackets. ① ([\vec{L},\vec{s},L_{z}]) ↓ [24 ↓ lovel ② ([\vec{L}\cdot\vec{s},Sz]) [[\vec{L}\cdot\vec{S},S_{Y}]] Term ③ ([\vec{L}\cdot\vec{s},J_{z}]) @ ([\vec{I}\cdot\vec{s},J_{y}]) \[[\vec{L}\cdot\vec{S},L^{2}]\] \[[\vec{L}\cdot\vec{S},L_{x}]\] \[[\vec{T}\cdot\vec{s},s^{2}]\] \[[\vec{L}.\vec{S},S_{n}]\] ⑥ \([\vec{L}\cdot\vec{S},J_{n}]\) ⑫ \([\vec{L}\cdot\vec{s},J^{2}]\) angular momentum quantumno. → Orbital quantum no. s spin \(\exists=|l-s|---|1+\beta|\) with ditt. of cnity. mj = -J to tj with diffu of unids 293/2 [L.S,Lz] = [LxSx + LySy + LzSz, Lz] = [LxSx, Lz] + [LySy, Lz] + [LzSz, Lz] = Lx[Sx, Lz] + [Lx, Lz]Sx + Ly[Sy, Lz] + [Ly, Lz]Sy + Lz[Sz, Lz] + [Lz, Lz]Sz = 0 [L.S,Lz] = -iħLySx + iħLxSy = iħ(LxSy - LySx) [L.S, Sz] = [LxSx + LySy + LzSz, Sz] = [LxSx, Sz] + [LySy, Sz] + [LzSz, Sz] = Lx[Sx, Sz] + [Lx, Sz]Sx + Ly[Sy, Sz] + [Ly, Sz]Sy + Lz[Sz, Sz] + [Lz, Sz]Sz = 0 = -iħLxSy + iħLySx = iħ(LySx - LxSy) => [L.S, Jz] = [L.S, Lz + Sz] = [L.S, Lz] + [L.S, Sz] = (-iħLySx + iħLxSy) + (iħLySx - iħLxSy) = 0 [L.S, L^2] = [L.S, S^2] = [L.S, J^2] = 0 => R = e^(iθ.n) Ques.19Page-53 (option-a) => Sz Ques.29Page-54 ↑ Z ↑ S,G (b) = |Ψ|² = C₁|Ψ₁> + C₂|Ψ₂>|^2 = C₁²|Ψ₁>² + C₂²|Ψ₂>² + C₁C₂<Ψ₁|Ψ₂> + C₂C₁<Ψ₂|Ψ₁> = C₁² + C₂² + 2C₁C₂Cosθ = C² (Cos²θ/2 + Sin²θ/2) + 2C²Sinθ/2Cosθ = C² (Cosθ + 1) P parallel ​ =∣∣ 2 = =∣∫ 0 π/2 ​ sinθe iϕ <11∣(θ)∣dθ∣ 2 =∣cos 2 θ ​ ∣ 2 =∣cos15°∣ 2 = 4 3 ​ =0.75 Ques.16Page-57 Ques.18Page-57 After Rotation ∣Ψ(θ)>= ( cos 2 θ ​ sin 2 θ ​ e −iϕ ​ ) θ=30° =( 2 3 ​ ​ 2 1 ​ ​ ) Ratio= 1 3 ​ ​ = 1 4