Quantum Theory - Particle Physics Group

CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 884.3 SummaryFor a Hamiltonian of the form H = P2 − V (r), which is symmetric under rotations, the2mangular momentum L = X× ⃗ P ⃗ is conserved [H, L i ] = 0 and the algebra [L i , L j ] = iε ijk L kleads to the following three commuting operators[H, L z ] = [H, L 2 ] = [L z , L 2 ] = 0. (4.68)The common eigenfunctions of L 2 and L z are the spherical harmonics with eigenvaluesL 2 Y lm = 2 l(l + 1)Y lm (4.69)andL z Y lm = mY lm (4.70)with l and |m| ≤ l integer. The eigenvalue of L 2 is (2l + 1) fold degenerate.The Schrödinger equation for the hydrogen atom can be solved by reducing the nonrelativistictwo-body problem to the one-body problem with reduced mass µ = m 1 m 2 /Mand a free center of mass motion with total mass M = m 1 + m 2 .With the formula for the Laplace operator∆ = 1 r∂ 2∂r 2r − 1 r 2 L 2 2 (4.71)and a separation ansatz in spherical coordinates the energy eigenvalues( ) 2E n = − Z2 R Z= −2µ , (4.72)n 2 a 0 nare determined by the termination condition of the power series solution to the radialequation (4.57), which is related to the differential equationxL ′′ (x) + (2l + 2 − x)L ′ (x) − (l + 1 − n)L(x) = 0 (4.73)for the associated Laguerre polynomialsL s r(x) = ∂ s x L r (x) = ∂ s x e x ∂ r x e −x x r , with r = n + l, s = 2l + 1 (4.74)by x = 2ρ = 2κr with κ =u nlm =√−2µE n 2= Zna 0. The normalized wave functions are√(n − l − 1)!(2κ 3 )2n((n + 1)!) 3 (2κr) l e −κr L 2l+1n+l (2κr) Y lm(θ,ϕ) (4.75)where n ∈ N is the principal quantum number, l < n the orbital quantum number and mthe magnetic quantum number. Due to the approximation of a pure Coulomb interactionand electrons without spin E n is n 2 -fold degenerate.