Summer Lecture Notes Hydrogen-like Atoms, Transition Theory Andrew Forrester September 7, 2006 Contents 1 Lecture 7 3 2 Preliminaries and Notation 3 3 Hydrogen-Like Atoms to Zeroth Order 4 3.1 The Hamiltonian . 4 3.2 Bound State Energy Eigenfunctions and Spectra . 5 3.3 Special Functions . 5 3.4 Useful Integrals and Expectation Values . 6 4 Approximation Methods for Bound States 8 4.1 Bound State Perturbation Theory . 8 4.1.1 Time-Independent, Non-Degenerate . 8 4.1.2 Time-Independent, Degenerate . 8 4.1.3 Time-Dependent (Non-Degenerate/Degenerate)? . 8 4.2 Variational Method . 8 4.3 Born-Oppenheimer Approximation (is this used inside other methods?) . 8 4.4 WKB (Wentzel-Kramers-Brillouin) Method . 8 5 Hydrogen-like Atom Effects and Corrections 9 5.1 General or External Effects . 9 5.1.1 (Electric Coupling?): Stark Effect . 9 5.1.2 (Magnetic Coupling): Zeeman Effect, Anomalous Zeeman Effect . 9 5.1.3 Angular Momentum Coupling . 10 5.1.4 van der Waals Effect . 10 5.1.5 Minimal Coupling(?) . 10 5.2 Internal Effects, or Corrections . 10 5.2.1 Reduced Mass Correction . 10 5.2.2 Relativistic Kinetic Energy Correction . 10 5.2.3 (Magnetic-(Electric/Magnetic) Coupling?): Spin-Orbit (LS) Coupling, Fine Structure . 10 5.2.4 (Magnetic Coupling?): Spin-Spin (SS) Coupling, Hyperfine Structure (A Per- manent Zeeman Effect) . 10 5.2.5 Finite Size of the Nucleus Correction . 10 5.2.6 Lamb Shift . 10 6 Examples 11 6.1 Helium Ground State . 11 6.2 Molecules: Born-Oppenheimer Approximation, Hydrogen Molecular Ion . 11 1 7 Transition Theory 12 7.1 (Bound State?) Perturbation Theory . 12 7.2 Scattering Theory . 12 7.2.1 Potential Scattering . 12 7.3 Collision Theory (subtopic of scattering?) . 12 7.4 Decays of Excited States . 12 2 1 Lecture 7 Will Try • Hydrogen Atom (Spectrum, Minimal Coupling, the Stark/Zeeman Effect, ...) (all corrections to the hydrogen atom spectrum and their relative magnitudes) • Transition Theory (Potential Scattering, Fermi’s Golden Rule, ...) Wanted • Hydrogen Atom Spectrum, Minimal Coupling, and The Stark/Zeeman Effect (all corrections to the hydrogen atom spectrum and their relative magnitudes) • Potential Scattering, Transition Theory, and Fermi’s Golden Rule • Equations of state and manipulations using Maxwell relations • Quasi-static/Adiabatic Processes and Heat Engines 2 Preliminaries and Notation You should know the values of the fine-structure constant α and the Bohr radius a (are they defined this way?): e2 2 α = k a = ~ ~c kme2 so α = ~ a = ~ mac mαc where k is Coulomb’s constant, e is the proton charge, ~ is Dirac’s constant (the reduced Planck’s constant, Planck’s constant h divided by 2π), and c is the speed of light in vacua. Z is the atomic number, or proton number (the number of protons in an atom or nucleus), and A is the atomic mass number, or nucleon number (the number of nucleons, both protons and neutrons, in an atomic nucleus). mN is a nucleus mass, mp is the proton mass, mn is the neutron mass, and me is the electron mass. rN is the nucleus position, pN is the nucleus momentum, re is the electron position, and pe is the electron momentum. The Bohr magneton is ~e µB = 2me 3 3 Hydrogen-Like Atoms to Zeroth Order Hydrogen-like atom bound state energy eigenfunctions and eigenvalues (energy levels, or spectra) to zeroth order (in what?) 3.1 The Hamiltonian We transform 2 2 p1 p2 q1q2 H(r1, r2, p1, p2) = + − k 2m1 2m2 |r1 − r2| p 2 p 2 Ze2 = N + e − k 2mN 2me |rN − re| to P2 p2 q q H(R, r, P, p) = + − k 1 2 2M 2µ |r| P2 p2 Ze2 = + − k 2M 2µ r where mn = Zmp + (A − Z)mn M ≡ mN + me 1 1 1 ≡ + µ mN me m r + m r R ≡ N N e e M r ≡ rN − re P = MR˙ p = µr˙ Corrections for various phenomena will change the Hamiltonian, and thus the energies, to various orders of magnitude (with respect to some variable...) If we have (missing k’s) Ze2 2 g2 V (r) = − + ~ r 2µ r2 2 2 then we get (α = k e or α = e ) ~c ~c 