Appendix

A. The Dirac Delta Function and the Normalisation of the Wavefunction of a Free Particle in Unbounded Space

The English physicist Dirac introduced a function which is extremely useful for many purposes of theoretical physics and mathematics. Precisely speaking, it is a generalised function which is only defined under an integral. We shall first give its definition, and then discuss its uses. The delta function (J function) is defined by the following properties (x is a real variable, - 00 ~x ~ 00):

1) J(x) = 0 for X::j:: 0 , (A.1)

b

2) JJ(x)dx=1 for a<O<b. (A.2) a

The J function thus vanishes for all values of x ::j:: 0, and its integral over every interval which contains x = 0 has the value 1. The latter property means, speaking intuitively, that the J function must become infinitely large at x = O. The unusual properties of the J function become more understandable when we consider it as the limiting case of functions which are more familiar. Such an example is given by the function

which is shown in Fig. A.1.

1 x2 --exp(--) I{ffu u2

-u u x

(A.3)

Fig. A.I. The function (1/Vnu) exp( _x2Iu 2)

plotted against the variable x. If we let the para­meter u become smaller, the value of the func­tion at x = 0 gets larger and larger and the de­crease to both sides gets steeper, until the func­tion has finally pulled itself together into a <5 function

376 Appendix

If we allow u to go to zero, the function becomes narrower and higher until it is finally just a "vertical line" . We thus have

for x =1= 0: lim _1_ e _x2!u 2 = 0 . U~O Vnu (A.4)

On the other hand, one can find in any integral table the following fact:

+ 00 1 J -x2!u2d - 1 --e x--00 Vnu ' (A.5)

independently of the value of u. If the limit u -+ 0 is calculated it becomes clear that because of (A.4), we can write the integral (A.5) with any finite limits a and b with a < 0 < b without changing its value. This is just the relation (A.2).

In many practical applications in quantum mechanics, the <5 function occurs as the following limit:

j;( ) _ I' 1 sin ux uX - lm----.

u~oo n x

sin(ux) n;x

x

(A.6)

Fig. A.2. Thefunction sin (ux)/ nx plotted against x. If we allow u to go to the limit infinity, the value of the function at x = 0 becomes larger and larger. At the same time, the position of the zero­crossing moves towards x = 0

The property (A.2) is found to be fulfilled when we take into account that

+r ~ sin(ux) dx = 1 . (A.7) -00 n x

The property (A.1) is not so obvious. To demonstrate it, one has to consider that for u -+ 00, x =1= 0, sin(ux)/x oscillates extremely rapidly back and forth, so that when we average the function over even a small region, the value of the function averages out to zero (Fig. A.2).

The <5 function has, in particular, the following properties:

A. The Dirac Delta Function and the Normalisation of the Wavefunction 377

for a continuous function f(x) ,

b S f(X)t5(x-xo)dx =f(xo), a < Xo < b is valid. (A.8) a

For a functionf(x) which is n times continuously differentiable,

b Sf (x) t5(n)(x-xo)dx = ( -1tj<n)(xo) , a < Xo < b (A.9) a

holds. Herej<n) and t5(n) mean the nth derivatives W.r.t. x. The proof of (A.8) follows immediately from (A.1, 2). The proof of (A.9) is obtained by n-fold partial integration. Furthermore,

1 t5(ex) = - t5(x) , e real

lei (A.10)

is valid. The relation (A. 1) is seen to be fulfilled on both sides. If we insert (A.10) in (A.2), we find

b

S t5(ex)dx, a

which, after changing variables using ex = x', becomes

b' 1 S -t5(x')dx', a' :;:: b' for e:;:: 0, a' e

which is thus, according to (A.2), equal to 11 I e I. We now turn to the question of the normalisation of wave functions in unbounded

space, where we can limit ourselves to the one-dimensional case without missing any essentials. We start with wavefunctions which are normalised in the interval L,

(A.11)

for which we have the normalisation integral

L12

S IIf/(x) 12dx = 1 . (A.12)

-L12

If we furthermore assume that If/(x) is periodic, If/(x+ L) = If/(x), the k's must have the form

k = 2nn 0 1 2 L ' n= ,± ,± , ... (A.13)

It is easy to convince oneself that

378

Ll2

I 'II,/(x) 'IIdx) dx = Ok,k' -Ll2

= {~ for k = k'

for k * k' .

Appendix

(A.14)

(A.15)

To prove this relation, the integral must be computed, taking account of (A.13). The integral yields

1 Ll2 1 - I e-ikx+ik'Xdx= (exp[i(k'-k)LI2]-exp[-i(k'-k)LI2]. L -Ll2 iL(k'-k) (A.16)

If we now abbreviate k' - k with'; and LI2 with U, we may write (A.16) in the form

sin (.; u)1 .; u . (A.17)

This is, however, apart from the factor 2 niL, just the function which appears in (A.6) on the right under the limit, if we identify'; with x. If we thus divide (A.16) by 2 niL and form lim, we obtain on the left side of (A.16)

L-->oo

1 Ll2 - lim I exp( - ikx+ ik'x)dx, 2n L-->oo -Ll2

which we may also write somewhat differently:

+( [_1 _ eikx] * [_1 _ eik'X] dx. - 00 V21T V21T

(A.iS)

(A.19)

The right-hand side of (A.16), using (A.17) and (A.6), goes to o(k'-k). We thus finally obtain

+00

I 'II'/(X) 'IIk'(X) dx = o(k' - k) , (A.20) - 00

where

1 Ok 'IIk(X) = --e' x.

V21T (A.2i)

Equation (A.20) with (A.21) generalises the relation (A.14) [with (A.i1)] to the case of wavefunctions without finite boundary conditions and thus to the corresponding case of continuous k values. As may be seen in all practical applications, the 0 function in (A.20) always occurs under further integrals over k or k' (or both), so that we have found a self-consistent formalism.

