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

2024
Views:

30

Pages:

6
Author:

Jabin F.

Properties of Pauli's spin operator: ⇒ Pauli's spin operators are Hermitian. σ_j^† = σ_j j = x, y, z Eigen values of Pauli's spin operators are real. = λ±1 Tr(σ_j) = 0 Det(σ_j) = -1 (Product of eigen value) σ_i * σ_j + σ_j * σ_i = 2 * δijk i, j, k = x, y, z [[σ²,σ_η]]=[[σ_x² + σ_y² + σ_z²,σ_η]]=0 ⇒ [[σ',σ_η]]=[0 (i=x,y,z)] σ_xσ_yσ_z = iσ_zσ_z = iσ_z² = i^ <1|1> = <ψ|ψ> = 1 ⇒ <1/1> = (1 0) (0 1) = 0 = <1|1> ⇒ σ_x|1> = (. (0 1) (1 0)) (1 0) = (0 1) = |i> ⇒ <1|i> = (1 0) (0 1) = (0) = 0 σ_y|1> = (. (0 -i) (i 0)) (1 0) = (0 i) = i|ψ> = ㅇ |i> = (. (0 -i) (i 0)) (0 i) = (-i 0) = -i|1> ← (σ,n)² = I . vector along a particular dir!! n = (Sx i + Sy j + Sz k) . (nx i + ny k + nz k) = Sx nx + Sy ny + Sz nz = ħ/2 (. (0 i) (i 0)) cos θ cos φ + ħ/2 sin θ sin φ (. (0 -i) (i 0)) ħ/2 cos θ (. (1 0) (0 -1)) For spin-1/2 Particle S_n = ±ħ/2 {S_n = ħ/2(matrix(cosθ sinθe^(-iφ), sinθe^(iφ) -cosθ))} S · n̂ = ħ/2 σ · n̂ σ · n̂ = (matrix(cosθ sinθe^(iφ), sinθe^(iφ) -cosθ)) Eigen value (σ · n̂) ⇒ |σ · n̂ - λI| = 0 -(cosθ - λ)(cosθ + λ) - sin²θ = 0 -cos²θ + λ² - sin²θ = 0 λ = ±1 λ² = 1 Eigen fh n̂|ψ(t)> = +i|ψ(t)> (matrix(cosθ sinθe^(iφ), sinθe^(iφ) -cosθ))(matrix(a, b)) = (matrix(a, b)) a cos θ + b sin θ e^(iφ) b sin θ e^(-iφ) = a(1 - cos θ) x = sin(θ/2) cos(θ/2) e^(-iφ) = a(1 - 1 + 2 sin²(θ/2)) b = a e^(iφ) tan(θ/2) b = for normalized |a|² + |b|² = 1 |a|² + |a|² tan²(θ/2) = 1 |a|²(1 + tan²(θ/2)) = 1 sin²θ/2 → |a|² = cos²(θ/2) a = cos(θ/2) b = sin(θ/2) e^(iφ) cos θ/2 sin θ e^(iφ) then Similarly |ψ(+)> = (matrix(cos(θ/2), sin(θ/2) e^(iφ))) |ψ(-)> = (matrix(sin(θ/2), -cos(θ/2) e^(iφ))) These are eigen kets of σ · n̂ For 2-Compo. n = 2 θ = 0° |ψ₂(+)> = (1), |ψ₂(-)> = (-i) For y-Comp. θ = 90° φ = 90° |ψᵧ(+)> = (matrix(1/sqrt(2), i/sqrt(2))) = 1/sqrt(2)(matrix(i, i)) |ψᵧ(-)> = (matrix(1/sqrt(2), -i/sqrt(2))) = 1/sqrt(2)(matrix(i, -i)) For X-Compo s = sx, = sy, = sz θ = 90°, φ = 0 |ψₓ(+)> = (matrix(1/sqrt(2), 1/sqrt(2))) = 1/sqrt(2)(matrix(1, 1)) The equation involves the ket vector |ψ>, which is equated to -1 and followed by an unclear symbol "(c)=" and then the square root of 1.