Summer Lecture Notes Hydrogen-Like Atoms, Transition Theory

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

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... using (bound state) perturbation theory first-order effect (linear effect) second-order effect (quadratic effect) (quadratic Stark/Zeeman effects, you particularly hear of second-order effects when the first-order effect is zero) third,cubic fourth,quartic fifth,quintic

5.1 General or External Effects Stark effect, Zeeman effect, and van der Waals effect. (Angular momentum coupling? jj-coupling?) (Any more?) Effect Description Perturbation Stark Uniform electric field E in z-direction H1 = q Φ(r) = eEz Zeeman Uniform magnetic field in z-direction

5.1.1 (Electric Coupling?): Stark Effect The Stark effect is the splitting and shift of a spectral line into several components in the presence of an electric field. (I don’t know if this term is reserved for atomic physics, or if it applies to any system, but I do know we apply it to the harmonic oscillator as well.) The amount of splitting itself is called the Stark shift. We have E in the z-direction. Our perturbing Hamiltonian is H1 = q Φ(r) = (−e)(−E · r) = eEzˆ · r = eEr cos θ = eEz Ground State Polarizability Quadratic Stark Effect: (Is the quadratic Stark effect the Stark effect? Harmonic Oscillator...) (Abers Problem 7.9) First order Stark effect in quantum mechanics hydrogen atom matches that of the Bohr hydrogen atom. Higher order Stark effects do not match in each theory. “The parity argument can be used whennever the unperturbed Hamiltonian is invariant under reflection, and it can be generalized to the statement that systems in nondegenerate states cannot have permanent dipole moments. The statement of nondegeneracy is important: it is only then that the states are also eigenstates of the parity operator, and them |φ(r)|2 is even, and the expectation value of z vanishes.” (From Gasiorowicz [1] pg 271)

5.1.2 (Magnetic Coupling): Zeeman Effect, Anomalous Zeeman Effect Fine Structure (here too?) (Hydrogen-like atoms, Sakurai’s contents say, relates here) The splitting of spectral lines by a weak external magnetic field is called the Zeeman effect. (Abers pg 178: “The states originally in the angular momentum ladders are no longer degenerate, since H is not rotationally invariant. But there is still some degeneracy remaining.”) The anomalous Zeeman effect appears on transitions where the net spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of odd if there’s an uneven number of electrons involved. If the magnetic field strength is too high, the effect is no longer linear; at even higher field strength, electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen-Back effect. An atom in a weak uniform magnetic field (that is, weak in comparison to the system’s internal magnetic field) has its energy shifted by a term linear in (the magnetic field, the atom’s angular momentum, the atom’s magnetic dipole moment?). The Land g-factor relates the magnetic dipole moment to the angular momentum of a quantum state. Quadratic Zeeman Effect: Diamagnetism (Abers Problem 7.10) Zeeman Effect for Intermediate Magnetic Fields (Abers Problem 7.18)

9 5.1.3 Angular Momentum Coupling Term symbols, selection rules

5.1.4 van der Waals Effect 5.1.5 Minimal Coupling(?) 5.2 Internal Effects, or Corrections Reduced mass effect (for hydrogen) µ m ≈ 1 − e ≈ 1 − 5.4 × 10−4 me mp Relativistic kinetic energy effect

1 2 H p 2 −4 2 0 ≈ 2 2 ≈ (Zα) = (0.53 × 10 )Z hH i me c (for hydrogen, less than the reduced mass eﬀect by an order of magnitude) Spin-orbit coupling (Lamb shift) Anomalous Zeeman eﬀect

5.2.1 Reduced Mass Correction 5.2.2 Relativistic Kinetic Energy Correction 5.2.3 (Magnetic-(Electric/Magnetic) Coupling?): Spin-Orbit (LS) Coupling, Fine Struc- ture In heavier atoms, jj coupling...(?)

5.2.4 (Magnetic Coupling?): Spin-Spin (SS) Coupling, Hyperfine Structure (A Per- manent Zeeman Effect) (Nuclear Spin - Electronic Spin) (A permanent Zeeman effect due to the magnetic field generated by the magnetic dipole moment of the nucleus.)

5.2.5 Finite Size of the Nucleus Correction (Abers Problem 7.7)

5.2.6 Lamb Shift

10 6 Examples

6.1 Helium Ground State 6.2 Molecules: Born-Oppenheimer Approximation, Hydrogen Molecular Ion See Abers section 7.6

11 7 Transition Theory

Does “transition” imply time-dependence? “A system in an initial state transitions to a diﬀerent ﬁnal state.” Or does it include the more abstract notion of perturbation of a model? “If we take this model and perturb it slightly, the states of the system will transition to another set of states.” Transition matrix, (relation to scattering amplitude, scattering matrix) Dyson Series, Born- Oppenheimer Approximation, Fermi’s Golden Rule Lippmann-Schwinger equation Transition rules (selection rules)

7.1 (Bound State?) Perturbation Theory ? Time-Independent (Non-Degenerate, Degenerate) Time-Dependent (Non-Degenerate, Degenerate)?

7.2 Scattering Theory Scattering matrix

7.2.1 Potential Scattering Scattering amplitude, Partial waves Born series, spherically symmetric potentials, optical theorem, refractive index

7.3 Collision Theory (subtopic of scattering?) 7.4 Decays of Excited States Lowest order transition rates, etc.

References

[1] Stephen Gasiorowicz: Quantum Physics, Second Edition, Wiley (1996)

[2] J. J. Sakurai: Modern Quantum Mechanics, Addison-Wesley (1994)

[3] Ernest S. Abers: Quantum Mechanics, Pearson, Prentice Hall (2004)