Let the wave functions in (A.21) depend not upon k, but uponp = hk; then we must observe (A.l0). In order to normalise the new wavefunctions

B. Some Properties of the Hamiltonian Operator, Its Eigenfunctions and Its Eigenvalues 379

correctly, we must set N equal to (i/l/h) (i/V21l) = (i/yh). The normalised wavefunc­tion is now given by

B. Some Properties of the Hamiltonian Operator, Its Eigenfunctions and Its Eigenvalues

We write the time-independent Schrodinger equation in the form

with the Hamiltonian

112 £ = - 2m \7 2+ V(r) , V(r) real.

(B.i)

The IfIn(r) are square-integrable eigenfunctions with the eigenvalues En. Here, IfIn = 0 is excluded. The eigenvalues En may be discrete or they may be continuous.

In the following, we denote by ifill and IfIv the wavefunctions on which the operator £ can act. We can now easily read off the following properties:

a) £ is a linear operator, i.e. the relation

holds, where cll and cvare some complex numbers. In particular, it follows from this that every linear combination of eigenfunctions of £ with the same eigenvalue E is itself an eigenfunction of £ with the eigenvalue E.

b) £ is Hermitian, i.e. the equation

(B.2)

is valid. It follows from (B.2) that for the operator of the potential energy, V*(r) = V(r). For the kinetic energy operator, (B.2) can be proved by double partial integration, taking into account the fact that the wavefunctions vanish at infinity.

c) The eigenvalues En are real. This is a consequence of (B.2), if one inserts for ifill and IfIv the same eigenfunction IfIn and utilises (B.i).

d) Eigenfunctions with different eigenvalues are orthogonal.

We take the following scalar products (different eigenvalues belong to the functions IfIm and IfIn):

380

J If/;!',(r) (£If/n(r)] dV = En J If/;!',(r) If/n(r) dV,

H£lf/m(r)]* If/n(r)dV = E;!',J If/;!',(r) If/n(r)dV.

Appendix

(B.3)

(B.4)

We subtract (B.4) from (B.3) and use (B.2) and the property that the eigenvalues are real:

The second bracket indicates the orthogonality of the wavefunctions. Furthermore, it can be shown that eigenfunctions with the same eigenvalue may

always be chosen to be orthogonal by using appropriate linear combinations.

Bibliography of Supplementary and Specialised Literature

1. General Physics Textbooks

M. Alonso, E. J. Finn: Fundamental University Physics (Addison-Wesley, Reading 1972)

R. P. Feynman, R. B. Leighton, M. Sands: Lectures on Physics (Addison-Wesley, New York 1980)

D. Halliday, R. Resnick: Physics I and II, 3rd ed. (Wiley, New York 1978) J. N. H. Hume, D. G. Ivey: Physics 1 and 2 (Wiley, New York 1974)

2. Atomic Physics Textbooks

M. Born: Atomic Physics (Hafner, New York 1966) U. Fano: The Physics oj Atoms and Molecules; an Introduction to the Structure oj

Matter (University of Chicago Press, 1972) M. Karplus, R. M. Porter: Atoms and Molecules: an Introduction jor Students oj

Physical Chemistry (Benjamin, New York 1970) R. L. Sproull, W. A. Phillips: Modern Physics: the Quantum Physics oj Atoms,

Solids, and Nuclei (Wiley, New York 1980)

3. Quantum Mechanics Textbooks

J. Avery: Quantum Theory oj Atoms, Molecules, and Photons (McGraw-Hill, New York 1972)

H. A. Bethe, E. F. Salpeter: The Quantum Mechanics oj One- and Two-Electron Atoms (Plenum, New York 1977)

A. Bl>hm: Quantum Mechanics (Springer, Berlin, Heidelberg, New York 1979) C. Cohen-Tannoudji, B. Diu, F. Laloe: Quantum Mechanics I and II, 2nd ed. (Wiley,

New York 1977) D. S. Davydov: Quantum Mechanics, 2nd ed. (Pergamon, Oxford 1976) R. Eisberg, R. Resnick: Quantum Physics (Wiley, New York 1974) L. D. Landau, E. M.Lifshitz: Quantum Mechanics, 3rd ed. (Pergamon, Oxford 1982) J. L. Martin: Basic Quantum Mechanics (Oxford University Press, Oxford 1981) E. Merzbacher: Quantum Mechanics, 2nd ed. (Wiley, New York 1970) A. Messiah: Quantum Mechanics I and II (Wiley, New York 1958) R. Shankar: Principles oj Quantum Mechanics (Plenum, New York 1980) E. H. Wichmann: Quantum Physics (Vol. IV of Berkeley Physics Course) (McGraw­

