Appendix: Special Mathematical Functions

Home , Displacement operator, Sherman function

For completeness this appendix briefly lists without further discussion the definiti• tions and some important properties of the special functions occurring in the book. More detailed treatments can be found in the relevant literature. The "Handbook of Mathematical Functions" [AS70], the "Tables" by Gradshteyn and Rhyzik [GR65] and the compilation by Magnus, Oberhettinger and Soni [M066] are particularly useful. Apart from these comprehensive works it is worth mention• ing Appendix B in "Quantum Mechanics I" by Messiah [Mes70], which describes a selection of especially frequently used functions.

A.I Legendre Polynomials, Spherical Harmonics

The Ith Legendre Polynom P,(x) is a polynomial of degree 1 in x,

1 d' 2 , P,(x) = 2'11 dx'(x - 1), 1 = 0, 1, ... (A.1)

It has 1 zeros in the interval between -1 and +1; for even (odd) I, P,(x) is an even (odd) function of x. The associated Legendre junctions P"m(x) , Ixl ~ 1, are products of (1 - x2)m/2 with polynomials of degree 1 - m (m = 0, ... , I) ,

(A.2)

The spherical harmonics Yi,m(fJ, c/» are products of exp (imc/» with polyno• mials of degree m in sin fJ and of degree 1- m in cos fJ, where the fJ-dependence is given by the associated Legendre functions (A.2) as functions of x = cos fJ. For m ;::: 0, 0 ~ fJ ~ 7r we have

Yi,m(fJ, c/» = (_1)m [(2~: 1) ~~: :;!! f/2 P',m(cos fJ)eim.p (A.3) = (_I)m [(2/+1) (/_m)!]1/2 . mfJ dm n( fJ) im.p 47r (1 + m)! sm d(cos fJ)m .q cos e

The spherical harmonics for negative azimuthal quantum numbers are obtained via (A.4) 302 Appendix

A reflection of the displacement vector

x = r sin 0 cos , y = r sin 0 sin , z = r cos 0 at the origin (cf. (1.67» is achieved by replacing the polar angle 0 by 7r-0 and the azimuthal angle by 7r+¢. This does not affect sin 0, but cos 0 changes to - cos O. In the expression (A.3) for Yi,m spatial reflection introduces a factor (_I)I-m from the polynomial in cosO and a factor (_I)m from the exponential function in ¢. Altogether we obtain (A.5) The integral over a product of two spherical harmonics is given by the or• thonormality relation (1.58). The integral over three spherical harmonics is a pro• totype example for the Wigner-Eckart theorem, which says that the dependence of the matrix elements of (spherical) tensor operators in angular momentum eigenstates on the component index of the operator and the azimuthal quantum numbers of bra and ket is given by appropriate Clebsch-Gordan coefficients (see Sect. 1.6.1). For the spherical harmonics YL,M as an example for a spherical tensor of rank L we have

J Yi:m(Q) YL,M(Q) Yi',m,(Q) dil , , [(2I'+I)(2L+I)]1/2 , = (I, miL, M, 1 ,m ) 47r(21 + I) (1, OIL, 0,1 ,0) (A.6)

The special Clebsch-Gordan coefficient (I, OIL, 0, 1',0) is given by [Edm60] (I,OIL,O, 1',0) = v'2T+1 (_1)(I-L-I')/2

X [(J -2l)!(J -2L)!(J -2I')!] 1/2 (J + I)! (J/2)! x (J/2 _ l)!(J /2 _ L)!(J/2 _ I')! (A.7)

The sum J = 1 + L + I' of the three angular momentum quantum numbers must be even. The Clebsch-Gordan coefficient (A.7) vanishes for odd J. Explicit expressions for the spherical harmonics up to I = 3 are given in Sect. 1.2.1 in Table 1.1. For further details see books on angular momentum in quantum mechanics, e.g. [Edm60,Lin84].

A.2 Laguerre Polynomials

The generalized Laguerre polynomials L~(x), v = 0, 1, ... are polynomials of degree v in x. They are given by

(A.8) A.3 Bessel Functions 303

and have v zeros in the range 0 < x < 00. The ordinary Laguerre polynomials LII(x) correspond to the special case 0 =O. In general 0 is an arbitrary real number greater than -1. The binomial coefficient in (A.S) is defined as follows for non-integral arguments:

( z) r(z+ 1) (A.9) y = r(y + 1) r(z - y + 1) . Here r is the Gamma function. It is defined by

oo r(z + 1) = L e e-t dt (A. 10)

and has the property

r(z + 1) = zr(z) (A.11)

For positive integers z = n we have r(n + 1) = n!. For half-integral z we can derive r(z) recursively from the value r(1/2) = ..fo via (A.ll). The orthogonality relation for the generalized Laguerre polynomials reads

oo -% a a a d r(v+o+ 1) c L e X L,.(x)LII(x) X = , V,.,II. (A.12) o ~ The following recursion relation is very useful, because it enables the numerically efficient evaluation of the Laguerre polynomials for a given index 0:

(v+1)L~+1(x) - (2v+o+1-x)L~(x) +(v+O)L~_l(X) = 0 , (A. 13) v = 1, 2, ...

