Anti Commutator | Lecture Note - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

430

Pages:

5
Author:

Jabin F.

Anti Commutator {sigma_x, sigma_y} = sigma_x sigma_y + sigma_y sigma_x = 1 / 2i [(2i sigma_x) sigma_y + sigma_y (2i sigma_x)] = 1 / 2i [[sigma_y sigma_z] sigma_y + sigma_y [sigma_y, sigma_z]] = i / 2i (sigma_y sigma_z sigma_y - sigma_z + sigma_z - sigma_y sigma_z sigma_y) = 0 => {sigma_x, sigma_y} = sigma_x sigma_y + sigma_y sigma_x = 0 (1) (307) => {sigma_y, sigma_z} = sigma_y sigma_z + sigma_z sigma_y = 0 => {sigma_z, sigma_x} = sigma_z sigma_x + sigma_x sigma_z = 0 :: [sigma_x, sigma_y] = 2i' sigma_z sigma_x sigma_y - sigma_y sigma_x = 2i sigma_z sigma_x sigma_y + sigma_y sigma_x = 0 => sigma_x sigma_y = (i sigma_z) Similarly sigma_y sigma_z = (i sigma_x) sigma_z sigma_x = (i sigma_y) Ques.16 [Lx, Ly, Lz] = ? page-52 = Lx [Ly, Lz] + [Lx, Lz] Ly = Lx (iħLx) + (-iħLy) Ly = -iħ(Lx^2 - Ly^2) (option-b) Ques.22 [Ln, Py] = iħPz (option-d) Rage 53 Ques.14 [Ln, Ln LyLz] = ? page-57 = Ln[Ln, LyLz] + [Ln, Lu] LyLz = LnLy [Ln, Lz] + Ln [Lx, Ly] Lz = LnLy (-iħLy) + Lu (iħLz) Lz = -iħ(Lx Ly^2 + Ly Lz^2) = iħ(Lz^2 - Ly^2) (option-a) = = + i = 2ħ<1, 1> + 2ħ<1, 1> = 2ħ(<1, 1> + <1, 1>) = 4ħ<1, 1> (option-c) Ques.25 <1,0| (Lx + iLy) |1, -1> = <1,0| L+ |1, -1> = <1,0| √2ħ |1, 0> = √2ħ (option-c) Ques.11 Y2,1 = √(1/15) (y^2 - x^2 - 2i xy) L^2 Y2,1 = (l(l + 1))ħ^2 Y2,1 L^2 Y2,1 = (2(2 + 1))ħ^2 Y2,1 = 6ħ^2 Y2,1 (1) Ly Y2,1 = (ħ/2i) (∂/∂x - i ∂/∂y) (√(1/15) (y^2 - x^2 - 2i xy)) = ħ/2i √(1/15) (2y - 2x + 2i y) = ħ √(1/15) (2y) = 2ħ √(1/15) y = 2ħ √(3/15) √(1/5) y = 2ħ √(1/5) Y1,1 (2) L_z Y_l,m = m ħ Y_l,m m is not same in given fh, so not eigenfn. L_z Y_1,1 = 1/sqrt(2) (L_z Y_2,2 - L_z Y_2,-2) = ħ/sqrt(2) (Y_2,2 + Y_2,-2) ≠ 2ħ/sqrt(2) Y_1,1 (ii) from (i) and (ii) -> (option-b)

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