Hill, New York 1971)

382 Bibliography

4. Specialised Literature

Chapter 1

J. L. Heilbron: Elements oj Early Modern Physics (University of California Press, Berkeley 1982)

B. L. van der Waerden (ed.): Sources oj Quantum Mechanics (Dover, New York 1967)

Chapter 2

S. G. Brush: Statistical Physics and the Theory oj Atomic Spectra (Princeton University Press, Princeton 1983)

J. W. Gibbs: Statistical Mechanics (Oxbow Press, Woodbridge 1981) Sir J. Jeans: Introduction to the Kinetic Theory oj Gases (Cambridge University Press,

Cambridge 1982) C. Kittel, H. Kroemer: Thermal Physics (Freeman, San Francisco 1980)

Chapter 3

P. Richard (ed.): Atomic Physics: Accelerators (Academic, New York 1980) S. Villani (ed.): Uranium Enrichment (Springer, Berlin, Heidelberg, New York 1979)

Chapter 4

M. A. Preston, R. K. Bhaduri: Structure oj the Nucleus, 2nd ed. (Addison-Wesley, New York 1975)

E. Segre: Nuclei and Particles, 2nd ed. (Benjamin, Reading 1977)

Chapter 5

R. P. Feynman, R. B. Leighton, M. Sands: Lectures on Physics (Addison-Wesley, New York 1980)

L. D. Landau, E. M. Lifshitz: Statistical Physics, 3rd ed. (Pergamon, Oxford 1982)

Chapter 6 Atomic Physics Textbooks

Chapter 7 Quantum Mechanics Textbooks

Chapter 8

R. W. Dixon: Spectroscopy and Structure (Methuen, London and Wiley, New York 1965)

J. L. Heilbron: Elements oj Early Modern Physics (University of California Press, Berkeley 1982)

H. G. Kuhn: Atomic Spectra, 3rd ed. (Longmans, London 1969) A. Sommerfeld: Atombau und Spektrallinien, 4th ed. (Vieweg, Braunschweig 1924) B. L. van der Waerden (ed.): Sources oJ Quantum Mechanics (Dover, New York 1967) H. E. White: An Introduction to Atomic Spectra (McGraw-Hill, New York 1954)

Chapters 9, 10 Quantum Mechanics Textbooks

Chapters 11, 12

H. G. Kuhn: Atomic Spectra, 3rd ed. (Longmans, London 1969) H. E. White: An Introduction to Atomic Spectra (McGraw-Hill, New York 1954)

Bibliography 383

Chapters 13, 14

A. Abragam: The Principles oj Nuclear Magnetism (Oxford University Press, Oxford 1961)

R. McWeeny: Spins in Chemistry (Academic, New York 1970) C. P. Slichter: Principles oj Magnetic Resonance, Springer Ser. Solid State Sci., Vol. 1,

2nd ed. (Springer, Berlin, Heidelberg, New York 1978)

Chapter 15

N. F. Ryde: Atoms and Molecules in Electric Fields (Almqvist and Wicksell, Stock­holm 1976)

Chapter 16

E. U. Condon, G. H. Shortley: The Theory oj Atomic Spectra (Cambridge University Press, Cambridge 1979)

1. 1. Sobelman: Atomic Spectra and Radiative Transitions, Springer Ser. Chern. Phys., Vol. 1 (Springer, Berlin, Heidelberg, New York 1979)

Chapter 17

G. Herzberg: Atomic Spectra and Atomic Structure (Dover, New York 1944) H. G. Kuhn: Atomic Spectra, 3rd ed. (Longmans, London 1969) H. E. White: An Introduction to Atomic Spectra (McGraw-Hill, New York 1954)

Chapter 18 .

B. K. Agarwal: X-Ray Spectroscopy, Springer Ser. Opt. Sci., Vol. 15 (Springer, Berlin, Heidelberg, New York 1979)

C. Kunz: Synchroton Radiation, Topics Curro Phys., Vol. 10 (Springer, Berlin, Heidelberg, New York 1979)

Chapter 19

M. Cardona, L. Ley (eds.): Photoemission in Solids I, II, Topics Appl. Phys., Vols. 26,27 (Springer, Berlin, Heidelberg, New York 1978, 1979)

Chapter 20

A. J. Freeman, R. B. Frankel: Hyperjine Interactions (Academic, New York 1967) H. Kopfermann: Nuclear Moments (Academic, New York 1958)

Chapter 21

H. Haken: Laser Theory (Springer, Berlin, Heidelberg, New York, Tokyo 1984); Licht und Materie II, Laser (Bibliographisches Institut, Mannheim 1981)

D. C. O'Shea, W. R. Callen, W. T. Rhodes: Introduction to Lasers and Their Applica-tions (Addison-Wesley, Reading 1978)

Chapter 22

A. Corney: Atomic and Laser Spectroscopy (Oxford University Press, Oxford 1977) W. Demtroder: Laser Spectroscopy, Springer Ser. Chern. Phys., Vol. 5 (Springer,

Berlin, Heidelberg, New York 1982)