Note: The Laguerre polynomials defined by (A.S) correspond to the definitions in [AS70, GR65 und M066]. The Laguerre polynomials in [Mes70] contain an additional factor r(v+o+ 1).

A.3 Bessel Functions

The Bessel functions of order v are solutions of the following second-order differential equation:

2J2w dw 2 2 z dz2 + z dz + (z - v )w = 0 (A. 14)

The ordinary Bessel function JII(z) is the solution which fulfills the following boundary condition near the origin z =0:

J (z) %;0 (!Z)" (vf= -1, -2, -3, ...) (A. 15) II r(v + 1) , 304 Appendix

As a power series in Z we have

(Z)V 00 (_lz2)k Jv(z) = i L k!r(~+k+ 1) (A.I6) k=O

For Izl --+ 00 the asymptotic form of Jv is

Jv(z) Izl~oo ifcos [z - (v+ 4) i] (A.I7)

From the Bessel function J v with the asymptotic behaviour (A.17) and a second solution of (A.14), which oscillates like a sine asymptotically, we can construct two linear combinations asymptotically proportional to exp ±i(z - ... ). They are called the first and second Hanke/ functions, H~l) and H~2). Their asymptotic form is

Hv(1) (z) Izl-+oo= y;-;{2 exp {.+1 [z- ( v+i1) '27r]} ' (A.18) (2) Izl-+oo 1) Hv (z) = y;-;(2 exp {.-1 [z - ( v + '2 "27r]}

Near z = 0 we have (for ~1I > 0)

H~l)(z) = _H~2)(z) = -2.. (~(1I)~, z --+ 0, ~1I > 0 . (A.19) 7r i Z

The modified Besse/functions Iv(z) of order 1I are connected to the ordinary Bessel functions by the simple relation

iVIv(z)=Jv(iz), (-7r

They are hence solutions of the differential equation

2 J2w dw 2 2 Z - + z- - (z + 1I )w = 0 (A.21) dz2 dz ' and their behaviour for small Izl is, as for Jv,

z-+o (!zV Iv(z) = r(lI+l) , (lIf-l,-2,-3, ...) (A.22)

For Izl --+ 00 the asymptotic form of Iv is

Izl~oo ~ I v ()z ;;;--- , (I arg(z) 1< 7r/2) (A.23) y27rz

For non-integral values of 1I the modified Bessel functions Iv(z) and Lv(z) defined by (A.20), (A.16) are linearly independent, and there is a linear combi• nation A.3 Bessel Functions 305

K ( ) = ~ Lv(z) - Iv(z) Z (A.24) v 2 S10.() V7r , which vanishes asymptotically,

1%1::'00 [7r -% K ( ) 37r/2) (A.25) v Z - V2z e , (Iargzl <

The Bessel functions of half-integral order v = 1+ 1/2, 1 = 0, 1, ... play an important role as solutions of the radial SchrOdinger equation (1.74) with angular momentum quantum number 1in the absence of a potential. The connection to the radial SchrOdinger equation becomes clear when we write the equations (A.I4) and (A.2I) as differential equations for the function

¢(Z) = v'Z w(z) . (A.26) (A.I4) then becomes (with v = I+!)

cP¢ _ 1(1 + 1) ¢ + ¢ = 0 (A.27) dz2 z2 ' and (A.2I) becomes

cP¢ _ 1(1 + 1) ¢ _ ¢ =0 (A.28) dz2 z2

If we write z as kr (for E=h2k2/(21-') > 0) or as ",r (for E= -h2",2/(21-') <0), then (A.27) or (A.28) respectively is just the radial SchrOdinger equation (1.74) for V == o. For the modified Bessel function K '+1/ 2 of half-integral order 1+ 1/2 there is a series expansion

K () [7r -% ~ (I + k)! -k 1+1/2 z = V2z e ~ k!(1 _ k)! (2z) . (A.29)

