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

2024
Views:

149

Pages:

6
Author:

Jabin F.

#Harmonic Perturbation V(t) = { Voe^(iωt) + Vo*e^(-iωt) 0 < t < t' { 0 otherwise Pi→f = -i ∫(e^(iωt') dt') t→∞ = 2πħ ||^2 δ(Ef - Ei + ħω) + πħ ||^2 δ(Ef - Ei - ħω) Pi→f = πħ/2 ||^2 [δ(Ef - Ei + ħω) + δ(Ef - Ei - ħω)] Pi→f = 2πħ/ħ ||^2 (Ef - Ei + ħω) + 2πħ/ħ ||^2 (Ef - Ei - ħω) Pi→f = { Non-zero, if Ef - Ei ± ħω = 0 { Zero, otherwise Note: The symbols used in the image are standard quantum mechanical notation. ħ is the reduced Planck constant |i> and |f> represent initial and final states, respectively Vo is a constant related to the perturbation ω is the angular frequency of the perturbation δ is the Dirac delta function Nonzero, if DE = Ef - Ei = ±ħωOtherwise Pif = { 0 Absorption Pif = |(-i/ħ) ∫(from 0 to t) e^(iωft') dt'|^2 Emission Pif = |(-i/ħ) ∫(from 0 to t) e^(-iωft') dt'|^2 Pif = |(-iωf/ħ) ∫(from 0 to t) e^(iωft') dt'|^2 Pif = |(-i/ħħ) ∫(from 0 to t) (d/dt') e^(iωft') dt'|^2 Pif = |(-i/ħħ) [e^(iωft') ](from 0 to t) + (iωf/ħħ) ∫(from 0 to t) e^(iωft') dt'|^2 Pif = |(-1/ħħ) e^(iωft) + (1/ħħ) + (ωf/ħ) ∫(from 0 to t) e^(iωft') dt'|^2 Pif = |(1/ħħ) [ - e^(iωft)] + (ωf/ħ) ∫(from 0 to t) e^(iωft') dt'|^2 E1, E2 (σT) = (σT)2 S-wave = 0 4πT / K1 * sin δ0 = 4πT / K2 * sin δ2 1/K1² * sin²30 = 1/K2² * sin²60 K2² = sin²60 / sin²30 * K1² K2² = (3/4) / (1/4) * K1² = 3K1² => E2 = ħ²K2² / 2m = 3ħ²K1² / 2m = 3E1 = 3 * 0.1 MeV = 0.3 MeV. (option-b) Question 12 S-wave dσ/dΩ = |e^(iδ0)δ1 + 3e^(iδ1)sinδ1cosθ|² P-wave dσ/dΩ = |e^(iδ0)δ1 + 3e^(iδ1)sinδ1cosθ|² d/dθ (dσ/dΩ) = 0 2|e^(iδ0)δ1 + 3e^(iδ1)sinδ1cosθ| * [0 + 3e^(iδ1)cosδ1cosθ - 3e^(iδ1)sinδ1sinθ] = 0 |e^(iδ0)δ1 + 3e^(iδ1)sinδ1cosθ| * [3e^(iδ1)cosδ1cosθ - 3e^(iδ1)sinδ1sinθ] = 0 e^(isδ) sinδ + 3e^(isδ) sinδ cosθ = 0 3e^(isδ) sinδ cosθ = -e^(isδ) sinδ => sinδ * (3e^(isδ) cosθ + e^(isδ)) = 0 => sinδ = 0 or 3e^(isδ) cosθ + e^(isδ) = 0 => δ = nπ => 3e^(isδ) cosθ = -e^(isδ) => 3 cosθ = -1 => cosθ = -1/3 for real cosθ = -2/3 * cos²δ₀ / (2 + cos²δ₀) cosθ = -2/3 Page - 92 V(x,y,z) = (1/2)K(x^2 + y^2 + z^2) H0 = p_x^2 / 2m + p_y^2 / 2m + p_z^2 / 2m + (1/2)K(x^2 + y^2 + z^2) E_n^0 = (n_x + n_y + n_z + 3/2)ħω = (n + 3/2)ħω n_x, n_y, n_z = 0, 1, 2, 3, ... E = (n' + 1 + 3/2)ħω n' = 0, 1, 2, ... ∞ l = 0, 1, 2, 3, ... ∞ (n', l) (m_l) E^(0) = (3/2)ħω g = 1 (000) = (5/2)ħω g = 3 (100), (010), (001) (7/2)ħω g = 6 n' = 0, J = 2 m_l = ±2, ±1, 0 m_l = 1, d = 0 2n' + J = 2 For second n = 2 2n' + l = 2 Distinct value of m_l for II excited state = 5 So no. of distinct levels = 5 (1,0,0) (010) (001)