384 Bibliography

D. C. Hanna, M. A. Yuratich, D. Cotter: Nonlinear Optics of Free Atoms and Molecules, Springer Ser. Opt. Sci., Vol. 17 (Springer, Berlin, Heidelberg, New York 1979)

V. S. Letokhov, V. P. Chebotayev: Nonlinear Laser Spectroscopy, Springer Ser. Opt. Sci., Vol. 4 (Springer, Berlin, Heidelberg, New York 1977)

D. C. O'Shea, W. R. Callen, W. T. Rhodes: Introduction to Lasers and Their Applica­tions (Addison-Wesley, Reading 1978)

K. Shimoda (ed.): High-Resolution Laser Spectroscopy, Topics Appl. Phys., Vol. 13 (Springer, Berlin, Heidelberg, New York 1976)

H. Walther (ed.): Laser Spectroscopy of Atoms and Molecules, Topics Appl. Phys., Vol. 2 (Springer, Berlin, Heidelberg, New York 1975)

Chapter 23

L. Pauling: The Nature of the Chemical Bond (Cornell University Press, Ithaca 1960) K. S. Pitzer: Quantum Chemistry (Prentice-Hall, Englewood Cliffs 1961) F. O. Rice, E. Teller: The Structure of Matter (Wiley, New York 1961) M. Karplus, R. M. Porter: Atoms and Molecules: an Introduction for Students of

Physical Chemistry (Benjamin, New York 1970)

Subject Index

Abbe's theory 20 Absorption 14, 53, 92, 96f. - in the laser mechanism 334, 338 Absorption coefficient 36, 286 Absorption edge 286 Actinides 25 Adiabatic invariance principle 295 Alkali atoms - , general 27 -, level scheme 188f. -, spectra 163f. -, term diagrams 164ff. a particles 10, 37 Amplitude fluctuations in the laser 340 Amplitude stability in lasers 340 Angular momentum 110, 197 -, addition of 270ff. -, commutation relations 133, 149 -, conservation of 40 -, coupling mechanism 270 -, creation and annihilation operators 149 - decoupling 322 - eigenfunctions 147ff. - , in spherical polar coordinates 151 - of light quanta 206 - quantum number 156f., 178 - , vector model 270 Annihilation operator - for spherical harmonic functions 149 - for the harmonic oscillator 137 Antineutrino 104 Antisymmetry of wavefunctions 301, 306f. Areas, law of 111 Aston's mass spectrometer 68 Atom -, angular momentum, total 316 -, as a wave 72 -, general 1, 5 -, ground state 300 -, heavy 276 - in an electric field 235f. - in a magnetic field 197f. -, mass of 26 -, -, absolute 6 -, -, relative 5f., 25, 32 -, paramagnetic 198 -, radius 15, 20 -, shell structure 279 -, size 10, 15 -, volume of 27

Atomic beam resonance (Rabi's method) 325f. Atomic model -, of Bohr 53 -, of Prout 2 - , of Rutherford 3, 39, 96 Atomic nucleus -, combined motion of 313 -, general 25, 313f. -, magnetic moment of 313ff. -, paramagnetism of 313 -, spin 313f. Atomic number (nuclear charge number) 25ff.,44 Atomic spectra, influence of the nucleus 313f. Atomic weight -, determination of 102 -, general 5, 25, 32 Atomism -, general -, of electricity 2 -, of energy 3 -, of heat 2 Auger coefficient 289 Auger effect 288f. Auger electron 289 Avogadro's hypothesis 2, 5 Avogadro's number 6

Backscattered electrons 61 Balmer formula 3, 94, 100 Balmer series 94ff., 103, 109ff., 187 Band spectra 91 Beat frequency 345 Beer's law 13 Benzene 372 Bergmann series 169 Binding energy 99, 111, 269, 284, 290, 354, 359 -, of inner electrons 284 Biot-Savart, law of 185 Black body radiator 49 -, energy density of 49 Bloch equations 231ff. Bohr-Sommerfeld theory 112f. Bohr magneton 176,215,314 Bohr model - for hydrogen-like systems 98 - for the hydrogen atom 91ff. Bohr postulates 96f. Bohr radius 98 Bohr's atomic model 52 -, spin-orbit splitting in 184

386

Bohr's correspondence principle 112f.,206 Bolometer 200 Boltzmann constant 7 Boltzmann distribution 8, 11, 54, 265 Boundary conditions for eigenvalue equations

123, 130 Boyle's law 10 Brackett series 94f. Bragg reflections, law of 17 Bragg's rotating crystal method 15f., 18 Breit-Rabi formula 324 Bremsstrahlung 48 Brownian motion 8

Cadmium atom, normal Zeeman effect 201, 205 Canal rays 20f. Carbon 370 Carbon atom -, Grotrian diagram 273 -, valence of 301 Cathode rays 20f., 35 Cavity radiation 3, 49ff. Centimetre waves 197 Central field -, m degeneracy 159 -, radial part of the wavefunction 153ff. Centrifugal acceleration 28f. Centrifugal force 28, 97f., 203 Charge cloud 166 Chemical bonding 353ff. -, heteropolar 353 - ,homopolar 353 Cloud chamber 37, 39 Coherence length 338 Coherence of laser light -, spatial 331 -, temporal 331 Coherent light 247 Coherent spin motion 232f. Coincidence circuit 61 Collective radiation 334 Collisional broadening 265 Collision radius 36 Colour centres 200 Commutation relations - and simultaneous measurability 131ff., 145, 224