The derivative of K'+l/2 can be expressed in terms of K'+1/2 and K ,-l/2, d 1+ 1 -d K ,+1/2(Z) = ___2 K ,+1/2(Z) - K ,- 1/2(Z) (A.30) z z The spherical Bessel function j,(z) is defined as

j/(z) = IfJ ,+l/2(Z) . (A.3I) For small z we have, according to (A.I5),

• %--+0 zl J/(Z) = (21 + I)!! (A.32) 306 Appendix

and asymptotically according to (A.l7) . Izl-'oo. ( 7r) ZJl(z) = SIll Z -12 (A.33)

With (A.26) we see that zj,(z) is a solution of (A.27), the radial SchrOdinger equation for positive energy. The linearly independent solution, which differs asymptotically from (A.33) in that the sine is replaced by a cosine, is Z n,(z), where n, is the spherical Neumann function,

(A.34)

For the derivatives of the spherical Bessel functions we have the simple formula

d.() . () l+l.() -d )IZ =)1-1 Z ---liZ, 1 ~ 1 (A.35) Z Z

A.4 Whittaker Functions, Coulomb Functions

Whittaker functions appear as solutions of the radial Schrooinger equation in the form (A.28) when it contains an (attractive) Coulomb potential -2,/Z in addition to the centrifugal potentiall(l + 1)/ z2,

2 d

(A.37)

For positive energies (cf. (A.27» the radial SchrOdinger equation including a Coulomb potential has the form

(A.38) where a negative 1] corresponds to an attractive and a positive 1] to a repulsive Coulomb potential. Two linearly independent solutions of (A.38) are the regular F ,(1], z) G (1], z). Coulomb function and the irregular Coulomb function ' Their asymptotic (z -+ +00) behaviour is

z-.oo . ( F'(1],z) = SIll z -1] In2z -l27r) +0"1 (A.39) G,(1], z) z-.oo= cos ( z -1] In2z -127r) + 0"1 References 307

The constants 171 are the Coulomb phases,

171 = arg r(l + 1 + i1]) . (A.40)

The regular Coulomb function can be expressed in terms of the confluent hypergeometric series,

E', (1J z) = 21 e-~ 11"'1 Ir(l + 1 + i1])1 e-iz zl+1 F(l + 1 - i1] 21 + 2' 2iz) (A.41) 1 , (21 + 1)! ' , .

The confluent hypergeometric series F is defined by

~ r(a + n) r(b) zn (A.42) F(a, b; z) = ~ r(a) r(b + n) n!

For small arguments z (and fixed Coulomb parameter 1]) we have

E',( ) z=:O i -t1l"'1 Ir(l + 1 + i1])1 1+1 1 1], z - e (21 + 1)! z (A.43)

For 11]1 -t 00, which corresponds to approaching the threshold according to (1.117), we have

(A.44)

In order to obtain a formula for the regular Coulomb function of small argument z = kr close to threshold we combine (A.43) and (A.44) to

E',( k) k-+°kT -+O J 7r (2kl1]1 r)l+l -t1l"('1+I'1il (A.45) 1 1], r 211]1 (21 + 1)! e

References

[AS70] M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1970. [Edm60] A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, 1960. [Lin84] A. Lindner, Drehimpulse in der Quantenmechanik, Teubner, Stuttgan, 1984. [GR65] I.S. Gradshteyn and I.M. Ryzhik, Tables ofIntegrals, Series and Products, Academic Press, New York, 1965. [Mes70] A. Messiah, Quantum Mechanics, vol. I, North Holland, Amsterdam, 1970. [M066] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theoremsfor Special Functions of Mathematical Physics, Springer-Verlag, Berlin, Heidelberg, 1966. Subject Index

