Physics Exam: Non-Degenerate Perturbation, Scattering, Hydrogen Energy & Wave Functions, Exams of Quantum Physics

Download Physics Exam: Non-Degenerate Perturbation, Scattering, Hydrogen Energy & Wave Functions and more Exams Quantum Physics in PDF only on Docsity! Physics 472 Final Exam Spring 2007 Name: Non-Degenerate Time-Independent Perturbation Theory If a level n with energy E(0)n and eigenfunction ψ (0) n is non-degenerate, then the perturbed energy and eigenfunction are given by En = E (0) n + 〈ψ(0)n |H ′|ψ(0)n 〉+ ∑ k 6=n |〈ψ(0)k |H ′|ψ(0)n 〉|2 E (0) n − E(0)k + . . . ψn = ψ (0) n + ∑ k 6=n ψ (0) k 〈ψ(0)k |H ′|ψ(0)n 〉 E (0) n − E(0)k + . . . where H ′ is the perturbation. Time-Dependent Perturbation Theory If a system initially in an eigenstate with energy E1 is subjected to a time-dependent pertur- bation H ′(t), the first order probability amplitude for a transition to an eigenstate with energy E2 is C (1) 2 (t) = − i h̄ ∫ t 0 dt′ei(ω2−ω1)t ′〈2|H ′(t′)|1〉 , where ω2 − ω1 = (E2 − E1)/h̄. Scattering R`(r) r→∞→ A` sin(kr + δ` − `π/2) r (Scattering wave function behavior) dσ dΩ = |f(θ)|2 f(θ) = 1 k ∞∑ `=0 (2` + 1)eiδ` sin δ`P`(cos θ) (Partial wave expansion) σT = ∫ dΩ|f(θ)|2 = 4π k2 ∞∑ `=0 (2` + 1) sin2 δ` fB(θ) = −2m h̄2q ∫ ∞ 0 dr r V (r) sin(qr) q = 2k sin(θ/2) (Born approximation) Hydrogen Energies and Wave Functions En = −13.6 eV n2 ψ100(~r) = R10(r)Y 0 0 (θ, φ) ψ200(~r) = R20(r)Y 0 0 (θ, φ) ψ21m(~r) = R21(r)Y m 1 (θ, φ) Harmonic Oscillator Energies and Wave Functions En = h̄ω(n + 1 2 ) ψn(x) = ( mω πh̄ )1/4 1√ 2nn! Hn(ξ)e −ξ2/2 ξ = √ mω h̄ x H0(ξ) = 1 H1(ξ) = 2ξ H2(ξ) = 4ξ 2 − 2 H3(ξ) = 8ξ3 − 12ξ