- for angular momentum operators 133, 149 - for creation and annihilation operators in the ; harmonic oscillator 136

Compton effect 58, 60, 61 Compton-scattered radiation 58 Compton shift 60 Compton wavelength 60f. Conduction electrons 200 Contact interaction 318 Correspondence principle 112f., 206 Coulomb field 39 Coulomb force 97f.,203 Coulomb interaction energy 304, 307

Coulomb potential 294 -, I degeneracy 159

Subject Index

-, radial part of the wavefunctions 155ff. Coulomb repulsion 39 Coulomb scattering 37f. Counter-field method 56 Covolume 15, 18 Creation operator - for spherical harmonics 149 - for the harmonic oscillator 137

d series 169 D term 169, 271 Dalton's law 5 Davisson-Germer experiment 70f. de Broglie hypothesis 72 de Broglie relations 77, 118 de Broglie waves 71, 80, 117, 121 Debye-Scherrer method 18, 73 Decay time, atomic 264 Decoupling of spin and orbital angular momenta 208f.

Degeneracy 11 0 - and perturbation theory 241 -, I degeneracy in hydrogen 159, 243 -, m degeneracy with spherical symmetry 159 Degenerate wavefunctions -, orthogonality 356 - , transformation behaviour 254 Delta function 83, 375ff. Density of states 8 Deuterium 101 Diamagnetism 178 Diffraction - , of an electromagnetic wave 47 -, of an electron beam by a slit 73, 81f., 84 Dipole moment, matrix element 242, 245, 251, 256ff.

Dipole, oscillating 260ff. Dirac equation 223 Dirac theory 190f. Directional degeneracy 204 Directional focussing 30 -, Dempster method 30 Directional quanti sat ion 177f., 181ff., 184, 197 - of a magnetic moment 173 Dissociation energy 254 Diving orbits 111, 166 Doppler broadening 187, 191, 266, 346, 350 Doppler-free saturation spectroscopy 346ff. Doppler-free spectroscopy 192 Doppler-free two-photon absorption 348f. Doppler shift 266, 346 Double focussing 30 Double resonance 210ff., 266 Doublet splitting 184, 187 Doublet state 273 Doublet structure 173, 183 f. Dualism, wave-particle 61f., 82, 83 Dye lasers 332

Subject Index

Eccentricity of elliptical orbits 11 0 Ehrenfest's adiabatic invariance principle 295 Eigenfunction -, general 130

of angular momentum 147ff. of the harmonic oscillator 138f. of the particle in a box 117ff. see also Wavefunctions

Eigenvalues -, general 130 - , see also Energy (eigenvalues) Einstein coefficients 54 Einstein-de Haas effect 173 Einstein-de Haas experiment 180f. Electric dipole moment 236 Electric potential 213 Electrolysis 7, 34 Electromagnetic spectrum 91 Electron 2, 25, 44, 60, 65, 104 -, angular momentum of 197 - at the nucleus 318 -, charge of 66 - collisions 107 - configuration 26, 163, 272, 294, 296, 298f. - -, ground state 300 - density distribution 19 - -electron repulsion 269f. -, equations of motion for 203 - , g factor of 179, 320 -, magnetic moment of 178ff. - microscope 20 -, production of free 65 -, radial density distribution 295 -, size of 65 - specific charge elm 67ff. - , wave character 70 - radius, classical 65f. Electron spin 176ff., 215ff. - , ordering of in the atomic ground state 300

- resonance 197ff., 210f. Electron volt 93 Electronic shells of ions 293 Electronic structure of atoms 293 Electrons -,free 107 -, inner 284 -, neutrinos associated with 104f. -, outer 284 Elementary charge 7,43,66 Elementary wave 17 Elliptical orbit 110 Emission 92, 96f. -, in the laser mechanism 334, 338 -, induced 53 -, spontaneous 53 Energy (eigenvalues)

of the harmonic oscillator 139 of the hydrogen atom 157 of the particle in a box 119

Energy filter 28 Energy level 96ff. Energy-mass equivalence 6, 55 Energy quantisation 52 Energy shift, relativistic 112 Energy (splitting)

of an electron in an electric field 237 - of a spin in a magnetic field 218f. - of orbital angular momentum in a magnetic field 215

- , see also Interaction energy Energy term diagram 96 Energy terms, notation 188 Equation of state 7 Equivalence principle 69 ESCA 290 ESR apparatus 199f. ESR spectroscopy 197 Ethylene 372 Exchange integral 356, 366

387

Exchange term in the Hartree-Fock method 309 Excited states 302 Expectation value 125f., 128 - , see also Observation value, mean value Extinction coefficient 286 Extremal principle, equivalence to the Schrodinger equation 309f.

f series 169 Fabry-Perot interferometer 201, 343f. Faraday constant 7, 66 Fermi contact interaction 318 Field emission microscope 20f. Field ionisation 113 Fine structure 173ff., 183ff., 187,267,313

of the hydrogen atom 189 of x-ray absorption edges 288 of x-ray spectra 284f. interaction 189, 316