above threshold ionization (ATn 251f bound states in the continuum 140,159 absorption (of particle flux) 206 bra 3 absorption cross section 206, 217 bra-ket notation 3 absorption of photons 106, 111, 127, 249 branching ratios 161 ac Stark effect 184,185,256,257 break-up amplitude 234ff action 45ff, 118f, 124, 292, 293 break-up channel 231ff action-angle variables 285 Breit-Wigner resonance 36 amplitude see also break-up amplitude, partial Brillouin's Theorem 81 wave amplitudes, scattering amplitude, spin• flip amplitude 268 canonical equations (of classical mechanics) angular momentum see also orbital angular 260 momentum, total angular momentum 49ff centre-of-mass coordinate 59 angular momentum quantum number 10, 64, centrifugal potential 13,14,17,21,23,39,118, 66,119,126,185,204,222,269,305 119 annihilation operator 101,265 channel see also closed channel, coupled anomalous Zeeman effect 174, 176 channels, eigenchannels, equivalent chan• anti-particle 63 nels, Landau channels, open channel anticommutation relations 62 30,129,215f antisymmetric wave functions 70 channel spin 134 antisymmetrizer 71,130 channel thresholds 32, 146ff, 226f associated Legendre functions 301 channet'wave functions 30,35,130,216ff asymmetric coplanar (e,2e) reaction 241 chaos 273 atomic units 60,168,177 chaotic classical dynamics 180, 274ff autoionizing resonances, states 134ff, 146ff, Clebsch-Gordan coefficients 51,54, 109, 110, 157ff,179 126, 174,302 averaging over initial states 108, 125,214, 226 close coupling 129ff azimuthal quantum number 10, 14f, 75,126, close-coupling equations 133, 207 166,170,174,178,195 close-coupling expansion 205,229 closed channel 32 background phase shift 34 coherent states 264ff,295 Baker-Campbell-Hausdorff relation 266,295 commutator 6,261ff basis 2 commuting operators 6 Bessel functions 304 compatibility equation (in MQDT) 151,157 Bessel functions of half-integer order 305 complete set of commuting observablcs 6 Bethe theory 220 completeness relation 4 Beutler-Fano function 137,149,159-161 configuration 77,80 bifurcation 290 configuration interaction (Cn 85ff billiard 296 confluent hypergeometric series 200, 307 Bohr magneton 173 constants of motion 9,76,80 Bohr radius 18,22,24,60,61,219 continuity equation 193, 206 Bohr-Sommerfeld quantization 47,57,119 continuous effective quantum number 122, Born approximation 195, 197, 218, 220, 240f 125,147,149,152,153,156,161 bosons 71 continuum see also quasi-continuum 27 bound states 9,14 continuum threshold 27, 32, 48, 117, 123, 124 310 Subject Index continuum wave fWlCtions 9,231 dipole polarizability 165, 187, 198 coordinate representation 7,261 dipole transitions 109 coordinates see cylindrical coordinates, hyper• dipole transitions, magnetic 110 spherical coordinates, parabolic coordinates, dipole transitions, selection rules for 108 scaled coordinates, spherical coordinates Dirac equation see also radial Dirac equation core electrons 88, 130 63,66,173,188 Coulomb eigenfunctions 27,56 Dirac equation, radial 65 Coulomb functions, energy normalized 29 Dirac's Hamiltonian 62,84 Coulomb functions, irregular 21,306 Dirac-Fock method 85 Coulomb functions, regular 21, 27, 204, 228, direct potential 82, 133 306 dispersion (of wave packet) 263,269ff Coulomb gauge 98 displacement operator 6 Coulomb parameter 21,220,240 distorted waves 203,221 Coulomb phases 22,204, 307 distorted-wave Born approximation (DWBA) Coulomb potential see also modified Coulomb 201ff, 221, 240 potential 24ff, 39, 65, 200£, 220, 245, 270 double scattering experiments 215 Coulomb potential, radial eigenfunctions in a downhill equation 169f 25 dressed states 136 Coulomb principal quantum number 24,171 DWBA (distorted wave Born approximation) Coulomb scattering amplitude 201 201ff, 210, 221, 240 Coulomb shell 24 Coulomb wave for three particles 240f (e, 2e) reactions 231ff coupled channel equations 31ff,133,138,156, effective Hamiltonian 206 179,215,222,229 effective one-body Hamiltonian 82 coupled channels 29,129f effective potential 13,206,216 coupling operators 31 effective quantum number see also continuous creation operator 101,265 effective quantum number 118,127,146, cross section see also differential cross section, 161 integrated cross section, total cross section effective range theory 198 107,111,112, 127ff, 206, 224, 226 effective SchrOdinger equation 205 cross section for photoabsorption, photo- eigenchannels 152,224 ionization 