University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

114

Pages:

5
Author:

Jabin F.

Some Mathematical Terms: Dot Product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = Σ aᵢbᵢ (Einstein summation, summation on repeated index is assumed) → Repeated index must be present on same quantity on two multiplied or two divided quantity, then only summation is assumed. δᵢᵢ = δ₁₁ + δ₂₂ + δ₃₃ = 1 + 1 + 1 = 3 δᵢᵢ = 3 Aᵢ AᵢAᵢ AᵢBᵢ BᵢBᵢ Ca = Σ aᵢ for i = 1, 2, 3 a₃ for i = 3 Σᵢⱼ aᵢbⱼ = a₁b₁ + a₁b₂ + a₁b₃ + a₂b₁ + a₂b₂ + a₂b₃ Σᵢⱼ aᵢbⱼ = Σᵢ aᵢ Σⱼ bⱼ = a₁b₁ + a₂b₂ Cross Product: a x b = | i j k | | a1 a2 a3 | | b1 b2 b3 | (a x b)₁ = a₂b₃ - a₃b₂ (a x b)₂ = a₃b₁ - a₁b₃ (a x b)₃ = a₁b₂ - a₂b₁ Another formula of cross product: (a x b)_k = ε_ijk aᵢbⱼ b x (a x b) = a(b.a) - b(a.b) b x (a x b) = a(b.a) - b(a.b) + (b.b)a + (a.b)bP-101 If L i are the components of angular momentum L , then the quantity ∑ i,j,k=1 3 [[L i ,L j ],L k ] equals to: Solution: Let [L i ,L j ]=L k [[L i ,L j ],L k ]=[L k ,L i ]=ϵ ijk L j L k =ϵ ijk ϵ klm L m L l =−ϵ jik ϵ jlm L m L l =−δ il δ km L m L l =−L i L i =−L 2