Lecture 31: The Hydrogen Atom 2: Dipole Moments

Lecture 31:The Hydrogen Atom 2:Dipole MomentsPhy851 Fall 2009

Electric Dipole Approximation The interaction between a hydrogen atomand an electric field is given to leading orderby the Electric Dipole approximation: Semi-Classical’ Approx:r rVE D E (rCM ) Electric field is classical COM motion is classical The dipole moment of a pure dipole:– Vector quantity– Points from - to .– Magnitude is charge distance-rr r r r-R rr rd q (r r ) For Hydrogen atom this gives:rrD eR

Dipole Moment Operator The electric dipole moment is an operator inH(R), which means that its value depends onthe state of the relative motion:rrD eRr rVE D E (rCM )r rVE e R E (rCM ) Choosing the z-axis along the electric fielddirection gives:VE e Z E (rCM ) Expanding onto energy eigenstates gives: n 1 n ' 1VE ll' nlm (V )n 1 n ' 1 l 0 l ' 0 m l m ' l 'E nlm ; n 'l 'm '(VE )nlm;n 'l 'm ' d nlm;n 'l 'm ' E (rCM )n' l ' m'

Dipole-Moment Matrix ElementsZ nlm;n 'l 'm ' nlm R cos Θ n' l' m' Separate radial and angular Hilbert spaces:d nlm;n 'l 'm ' e nl R n' l'(R)lm cos Θ l' m'(Ω) SELECTION RULES:– Arfken, 3rd ed., 12.213lm cos Θ l' m'(Ω)22 (l 1) 2 m 2 l m δ m,m 'δ l ,l ' 1 δ l ,l ' 1 22 4(l 1) 1 4l 1 The important thing to remember is thatd nlm;n 'l 'm ' δ m ,m 'δ l ,l ' 1 Electric Dipole Forbidden Transitions43spd2Examples:1f

Charged particle in a Magnetic Field EM fields are described by both a scalarpotential, and vector potential, A To include such EM fields, we can make thetransformation:vvv vP P qA( R) Here q is the charge and A(R) is the vectorpotential The Hamiltonian of an electron thenbecomes:– Units of B are Gauss (G):r r 2r1 rH P eA( R) eΦ( R)2me[] This is known as the ‘minimal couplingHamiltonian’

Vector potential of a uniform B-fieldr rr For a uniform B-field, B ( r ) B0 we have:r r1r rA(r ) r B02 Proof:r rr r rB(r ) A(r )r r1r r rB( r ) r B02r r rr r rr r r1 r r r r B0 B0 r r B0 B0 r2rr1 0 3B0 0 B02r B0([([ ))( ) (]))]( r r r e2 r r r e r r 2e r r r2 P 2 R B0 P 2 P R B0 R B0 P 4 R B0[]2rre 2 2 r r 2 2 P eL B0 R B0 R B0 4 ( )()2

An electron in a uniform B-field Putting this in the Hamiltonian gives:rP2e r re2 2 2H L B0 B0 R eΦ( R)2me 2me8me Choosing B along the z-axis gives:rP2eB0e 2 B02 22H Lz X Y ) eΦ( R)(2me 2me8me e Lz B02me“Paramagnetic term”e 2 B02 22X Y()8me“Diamagnetic term” Generates linear Zeeman effect Generates quadratic Zeeman effect

Paramagnetic Term: Magnetic DipoleInteraction A loop of current, I, and area, a, creates amagnetic dipole:µ Ia The orbital motion of a single electronconstitutes a current– For a circular orbit we haveev a π r 2I ,2π rev rIa 2 An electron therefore has a magnetic dipolemoment associated with its orbital motionev r 2 ee r re r rme vr p r r p2me 2me2mere rµ L2me The paramagnetic term is therefore theenergy of the orbital dipole moment in theuniform field: r rVB µ B0eB0VB Lz2me

Dipole Energy scaleeB0nlm VB nlm hm2me The energy shift between different m statesis very small compared to Hydrogen levelspacing Order of magnitude:VBB0eh 19 34 30 J 23 J 10 10meTT Strongest man-made B-fields 40 TVB 10 22 J E1 (2.18 10 18 J )Energy scale ofbare levels

Diamagnetic Term An electron in a uniform field will naturallyundergo circular motion in the planeperpendicular to the field– Cyclotron motion Thus the B-field induces a current This leads to an induced magnetic moment,which must be proportional to B0rrµinduced B0 The energy of this magnetic moment in theuniform B field therefore scales as B2rrE µinduced B0 B02VB 2e 2 B02 2 X Y 28me() Order of magnitude:VB 2B02e 2 a02JJ 10 38 20 30 2 10 28 28meTTVB 2 10 26 J VB (10 22 J ) E1 (10 18 J )– The diamagnetic term can be neglected unlessthe B-field is very strong

Zeeman Effect The Hamiltonian of a Hydrogen atom in a uniformB-field is– Can neglect diamagnetic termeBH H0 Lz2µ H 0 nlm En nlm‘Bare’ HamiltonianEigenstates are unchangedH n, l, m E n, l, m Energy eigenvalues now depend on m:En , m h 2 1 eB m222 µa0 n2µThe additional term is called the Zeeman shift– We already know that it will be no larger than 10-22J 10-4eV– E.g. 100 G field: EZeeman 10-25 J EZeeman/EI 10-25 18 10-7 To get the correct Zeeman shift, we will also needto include spin.– We will do this next semester using perturbationtheory and the Wigner-Ekert Theorem

and an electric field is given to leading order by the Electric Dipole approximation: The dipole moment of a pure dipole: -Vector quantity -Points from - to . -Magnitude is charge _ distance For Hydrogen atom this gives: Electric Dipole Approximation d q(r r ) rrr V E DE(r CM) rr Semi-Classical' Approx .