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

2024
Views:

261

Pages:

5
Author:

Jabin F.

General Formulae of PB with angular momentum: {xi, Lj} = εijk xk where εijk is Levi-Civita tensor of rank 3. {Li, Lj} = εijk Lk Q. Prove that {x, Lz} = -y Solution: {x, Lz} = {x, xpy - ypx} = x ∂(xpy - ypx)/∂x - (xpy - ypx) ∂x/∂x = x(y + x) - (xpy - ypx) = xy + x² - xpy + ypx = -y Evaluation PB of x ^ 2 , p ^ 2 , x ^ p ^ , p ^ x ^ Solution: {x̂², p̂²} = {x̂, [p̂, x̂]}{p̂} + {x̂, p̂}{x̂} = -iħ{x̂, p̂} = -iħ AᵢEᵢjEjkBjk = Aᵢ(δᵢⱼBⱼ) = Aᵢ²Bᵢ P-30 Problem: PB {x̂², p̂²} is equal to: Solution: {x̂², p̂²} = {x̂, [p̂, x̂]}{p̂} + {x̂, p̂}{x̂} + {p̂, [x̂, p̂]}{x̂} + {p̂, x̂}{p̂} = xᵢεᵢⱼk ∂k{p̂, x̂} + {x̂, p̂}xᵢ + pᵢεᵢⱼk ∂k{x̂, p̂} + {p̂, x̂}pᵢ = -iħxᵢ + iħxᵢ + -iħpᵢ + iħpᵢ = 0 P-31 Value of Poisson bracket {{Li, xj}, {pk, Li}} Solution: {{Li, xj}, {pk, Li}} = {εijk xk, εijk Pl} = εijk εijk {xk, Pl} = (εijk)² δkl = 6 δkl