- splitting 187,209,268,282, 350 Fluctuating forces in the laser 339 Fowler series 102 Fractional distillation 33 Franck-Hertz experiment 107 Fraunhofer lines 98 Frequency of an electromagnetic wave 93 Fresnel biprism 72 Fresnel diffraction 72

g factor 176, 200, 276, 316 for orbital magnetism 208

- of the electron 179f. - of spin magnetism 208 -, precision measurement 321 g series 169 Gamma radiation 106 Gas centrifuge 33 Gas constant, universal 7 Gas discharge 21f., 27, 35 Gas discharge tube 23

388

Gas laser 346ff. Gaussian line 265 Giant atoms 112 Grating spectrometer 201 Grotrian diagram 167f., 273f.

of the carbon atom 273 of the lithium atom 167 of the mercury atom 275 of the nitrogen atom 273 of the potassium atom 170 of the sodium atom 168

Ground state terms 298f. Group velocity 80 Gyromagnetic ratio 173,177, 180f., 205, 314, 316

of nuclei 199 - of the electron 200

Half-intensity thickness 286 Hamilton function for an electron in an electromagnetic field 213

Hamilton operator -, general 122, 379 - for an electron in an electromagnetic field 214 - for the many-electron problem 304ff. Hanle effect 350ff. Hartree-Fock method 294, 303ff. Hartree method 303ff. Heisenberg uncertainty relations 83ff. Heitler-London method 364 Helium atom 270 -, Lamb shift 192 -, spectrum 267f. -, term diagram 268 Helium ion 269 Helium-neon laser 333 hie measurement 57 Hermite polynomials 139 Hertzian dipole 97,204, 262f., 281 Hertzian vector 261 High-frequency mass spectrometer 32 Hund-Mulliken method 369 Hund's rules 301£f. Huygen's principle 17 Hybridisation 301, 370ff. Hydrogen atom -, Bohr's model 91£f. -, fine structure 189f. -, hyperfine structure 318ff. -, - in ESR 325 -, Lamb shift 190ff., 318 -, shell structure 295 -, spectrum 93ff., 113 -, structure of the H ex line 192 Hydrogen atom - , eigenvalues 157 - , I degeneracy 159 -, radial part of the wave functions 155ff. Hydrogen isotopes 33 Hydrogen-like spectra 102 Hydrogen molecule 95, 361ff.

ion 353ff. Hyperfine components 316 Hyperfine constant 318 Hyperfine interaction 316ff. Hyperfine interaction energy 324 Hyperfine levels 317 Hyperfine multiplet 319 Hyperfine structure 200, 313

Subject Index

-, in an external magnetic field 321 ff. -, in electron spin resonance 325 - multiplets 350 -, of the sodium ion 320ff. -, of the hydrogen ion 320ff. Hypothesis of Prout 25

Ideal-gas equation 10 Impact parameter, Rutherford scattering 39 Incoherent spin motion 232f. Induced emission in the laser 334 Infrared 114 Interaction cross section -, ex particle-atom 37 -, differential 42 -, general 11, 14, 66 -, macroscopic 43 - of an atom 35f. -, total 13, 43 Intercombination -, forbidden 271 -, lines 272, 275 Interference 17, 47, 72f. -, of neutrons 73 f. Interference spectrometer 201, 313 Internal pressure 15 Interval constant 318 Interval rule 318, 319, 328 Intrinsic angular momentum see Spin Intrinsic volume 15 Invariance of the Hamiltonian 252 Inversion -, critical 335, 337 -, (un)saturated 339 Ion 29 Ionization 99, 107,277 Ionization energy 25, 27, 96f., 164, 269, 298f., 368

Ionization of inner electron shells 284 Ionization potential 108, 163, 296 Isotope 25, 32, 313 Isotope effect 313 Isotope separation 33f. Isotope shift 101 Isotopic composition 30, 32 Isotopy 25, 30

jj coupling 274ff. Jordan's rule 129, 214

Kedge 284 K series 282ff.

Subject Index

K shell 170 Kepler's laws 11 Of. Kinetic energy 99 Kinetic theory of gases 2, 10 Kirchhoff's law 55 Klystron 190, 200

L series 282ff. L shell 170 { degeneracy, lifting of 163 ff. Laguerre polynomials 157 Lamb dip 347 Lamb shift 190ff., 320, 348 Lande factor 176,205,208,226,276 Laplace operator 122 -, in spherical polar coordinates 146 Larmor frequency 177,199,203 - , of the nuclear spin 326 Laser amplitude, complex 338ff. Laser conditions 335 Laser equation 338 Laser light 113 Lasers 331ff. -, in the photon picture 332, 334f. -, in the wave picture 332, 337ff. -, stationary states 337 Laser threshold 337 Lattice constant 8, 16 Lattice, face-centred cubic 8 Lattice planes 16 Laue method 16f., 74 Law of constant and multiple proportions 2 LCAO method 370, 373 Lead, x-ray absorption cross section 288 Legendre functions, associated 147 Legendre polynomials 147 -, the simplest 148 Lenard's cathode ray tube 35 Lenard window 35 Letokhov, method of 347 Level crossing 350 Level-crossing spectroscopy 350 Lifetime 267, 352 Lifting of {degeneracy 163ff. Light as a particle 48 Light as a wave 47 Light emission, exponential decay in spontaneous