107,111,112, 127ff eigenphases 152,224, 229f current density 192,236 eigenstate 4 cusp 227 eigenvalue 4 cylindrical coordinates 164, 176ff Einstein coefficients 106 cylindrical principal quantum number 177 elastic scattering cross section 206, 216, 220, 226 Darwin term 68, 113 electric field, atoms in a 164ff degeneracy, additional 15,24 electric field strength in atomic units 168 degenerate eigenstates 5,166,187ff electromagnetic field, quanti7.ation of 98 degenerate eigenvalue 4 emission, induced 105 degenerate multiplet 108 emission, spontaneous 98, 104 density functional 92f energy see internal energy, kinetic energy, density matrix 213,225,294 potential energy, quasi-energy, reduced density of final states 96 energy, scaled energy density operator 213,261,262 energy normalized (continuum) states 28f,35, diagonalizing in a subspace 44ff 97,135,146,197,204 diamagnetic term 176ff energy shell 277 differential cross section see also triple dif• ensemble 213 ferential cross section 192,196,199,201, ensemble of random matrices 281 203,208,211,214,216,226,228,246 equivalent channels 131 dipole approximation 103, 104, 11 0, 182, 253, equivalent integral equation see also Lipp• 268 mann-Schwinger equation 193f, 202, 232 dipole operator, electric 104 excess photon ionization (EPI) 251f Subject Index 311 exchange potential 83,133 Gutzwiller's trace formula 290 exothermic reaction 227 gyromagnetic ratio 173,175 expectation value 5, 8,42 expectation value, statistical 213 half waves 47 exponential divergence of classical trajectories Hamiltonian (operator) see also Dirac's Hamil• 273ff tonian, effective Hamiltonian, internal Hamiltonian, non-Hermitian Hamiltonian 7,8,26lf Fenni energy 88 Hamiltonian function (of classical mechanics) Fenni momentum 88 259,260 Fermi sphere 88 Hankel function 233,304 Fermi wave number 88 harmonic oscillator 14ff, 100,264,269,295, fennions 71 296 Feshbach resonance 30, 32f, 135, 144,230 harmonic oscillator ground state 264,295 Feshbach's projection formalism 205ff Hartree-Fock equations 82, 84 field gauge 183,257 Hartree-Fock method 81ff field ionization 167,170 Hartree-Fock potential 129,207 field-free threshold 171 Heisenberg picture 8 fine structure 65f Heisenberg's uncertainty relation 6,97,267 fine structure constant 65, 108 helium see ortho-helium, para-helium Floquet states 183,257f helium iso-electronic sequence 86f four-component spinors 62 Helmholtz equation 233,246 Fourier transformation 261ff Hermitian conjugate operator 3 Fourier-transformed spectrum 293 Hermitian operator 3,5,7 free-particle Green's function 193 Hilbert space 2,5 frequency-dependent polarizability 185 Hohenberg-Kohn theorem 92 functional 42,92 Hund's rules 78 hydrogen atom 24,59f Gamma function 22,303 hydrogen atom in a magnetic field I 77ff, gauge see Coulomb gauge, field gauge, Lan• 287ff,297 dau gauge, symmetric gauge, transverse hydrogen atom in a microwave field 285ff gauge, radiation gauge hydrogenic atom in an electric field I 67ff gauge transformation 98,188 hydrogenic ion 61 Gaussian orthogonal ensemble (OOE) 281 Hylleraas-Undheim theorem 45 Gaussian unitary ensemble (GUE) 281 hyper-angle 234,236f Gell-Mann-Goldberger decomposition 203, hyper-radius 235f 210 hyperfine interaction 77 generalized eigenvalue problem 133 hyperfine structure 68 generalized Laguerre polynomials 14,25,302 hypergeometric series see confluent hyper• generalized oscillator strengths 220 geometric series generator (of a symmetry transformation) II hyperspherical coordinates 235,247 generalized spherical harmonics 54f, 64, 208 GOE (Gaussian orthogonal ensemble) 281 impulse approximation 243 GOE statistics 290 incoherent superposition 213 Golden Rule 96, 186, 217,244 induced emission 105f good quantum number 9,79,110, 22lf, 290 inelastic scattering cross section 216, 219f, Green's function 33ff,56,202,217,23Iff, 226,229 240,246,253 initial states, averaging over 108, 125,214, Green's operator 193 226 ground state 9 instability (of a classical trajectory) 274,292 ground state of harmonic oscillator 264 integrable limit 275 group 11,71 integrable system 277 GUE (Gaussian unitary ensemble) 281 integrated cross section 192, 198, 217, 227, GUE statistics 290 229,244 312 Subject Index interference of resonances 138ff, 158ff Lippmann-Schwinger equation 194, 197, 217, interference of wave packets 271 232,235 internal energies 32, 132, 134 local momentum 46 internal Hamiltonian 60, 23lf, 235 