1 Z2α2 E = − µc2 2 h i2 n − l − 1/2 + p(l + 1/2)2 + g2 Dirac’s fine-structure formula: −1/2 α2 E = m 1 + 2 n − j − 1/2 + p(j + 1/2)2 − α2 4 3.2 Bound State Energy Eigenfunctions and Spectra The solutions for the bound states of hydrogen-like atoms are Ψ(R, r, t) = χ(R) ψ(r) τ(t) = χ(R) τR(t) ψ(r) τr(t) where χ(R) = (2π)−3/2eiP·R/~ −iEtt/ −iERt/ −iErt/ τ(t) = e ~ = e ~ e ~ = τR(t) τr(t) ψ(r) = Rnl(r) Ylm(Ω) 1 = U (r)Θ (θ)Φ (φ) r nl lm m 2r l 2r = N e−r/na L2l+1 N P m(cos θ) eimφ, nl na n+l na lm l with " #1/2 2 3 (n − l − 1)! N = − nl na 2n[(n + l)!]3 s 2l + 1 (l − |m|)! N = (−1)m lm 4π (l + |m|)! 2 2 4 P 2 µZ e Et = ER + Er = − k 2M 2~2n2 In the center-of-mass system, we have, simply, 2 4 2 µZ e Er = En = −k 2~2n2 Note that Ψ is not normalizable, since it contains a momentum eigenstate χ, but ψ is normalized. The solutions for non-bound states, where E > 0 rather than E < 0, are the confluent hypogeometric functions. These are of interest in electron or proton scattering. (From [1] pg 213) 3.3 Special Functions m m Legendre polynomials Pl (Pl , m = 0) Associated Legendre polynomials Pl l |m| P (x) = 1 d (x2 − 1)l P m(x) = (1 − x2)|m|/2 d P (x) l 2ll! dxl l dx|m| l P0(cos θ) = 1 1 P1(cos θ) = cos θ P1 (cos θ) = sin θ 2 1 P2(cos θ) = (3 cos θ − 1)/2 P2 (cos θ) = 3 cos θ sin θ 2 2 P2 (cos θ) = 3 sin θ 3 1 3 P3(cos θ) = (5 cos θ − 3 cos θ)/2 P3 (cos θ) = 3 sin θ(5 cos θ − 1)/2 2 2 P3 (cos θ) = 15 sin θ cos θ 3 3 P3 (cos θ) = 15 sin θ Legendre and associated Legendre polynomials: These functions are the solutions for Θ(θ) and happen to be orthogonal. 5 Spherical harmonics m imφ mq 2l+1 (l−|m|)! Ylm(θ, φ) = Nlm Pl (cos θ) e Nlm = (−1) 4π (l+|m|)! Ylm (m = 0) Ylm (m 6= 0, |m| ≤ l) Y (θ, φ) = √1 0,0 4π q 3 q 3 ±iφ Y1,0(θ, φ) = 4π cos θ Y1,±1(θ, φ) = ∓ 8π sin θ e q 5 2 q 15 ±iφ Y2,0(θ, φ) = 16π (3 cos θ − 1) Y2,±1(θ, φ) = ∓ 8π cos θ sin θ e q 15 2 ±2iφ Y2,±2(θ, φ) = 32π sin θ e q 7 3 q 21 2 ±iφ Y3,0(θ, φ) = 16π (3 cos θ − 5 cos θ) Y3,±1(θ, φ) = ∓ 64π (1 − 5 cos θ) sin θ e q 105 2 ±2iφ Y3,±2(θ, φ) = 32π cos θ sin θ e q 35 3 ±3iφ Y3,±3(θ, φ) = ∓ 64π sin θ e Normalized spherical harmonics: Ylm(θ, φ) = Θlm(θ)Φm(φ) j j Laguerre polynomials Li (Li , j = 0) Associated Laguerre polynomials Li r di i −r j dj Li(r) = e dri (r e ) Li (r) = drj Li(r) L0(r) = 1 1 L1(r) = 1 − r L1(r) = −1 2 1 L2(r) = 2 − 4r + r L2(r) = −4 + 2r 2 L2(r) = 2 2 3 1 2 L3(r) = 6 − 18r + 9r − r L3(r) = −18 + 18r − 3r 2 L3(r) = 18 − 6r 3 L3(r) = −6 Laguerre and associated Laguerre polynomials: These functions are the solutions for pnl(r) and happen to be orthogonal. Radial functions 1/2 2r l −r/na 2l+1 2r h 2 3 (n−l−1)! i Rnl(r) = Nnl na e Ln+l na Nnl = − na 2n[(n+l)!]3 Rnl (l = 0) Rnl (1 ≤ l ≤ n − 1) 1 −r/a R1,0(r) = √ e 2 a3 1 r −r/2a 1 r −r/2a R2,0(r) = √ 1 − e R2,1(r) = √ e 2a3 2a 6a3 2a 2 2r 2r2 −r/3a 8 r r −r/3a R3,0(r) = √ 1 − + 2 e R3,1(r) = √ 1 − e 3 3a3 3a 27a 9 6a3 6a 3a 4 r 2 −r/3a R3,2(r) = √ e 9 30a3 3a l −λr Normalized radial solutions: Rnl(r) = r e pnl(r). 3.4 Useful Integrals and Expectation Values ∞ Z 2 √ e−x dx = π Γ = 0 ∞ Z 2 xn e−x dx = 0 Z ∞ D kE k+2 2 r = dr r [Rnl(r)] nlm 0 6 So a2n2 r2 = [5n2 + 1 − 3l(l + 1)] nlm 2Z a hri = [3n2 − l(l + 1)] nlm 2Z 1 Z = 2 r nlm an 1 Z2 = 2 2 3 ½ r nlm a n (l + ) 1 Z3 = 3 3 3 ½ r nlm a n l(l + )(l + 1) 7 4 Approximation Methods for Bound States 4.1 Bound State Perturbation Theory 4.1.1 Time-Independent, Non-Degenerate Non-Degenerate Perturbation Theory (1) 1 ∆n = n H n 1 2 X m H n ∆(2) = − n E0 − E0 m6=n m n 4.1.2 Time-Independent, Degenerate 4.1.3 Time-Dependent (Non-Degenerate/Degenerate)? 4.2 Variational Method Variational Method for the Stark Effect on the Ground State of Atomic Hydrogen (Abers Problem 7.16) 4.3 Born-Oppenheimer Approximation (is this used inside other methods?) 4.4 WKB (Wentzel-Kramers-Brillouin) Method 8 5 Hydrogen-like Atom Effects and Corrections Higher order (in what variable) effects..