262ff. Light pulse, ultra short 331 Light quantum 48, 52, 55 -, angular momentum of 206, 259 Light quantum hypothesis 55f., 60 Linear combination -, asymmetric 305 -, of eigenfunctions belonging to the same energy 241, 355

- , see also Superposition Line spectra 92 Linewidth of a laser 331,340 -, homogeneous 248, 264

-, inhomogeneous 265 -, natural 265, 347ff., 352 Lithium atom -, shell structure 294 -, spin-orbit splitting in 209 -, term diagram 167 Lorentz equation 69 Lorentz force 28, 203, 213 Lorentz formula 69 Lorentz invariance 69 Lorentz line shape 264

389

Loschmidt's number (Avogadro's number) 6f., 66 Losses, in the laser process 334, 339 LS coupling 270ff., 274, 275 Lyman series 95f., 101, 103 -, muonic transitions in 105

M series 282ff. M shell 170 Magnetic dipole moment 174ff., 314 -, nuclear 314 -, measurement of 325ff. -, sign of 319 Magnetic dipole, potential energy 174 Magnetic dipole radiation 260 -, field strength of 175 -, induction of 175 -, nuclear quantum number 313 Magnetic dipole transitions 198 Magnetic moment 197,205,267 - , directional quantisation of 173ff. -, measurement of 181ff. - of an electron 178ff. - of a spin 219 -, of atomic nuclei 313ff. -, of many-electron atoms 276f. - of the orbital angular momentum 174ff., 215, 219 -, with spin-orbit coupling 207ff. Magnetic moments

of inner electrons 183 - quantum number 147,159,178,197, 294ff. - spin quantum number 294 Magnetisation, macroscopic 173 Magnetomechanical parallelism 176 Many-electron atoms 267ff. Many-electron problem 165, 303ff. Mass-and-spring oscillator 52 Mass change, relativistic 69, 111f. Mass defect 31 Mass density 8 Mass number 5f., 27, 31 Mass of the atom 5 Mass of the photon 55 Mass ratio proton/electron 101 Mass separation by diffusion 33 Mass spectrometer 30f -, applications 32 Mass spectrometry 27, 30, 32 Mass unit 5

390

Matrix element -, of the perturbation operator 239 - of the spin 217f. -, optical 242, 245, 251, 256ff. Matter wave 3, 70 - , see also De Broglie waves Maxwellian collision cylinder 10 Maxwell's equations 47,259,261 Maxwell's theory 47 Maxwell velocity distribution 11,265, 346f. Mean free path 14 Mean value - ,general 125ff.

of the angular momentum 129 of the energy 128f. of the momentum 126f. of the position 125 f. of the spin 220ff.

Measureability, simultaneous 313ff., 145, 224 Measurement value = eigenvalue 131 - expectation value (mean value) 131 Mercury atom -, double resonance in 211 -, Grotrian diagram for 275 Mercury low-pressure lamp, spectrum of 275 Metastable 267 Methane 370f. Microwave resonator 200 Microwave spectroscopy 187 Microwaves 114, 197 Miller indices 16 Millikan experiment 66 Molecular beam resonance (Rabi method) 325f. Molecular radicals 200 Molecular weight determination 102 Mole mass (molecular weight) 8 Momentum eigenfunctions 130 Momentum eigenvalues 130 Momentum filter 29 Momentum operator 129, 214 Momentum uncertainty 85ff. Moseley diagram 283 Motion of the nucleus looff. Multiple excitations 276 Multiplet structure 183f. - of hydrogen 110 - of the Balmer series 111 Multiplicity 188,267,270,273 Multipole radiation 259ff. Muonic atom l04ff. -, orbital radius of 104 Muon neutrino (neutretto) 104

Natural linewidth 347ff. Neutrino 104 Neutron 7, 27, 44, 104 Nitrogen atom, Grotrian diagram 273f. Noble gas atoms 27 Noble gas configuration 163, 170,296 Noble gases 296f.

Noise effect in the laser 334, 337 Nomenclature for terms 273 Normalisation of the wavefunctions - for the harmonic oscillator 138 - for the particle in a box 83, 120 -, general 83 - in infinite space 83, 375ff.

Subject Index

Normalisation of the Laguerre polynomials 157 Normalisation of the spherical harmonics 151ff. Normalisation of the spin wavefunctions 218,221 Nuclear angular momentum 314 Nuclear charge 5,25,27, 43f., 282, 313 Nuclear charge 39 -, effective 286 Nuclear charge distribution 329 Nuclear density 44 Nuclear force 44 Nuclear g factor 314, 326 Nuclear magnetic resonance 199, 316, 327f. -, apparatus for 327 Nuclear magneton 219, 314f. Nuclear motion 100 Nuclear photoeffect 58 Nuclear potential 45, 165 -, effective 165f. Nuclear quadrupole moment 328ff. Nuclear radius 39, 44f. Nuclear spin 314 -, Larmor frequency 325f. -, measurement of 325ff. Nuclear structure 44 Nuclear structure of the atom 35 Nucleon 7 Nucleus 37 -, gyromagnetic ratio 326 -, magnetic moment 314 -, -, measurement of 325ff. Nuclide 29

Occupation numbers of atomic states 247, 249 -, in the laser 334ff., 346 Occupation probability (density) 81ff., 159ff., 357ff.

