Matrix Lecture Note - Edubirdie

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

2024
Views:

181

Pages:

6
Author:

Jabin F.

Matrix | s, ms> = ħ²(s+1) δss' δmm' = ħ ms δss' δmm' = √((s-m)(s+m+1)) ħ δss' δm'm+1 = √((s-m')(s+m'+1)) ħ δss' δmm'-1 Sms SmsH < s, ms > | s-1, ms >= sqrt((s+ms)(s-ms+1)) + (s+ms)/(s-ms) S = 1/2, m3 = 1/2, 支 S = 320 24 4 ms + 1/2 - 1/2 4 T 두 두 8 S' = 1/2, m3' = 1/2, -1/2 S'^2 = [3/2 0 ; 0 3/2] S = 3/2 + 1/4 S' 1/2 1/2 m3' 1/2 -1/2 ms + 1/2 -1/2 ..... S+ = [0 x] = [0 h] [0 0] [0 0] S- = S+^T = [h 0] [0 0] Sx = (S+ + S-) / 2 = [h/2 h/2] [h/2 0 ] => h/2 * ox = [h/2 h/2] => ox = [1 1] [h/2 0 ] [1 0] Sy = (S+ - S-) / 2i = [0 -ih/2] = -ih/2 * [0 -i] [ih/2 0 ] [i 0] h/2 * oy = [h/2 -ih/2] => oy = [1 -i] [ih/2 0 ] [i 0] Prob. The state of a particle is given by |ψ> = (1/sqrt(2))|1,+1> - (i/2)|1,0> + (i/sqrt(2))|1,-1> Where |1,m> is simultaneous eigenstate of L^2 and Lz find the expectation value of Lz and L^2. Sol" = for normalization 1/2 + 1/4 + 1/2 = 5/4 not Normalized Pn = |cn|^2 / Σ|cn|^2 = Σ Pm mħ^2 = 1/4 (ħ^2) + 1/4 (0) + 1/4 (-ħ^2) = 5/4 ħ^2 = 0 : Superposing term is not eigenstate of Lz that's why we used <ψ|Lz|ψ> |Lz|ψ> = (Lz+|ψ> + Lz-|ψ>) / 2 = 1/2 (ħ|1,1> + 0|1,0> - ħ|1,-1>) + 1/2 (ħ|1,0> + 0|1,-1> - ħ|1,1>) = ħ/2 (|1,1> - |1,-1>) <ψ|Lz|ψ> = <ψ|ħ/2 (|1,1> - |1,-1>) = <ψ|L^2|ψ> / <ψ|ψ> = 1/4 (ħ^2<ψ|1,1> + 2ħ^2<ψ|1,0> - ħ^2<ψ|1,-1>) + 1/4 (ħ^2<ψ|1,0> + 2ħ^2<ψ|1,-1> + ħ^2<ψ|1,1>) + ħ^2/4 <ψ|1,0> = 2√2/5 * (1/√2 - i/2 + i/2 * (1/√2 + i/√2) - i/√2 * i/2) = 2√2/5 * (-i/√2 + i/√2 - i/√2 + i/√2) ....

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