Eigen Value of Z component | Santa Fe College - Edubirdie

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

2024
Views:

78

Pages:

5
Author:

Jabin F.

Eigen function of z-component. The eigen value of sigma_z = ±1. The eigen value of S_z = ±ħ/2 = λ. sigma_z |ψ_z(+1)> = +1 |ψ_z(+1)> ( (1 0) (a) ) = +1 ( (a) ) ( (0 -1) (b) ) = +1 ( (b) ) a = a -b = b => 2b = 0 b = 0 |ψ_z(+1)> = ( (1) ) = |1/2, +1/2> = |↑> ( (0) ) |ψ_z(-1)> = ( (0) ) = |1/2, -1/2> = |↓> ( (1) ) => |1/2, +1/2> = |↑> = ( (1) ) = ψ_1/2 = spin up state ( (0) ) |1/2, -1/2> = |↓> = ( (0) ) = ψ_-1/2 = spin down state ( (1) ) These are eigen function of S_z, sigma_z, S^2 & sigma^2 Ques.7 page-51 Φ₁ = (1/√2) * (|↑↓> + |↓↑>) = c₁|↑↑> + c₂|↓↓> P(+) = |c₁|^2 = <1|Φ> = (1/2)^2 = 1/4 = 0.25 Ques.9 page-52 S_z = ħ/2, |ψ> = |↑↓> = ( (1) ) = |1/2> ( (0) ) = <ψ|S_z|ψ> = ( (1 0) / (0 0) ) ( (1) ) = 0 ( (0 1) / (0 0) ) ( (0) ) = <ψ|(ħ^2/4)σ_z^2|ψ> = (ħ^2/4) <1|1> = ħ^2/4 = <ψ|S_y|ψ> = ( (1 0) / (0 -i) ) ( (i 0) / (0 1) ) ( (1) ) = 0 ( (0 i) / (i 0) ) ( (0 i) ) ( (0) ) = <ψ|S_y^2|ψ> = <1|(ħ^2/4)σ_y^2|1> = ħ^2/4 ħ^2/4, ħ^2/4, ħ^2/4 (option-c) Ques.11 page-52 A = S_nS_yS_z = (ħ/2)^3 σ_nσ_yσ_z = ħ^3/8 iσ_z = <ψ|S_nS_yS_z|ψ> = <1|ψ> * ħ^3/8 iσ_z Ques.22 page-52 = (iħ^3/8) * <ψ|1|ψ> / <ψ|ψ> = (iħ^3/8) (option-a) Ques.27 page-53 A = λσβ = λβσn λ’ = ±λβ η = (i + j) / √2 (option-b) Ques.18 page-53 |ψ> = 0.8|↑> + 0.6|↓> = [(0.8), (0.6)], <ψ|ψ> = 1 A = 10σz + 5σn = 10[(i 0), (0 -i)] + 5[(0 i), (i 0)] = [(10 5), (5 -10)] = <ψ|A|ψ> = [(0.8 0.6)] [(10 5), (5 -10)] [(0.8), (0.6)] = [(0.8 0.6)] [(18), (-2)] = 7.6 Ans |ψ> = 0.8|↑> + 0.6|↓> <σ_z> = (0.8)^2 * (+1) + (0.6)^2 * (-1) = 0.64 - 0.36 = 0.28 <σ_n> = <ψ|σ_n|ψ> = (0.8<↑| + 0.6<↓|) * σ_n * (0.8|↑> + 0.6|↓>) = (0.8<↑| + 0.6<↓|) * (0.8|↑> + 0.6|↓>) = 0.8 * 0.8<↑|↑> + 0.6 * 0.2<↓|↑> = 0.48 + 0.12 = 0.6 = 10<σ_z> + 5<σ_n> = 10(0.28) + 5(0.6) = 7.6 Ques.1 page-55 eigen value ±ħ/2 (option-d) Ques.1 page-55 H = μσ * B = μ|B|σ * n̂ n̂ = B/|B| ω = ω_B/ħ e^(iHt/ħ) = e^(iωBt/ħ * σ * n̂) Compare with eiston e^(iHt/ħ) = I cos wt + i * σ * n̂ * sin wt = I cos wt + i * σ * (B / |B|) * sin wt (option-b) Ques. 4 page-55 η = (1 / √π) * ((i + i) / 3) for Sz' direct P_1(t) = 2/11 P_2(t) = |(3 / √π)|^2 = 9/11 (option-b)

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