Degeneracy of Nth Level of H Atom | Santa Fe College - Edubirdie

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

2024
Views:

227

Pages:

5
Author:

Jabin F.

En∝1/n^2 Ψnlm→|m> Ψ→Ψnlmχβ,ms for a given n, l=0,1,2,3-...n-1 for a given l, m=-l----l =(2l+1) no: of values with diff of unity The no. of energy eigen fn for given n and l =(2l+1) Degeneracy of n-th level = The no. of energy eigen fn of samen The no. of Lof energy eigermph for a particular n ΨTotal=Ψnlm(rθφ)Ψspin' fore, p=1/2 mp=+1/2,-1/2 |s,mp>=j|1/2,+1/2> |1/2,-1/2> gn=Σ(2l+1) , (excluding spin. l=0 degeneray) Σ(2l+1) , including spin l=0 Σ(l+1) = 1 + 3 + 5 + ... + 2(n-1) + 1 γ=0 = n[1 + 2n - 2 + 1] = n^2 gn = {n^2, n^2} excluding spin including spin excluding (γ=∞) The no. of radial nodes = (n-l-1) Ques. 15 H= -γBSy = -ωSy = -iħωσy / 2 page-61 |ψ(0)> = |↑> = |↑> = Sz|±> = ±1/2 |±> |↑(+)> = |↑(0)> = = <↓1h57m |ψ(t)> = e^{-iHt/ħ} |ψ(0)> = e^{iωtSy/ħ} |↑> = e^{iωtσy/2ħ} |↑> e^(iωtσy/2) |↑> = = e^(iωt/2)|↑> + iσy sin(ωt/2)|↑> = cos(ωt/2)|↑> + i sin(ωt/2)|↓> |ψ(t)> = cos(ωt/2)|↑> - sin(ωt/2)|↓> P₂(-ħ/2) = |-sin²(ωt/2)|² = sin⁴(ωt) (option-d) ques5 page-53 |ψ₁> = 1/sqrt(3)(|↑> + sqrt(2)|↓>) |ψ₂> = = i/sqrt(3)|↑> + sqrt(2/3)|↓> <ψ₂|ψ₁> = 0 (i/sqrt(3)<↑| + sqrt(2/3)<↓|)(c₁|↑> + c₂|↓>) = 0 => c₁/sqrt(3) + sqrt(2/3)c₂ = 0 c₂ = -c₁/sqrt(2) |c1|^2 + |c2|^2 = 1 |c1|^2 + |c1|^2/2 = 1 c1 = sqrt(2/3) => c2 = -1/sqrt(3) |ψ2> = sqrt(2/3)|↑> - sqrt(1/3)|↓> The probability of spin up state Pn = |sqrt(2/3)|^2 = 2/3 = 0.67 Ans. Pj = 1/3 = 0.33

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