local potential 82 internal states, wave functions 13Off, 216, long ranged potential 48,210,225,227, 238ff 221, 225, 231ff LS coupling 80,110,174 internal variables 130ff Lu-Fano plot 153ff invariance with respect to symmetry transfor• mations 11 inverted multiplet 79 magnetic field strength parameter 177 magnetic field, atoms in a 17Iff,287[f ionization see above threshold ioni7.ation, excess photon ionization, photoionization magnetic field, free electrons in a 176,188 irregular dynamics 273ff magnetic moment 172 irregular solution of the radial Schrooinger magneton 173 Maslov index 47,292 equation 23 mass polarization term 69, 115 iso-electronic sequence 62, 86 matrix see K -matrix, R-matrix, S-matrix, T -matrix, density matrix, random matrices, j j coupling 80 reactance matrix, scattering matrix, stability matrix K -matrix 222 matrix norm 274,295 Keldysch approximation 253ff matrix of an operator 5 Kepler ellipse 270ff, 288 MCDF (multi-configuration Dirac-Fock ket 3 method) 86 KFR (Keldysch-Faisal-Reiss) theory 255 MCHF (multi-configuration Hartree-Fock kicked rotor 275f, 279f method) 85 kinetic energy 7 mean field 74 kinetic momentum 98 mean level density 282 mean mode number 28lf I-mixing 180 mean oscillator strengths 125ff,186 Laguerre polynomials see also generali7.cd mean single-particle potential 74f Laguerre polynomials 302f microwave field 285f Lamb shift 66 minimal wave packet 267 Landau channels 178 mixed states 210, 213 f, 225, 261 Landau gauge 188 mode (of electromagnetic field) 99ff, 268f Landau states 177, 292 mode number 28lf Land~ factor 175 modified Bessel functions 18,304 Langer modification 124 modified Coulomb potential 20 Iff, 210 large components 64,67,188 momentum see also kinetic momentum, local Legendre functions see associated Legendre momentum, uncertainty in momentum 6 functions momentum of relative motion 59 Legendre polynomials 195,300 momentum operators 6, 7 level density 282 momentum representation 26lf Levinson's theorem 38, 199f momentum space see also wave function in Liapunov exponent 274,290, 295f momentum space 262 lifetime 97 Morse index 292 lifetime of an atomic state 105,114,167 Mott formula 246 light, non-classical 269 MQDT (multichannel quantum defect theory) light, speed of 62 146ff,228 line shapes 143 MQDT equation 155,156 linear operator 3,52 multi-configuration Dirac-Fock method linear Stark effect 165 (MCDF) 86 Liouville equation (of classical mechanics) multi-configuration Hartree-Fock method 260,261,268,294 (MCHF) 85 Subject Index 313 multichannel quantum defect theory (MQDT) oscillator width 14,55,264, 267 146ff,225 overlap 40, 55f, 72 multiphoton ionization 249ff overlap matrix 44, 130 multipole expansion 133 multipole matrix elements 133 para-helium 78 multipole moments 133 parabolic coordinates 167f paramagnetic interaction 176 parity see also z-parity 12,25,79,109,165 n-dimensional sphere 247 partial wave amplitudes 196,204,206,209, natural line width 105 211,245 nearest neighbour spacings (NNS) 282f partial waves 195, 203, 244 NNS (nearest neighbour spacings) 282f partial waves expansion 204, 206, 209,210, NNS distributions 283, 290f 221,224,226,228,245 non-classicallight 269 partial widths 161 non-Hennitian Hamiltonian 206 Paschen-Back effect 175f nonlocal potential 83 Pauli principle 71,78,129,198 norm of a wave function 2 Pauli spin matrices 52,63,112,212,245 norm of a matrix 274,295 periodic classical orbit 290ff normal Zeeman effect 173 periodic table of elements 75f nuclear spin 68 permutations 70f perturbation theory see also time-dependent perturbation theory 39ff,l64ff 174,184, observable 5 250 odd parity 79 perturbation theory for degenerate states 41, one-dimensional harmonic oscillator 264f 165,187 one-particle-one-hole excitation 73,81 perturbed Rydberg series 143,186,187 open channel 32 perturbed Rydberg series of autoionizing re• operator see annihilation operator, commuting sonances 158ff operators, creation operator, density operator, Peterkop theorem 238 dipole operator, displacement operator, phase shift see also eigenphases, Coulomb Hermitian conjugate operator, Hermitian phases 200,35, 123f, 147ff, operator, linear operator, reduced operator, 156ff, 196ff, 204ff, 226ff time evolution operator, unitary operator phase shifted reactance matrix 152,155 optical potential 206, 226 phase space 47,88,259, 273f optical theorem 193,244 phase space density 260,294 orbit, classical periodic 292f