Radial Equations and Spherical Harmonics in Quantum Mechanics - Prof. M. W. Bromley, Study notes of Quantum Mechanics

Download Radial Equations and Spherical Harmonics in Quantum Mechanics - Prof. M. W. Bromley and more Study notes Quantum Mechanics in PDF only on Docsity! Lecture 18 Outline - 3-D radial bits • Spherical harmonics recount [section 12.5] • Radial Equation [Section 12.6] • Infinite Spherical Well [Section 12.6] • (in brief) 3-D isotropic SHO [Section 12.6] • start on the H-atom [Section 13.1] Angular Momentum eigenstates • Simultaneous eigenstates ie. [L2, Lz] = 0 L2|` m〉 = ~2`(`+ 1)|` m〉 and Lz|` m〉 = ~m|` m〉 • for ` = 0, 1 2 , 1, 3 2 , . . . and m = −`, (−`+ 1), . . . , (`− 1), `. • separation of variables → spherical harmonics Y m`` (θ, φ) L̂2Y m`` = −~2 sin θ [ ∂ ∂θ ( sin θ ∂ ∂θ ) + 1 sin θ ∂ ∂φ ] Y m`` = ~ 2`(`+ 1)Y m`` L̂zY m` ` = [ −i~ ∂ ∂φ ] Y m`` = ~m`Y m` ` • ie. good quantum numbers Ĥ|n ` m`〉 = En|n ` m`〉, • when V (r) then [Ĥ, L̂2] = 0 meanwhile [Ĥ, L̂z] = 0 Radial Equation • Given separable solns: ψ(r, θ, φ) = R(r) Y m` (θ, φ), • do a (important) change of variable u(r) = rR(r): 1 R d dr ( r2 dR dr ) − 2mr 2 ~2 (V (r) − E) = `(`+ 1) R = u r , dR dr = 1 r2 ( r du dr − u ) , d dr ( r2 dR dr ) = r d2u dr2 − ~ 2 2m d2u dr2 + [ V (r) + ~ 2 2m `(`+ 1) r2 ] u = Eu • the radial Schrödinger eqn has an effective potential: Veff(r) = V (r) + ~ 2 2m `(`+ 1) r2 • With normalisation: ∫ ∞ 0 |R|2r2dr = ∫ ∞ 0 |u|2dr = 1 Infinite spherical well • Outside well ψ(r, θ, φ) = 0 = R(r) Y m` (θ, φ) • so want to find (bound state) eigenfunctions inside well: − ~ 2 2m d2u dr2 + [ V (r) + ~ 2 2m `(`+ 1) r2 ] u = Eu d2u dr2 = [ `(`+ 1) r2 − k2 ] u • where k = √ 2mE/~ (eigenenergies E unknown) • For ` = 0 we get a (familiar) 1-D eigenproblem d2u dr2 = −k2u thus u(r) = A sin(kr) +B cos(kr) • as r → 0 then cos(kr)/r → ∞, thus B = 0 Infinite sphere - ` = 0 solns • At well surface un(x = a) ∝ sin(kna) = 0 so kna = nπ... • allowed energies for ` = 0 are En,`=0 = ~ 2k2n 2m = n2 π2~2 2ma2 (∀n = 1, 2, 3 . . .) • normalisation ∫ ∞ 0 |u|2dr = 1 means A = √ 2/a, ψn00(r, θ, φ) = √ 2 a sin (nπ a x ) Y 00 (θ, φ) = 1√ 2πa sin ( nπ a x ) r • Energy En` depends on two quantum numbers • Wavefunction ψn`m depends on the three. Infinite sphere - Bessel piccies Isotropic SHO • Hamiltonian Ĥ = 1 2m (p̂2x + p̂ 2 y + p̂ 2 z) + 1 2 mω2(x̂2 + ŷ2 + ẑ2) • assuming ψ(r, θ, φ) = un`(r) r Y m` (θ, φ), gives radial SHO ( d2 dr2 + 2m ~2 [ E − 1 2 mω2r2 − ~ 2`(`+ 1) 2mr2 ]) u = 0 • as r → ∞ potential V (r) → 1 2 mω2r2 solns u(y) → e−y 2 2 with y ≡ √ mω ~ r sotry : u(y) = f(y)e −y2 2 • with a power series expansion f(y) = y`+1 ∑∞n=0 cnyn • reqd to terminate at integer k = 0, 1, 2 . . . (as before): E = (2k + `+ 3 2 )~ω = (n+ 3 2 )~ω Coulomb potential • Time-indep S.E. − ~2 2m ∇2ψ + V ψ = Eψ, • separable solns ψ(r, θ, φ) = u(r) r Y m` (θ, φ), − ~ 2 2m d2u dr2 + [ − e 2 4π0 1 r + ~ 2 2m `(`+ 1) r2 ] u = Eu • Coulomb potential here assumes infinitely heavy proton. • Continuum (scattering) states for E > 0, and • discrete bound states for E < 0 which we want: • introduce κ = √ −2mE/~ thus 1 κ2 d2u dr2 = [ 1 − me 2 2π0~2κ 1 κr + `(`+ 1) (κr)2 ] u • suggests introduce ρ = κr, and ρ0 for constant...