phase space factor 217,226,234 orbital angular momentum 9,12,113,221 phase space trajectory 259 orbital angular momentum quantum number photoabsorption, photoionization l06f,lllf, 64,66,74,110,117,208 125f,136f,141,145,16Of,171 ordinary Bessel functions 27,303 photons 101,127, 249ff, 269 ortho-helium 78 Poincar~ surface of section 278, 289 orthogonality of wave functions 2 Poisson bracket 260f orthogonality relation for generali7.ed Laguerre Poisson spectrum 28Off, 296 polynomials 303 Poisson statistics 290 orthonormal basis 2f polarizability see dipol-polarizability, orthonormality relation for spherical harmonics frequency-dependent polarizability 10,302 polarization of electromagnetic radiation 99, oscillator see harmonic oscillator, radially sym• 105 metric harmonic oscillator, one-dimensional polarization of electrons 211ff,225f harmonic oscillator polarization vector 212,245 oscillator shells 15f ponderomotive energy 254ff oscillator strengths see also generalized oscil• ponderomotive force 256 lator strengths, mean oscillator strengths, position variables 7 sum rules 111,136,141,145,160 positron 242, 243 314 Subject Index post-diagonalization 45 radially symmetric harmonic oscillator l4f potential see Coulomb potential, direct poten• radiation gauge 98, 182,252 tial, effective potential, exchange potential, radiative corrections 87 local potential, long ranged potential, non• random matrices 28Of, 284 local potential, optical potential, radial reactance matrix see also phase shifted potential, scalar potential, short ranged reactance matrix 152, 222f, 225 potential, vector potential recursion relation for Laguerre polynomials potential energy 7,9,263 303 pre-diagonalized states 41 reduced energy 137,148,149 principal quantum number 15,25,65,66,127, reduced mass 60, 69 171,180,185,270 reduced matrix element 109,174 principal quantum number, cylindrical 177 reduced operator 31,43 probability 5, 94ff, 208, 213, 255[f reference potential 22, 39 probability density 2, 260, 262, 294, 296 regular ground state multiplet 79 projection operator 4,71,205,213 regular solution of the radial Schrooinger equa- pseudo-resonant perturbation 145,149 tion 19,23,195 pseudostates 207,229 regularity (of classical dynamics) 276ff,284 pure states 210,213,225,261,262 relative distance coordinate 59 relative momentum 59 relativistic corrections to the Schrooinger equa- q-reversal 161,162 tion 66ff, 85, 87,113 QDT (quantum defect theory) 124,143 relativistic energy-momentum relation 62 quadratic Stark effect 165,166 relativistic (Dirac) Hamiltonian 62 quadratic Zeeman effect 177ff renormalization of the Hamiltonian 101 quadrupole transitions 110 renormalized radial wave functions 26,29, quantization see also Bohr-Sommerfeld quan• 127,145,146 tization 292 representation 6 quantization of the electromagnetic field see also coordinate representation, 99ff momentum representation, standard quantum defect function 121ff, 143, 147 representation quantum defect theory (QDn 124,143 residual antisymmetrizer 130 quantum defects 118,121,123,124,146,153, residual interaction 74 185,204 resonance see also autoionizing resonances, quantum fluctuations 269 Breit-Wigner resonance, window resonance, quantum number see angular momentum quan• Rydberg series of autoionizing resonances tum number, continuous effective quantum 3Off, 229, 280 number, Coulomb principal quantum num• resonance position 35,140, 146, 154, 159 ber, effective quantum number, good quan• rest energy 64,65,67 tum number, principal quantum number restricted Hartree-Fock method 84 quasi-continuum 124 retardation 85 quasi-energy 183,257f revival 272 quasi-Landau modulations 294 rotor see kicked rotor quasiperiodic motion 276,285 rounded cusp 227 Runge-Lenz vector 270 R-matrix 222 Russel-Saunders coupling 80 R-matrix method 163 Rutherford formula for the elastic scattering radial Dirac equation 65 cross section 201,219,246 radial eigenfunctions (in a Coulomb potential) 25 Rutherford scattering amplitude 201 radial Lippmann-Schwinger equation 197 Rydberg energy 24,60,61,117 radial potentials 221 Rydberg formula 118 radial Schrooinger equation 12f,16ff,23,47, Rydberg series see also perturbed Rydberg 55,195L208,210,305f series 118ff, 228 radial wave functions see also renormalized Rydberg series of autoionizing resonances radial wave functions 146ff,228,157ff 12f, 55f, 65,124,195, 222f Rydberg series of perturbers 153 Subject Index 315

S-matrix 223ff spin matrices see Pauli spin matrices scalar potential 98 spin-flip amplitude 208ff scalar product 1,2,52,183 spin-orbit coupling 53ff, 68, 77, 113, 175, 178, scaled action 292 207ff scaled coordinates, momenta 288 spinors 52,62ff,210,212,245 scaled energy 288, 292,297 spontaneous emission 98,I04ff scattering amplitude 192, 201, 209, 220, 224f, square well potential 17,20, 22ff 227,246 squeezed states 268f scattering channels 231 stability (of a classical trajectory) 274,292 scattering cross section see also differential stability matrix 274,295 cross section, elastic scattering cross sec• standard mapping 276 tion, inelastic scattering cross section 192, standard representation 63 219 Stark effect 165ff scattering length 198 Stark saddle 167 scattering matrix 223 static exchange potential 207 scattering plane 215 stationary (time-independent) Schr&linger SchrOdinger equation see also effective Schro.• equation 9,39,46,164,205,207 dinger equation, radial Schr&linger equation, statistical ensemble 213 stationary Schr&linger equation, time• strong coupling 154 dependent Schr&linger equation, time• Sturm-Liouville basis 86 independent Schr&linger equation 7ff subshells 75 Schr&linger picture 7, 8 sum rules l1Off, 126ff, 185 Seaton's formulation of MQDT 152 summation over final states 108,125,226 Seaton's theorem 123,205 superelastic scattering 227 secular equation 41,44 surface of section 278 selection rules 108ff symmetric gauge 171,176,187 self-consistency 84 symmetric wave functions 70 self-energies 83 symmetry group 12 semiclassical approximation 45ff,l24 shape parameter 137,142,148, l60f shell structure of atoms 74ff T-matrix 194 Sherman function 211,215 Theta function 281 shift in resonance position 36 Thomas-Fermi model 88ff short ranged potential 16,19,119,191,202, Thomas-Reiche-Kuhn sum rule 112,129 207 three-particle Coulomb wave 242 Sianai's billiard 296 threshold see also channel thresholds, ficld- single-particle density 82 free threshold, Wigner's threshold singlet state 78, 239 law 123,226ff Slater determinants 72,81 time evolution of coherent and squee7.ed Slater-type orbitals 86 states 266ff small components 64,67,188 time evolution operator 8,279 spatial coordinates 7 time-dependent perturbation theory 94ff,194, spectator modes 103 216,244 spectral rigidity 283f, 290f time-dependent Schr&linger equation 94,183, speed of light 62 253,257,261,279 sphere, n-dimensional 247 time-independent (stationary) SchrOdinger spherical Bessel functions 21,196,305 equation 9,39,46,193 spherical coordinates 10 time-independent Dirac equation 63 spherical harmonics see also generali7,ed torus 277 spherical harmonics 10,108, 30lf total angular momentum 76, 134, 208, 221 spherical Neumann functions 21,197,306 total angular momentum quantum number 76, spherical vector components 108,126,166 110,174,215,221 spin see also total spin, channel spin 5lff,78, total cross section 217 173ff, 207ff, 225f, 245 total elastic scattering cross section 192 316 Subject Index total inelastic scattering cross section 217, vanishing width (of resonance) 159,161 226f variation after diagonalization 45 total momentum 59 vector operator 6,109 total orbital angular momentum 76,134,172, vector potential 98,163,171,I87f,252,269 174f Volkov states 254 total spin 76 von-Neumann equation 261ff,268 trace formula (Gutzwiller's) 290 trajectory (classical) 259,273,276 wave function 1 transition amplitude 95,216,218,226,228, wave equation 99 253 wave function in momentum space 261, 294 transi tion operator 194, 218 wave number 19,216,220 transition probability 96,244 wave packet see minimal wave packct transverse gauge 98 wave vector 99,219,233f,24O triangle condition 51,54,77,109,110 weak coupling 154 triple differential cross section 237,239,242 Whittaker functions 18,306 triplet state 78, 239 width see oscillator width, natural line width two-channel quantum defect theory 151 width of a resonance 36,135,140,146,154, two-component spinors 52, 63, 210, 212, 245 158,167f two-electron excitations 134 wiggling 254 two-particle-two-hole excitation 74 Wigner distribution 283, 290 Wigner function 262f, 266, 268, 294, 295 uncertainty in momentum 7,267 Wigner representation of the von Neumann uncertainty in position 7,267 equation 263 uncertainty relation 6, 97,267 Wigner's threshold law 226 unit operator 5 Wigner-Eckart theorem 100f, 174, 198,302 unitary operator 8, 10 window resonance 138 unitary transformation 10,281 WKB method 46,57 unrestricted Hartree-Fock method 84 unstable (classical) dynamics 274 uphill equation 168f z-parity 178, 180 uphill potential 168f Zeeman effect 173ff uphill quantum number 170,171 zero-field threshold 179