One-electron states 163 Operator

for angular momentum 129f. - in spherical polar coordinates 151 for the energy 122f. for the momentum 129f. for the position 129f.

- ,spin 215ff. Operators - ,commuting 131 f. -, general 129ff. -, Hermitian 355, 379f. -, linear 355 Optical detection of ESR 210f. Optical pumping 204, 210ff. - in the laser 332, 335 Orbital 113

Subject Index

Orbital angular momentum 111, 173f., 178, 188, 267, 316 -, quantisation of 97 Orbital angular momentum quantum number 111,188,294

Orbital degeneracy 111f., 173, 188, 284 -, lifting of 111f., 163,269 Orbital magnetism 173ff., 180, 205 Orbital moment 203 Orbital radius 98 Orthogonality of wavefunctions 218, 238, 356 Orthohelium 268 Oscillator -, electromagnetic 51 -, quantum mechanical 134ff. -, zero point energy 52, 87, 138 Overlap integral 365

p series 169 p term 169, 271 Palladium 297 Parabola method of Thomson 27ff. Parahelium 267f. Parallel-plate interferometer 343 Paramagnetic ions 200 Paramagnetic states 200 Paramagnetism 178, 198, 297 Parity 253 Partition function 8 Paschen-Back effect 202, 209ff., 276 - in hyperfine structure 321f. Paschen series 95f. Pauli equation 222 Pauli principle 163, 269f., 272, 294f., 301, 305f., 362,364

Penetration depth of a particles 37 Perihelion, rotation of 111 Periodic system of the elements 2, 25ff., 293ff., 298f.

Permeability constant of vacuum 175 Perturbation theory - with degeneracy 241ff. - without degeneracy 238ff. Pfund series 95f. Phase diffusion in the laser 340 Phase velocity 80 Photocell 58 Photocurrent 56 Photoeffect 56, 62, 289 - ,inner 58, 288ff. -, outer 58 Photoelectric effect 56 Photoelectron 56 Photoelectron spectroscopy 290 Photon 47f., 49, 52, 60 -, mass of 55 -, spin of 206, 259 Photon avalanche 332, 335 Photon echo 247ff. Photon occupation number in the laser 334ff.

n bonding 372 Pickering series 103 Pions 104 n pulse, nl2 pulse 230, 247f. n transition 258 Planck's constant 51f., 84 -, precision measurement of 281 Planck's quantum of action 51f., 84 Planck's radiation formula 51,55 - , derivation by Einstein 52 -, transition probability per unit time 334 Platinum -, fine structure of the x-ray spectra 285 - , x-ray absorption 287 - , x-ray emission 287 Polaris ability 236 Polarisation

391

- as a quantum mechanical source term 259ff. -, Zeeman effect. 202, 204, 206, 210ff. Population inversion distribution 347 Position uncertainty 84, 86f. Positron 104 Potassium atom -, Grotrian diagram 170 -, shell structure 295 Potassium nucleus, magnetic moment 315 Potential - , effective, for the laser amplitude 340 -, electrical 213 Potential energy 99 Power broadening 347 Precession - ,general 199 - in a magnetic field 176ff. - of the spin 185, 220ff. Principal quantum number 96, 98, 104, 110,

156f., 188,267,273, 294f. -, effective 166 Principal series 168 Probabilistic interpretation of quantum mechanics 81ff.

Probability density 82ff., 123 - for the momentum 124 Proton 25, 44, 104 -, magnetic moment of 315, 325, 328 -, spin of 215, 219 Prout, hypothesis of 25 Pumping schemes in the laser 332 Pump power, critical 337 Pyrometry 49

Quadrupole interaction 329 Quadrupole mass filter 32 Quadrupole moment, electrical 261, 314, 328ff. Quadrupole resonance 330 Quadrupole shift of the hyper fine levels 330 Quantisation - of electron orbits 3 - of energy 49 Quantum beats 344ff.

392

Quantum defect 167 Quantum electrodynamics 179,191,193,320 Quantum jumps 106ff. Quantum number of the total angular momentum

183f. Quantum numbers 147, 156f., 269, 294 Quantum theory 3 -, of chemical bonding 353ff. Quantum yield 288 Quarks 66 Quartet state 273 Quintet state 273

Rabi atomic or molecular beam resonance 325f. Radial part of the wavefunction - in a central field 153 - in the hydrogen atom 155f. Radiation density 54 Radiation flux density 51 Radiationless processes 288 Radioactive decay 9, 22 Radio astronomy 95 Radio frequencies 199 Ramsauer effect 70 Rare earths 25, 298 Rate equations for the laser 334f. Rayleigh-Jeans radiation law 51, 55 Rayleigh scattered radiation 58 Recombination -, optical 332 - ,radiationless 336 Reduced mass 100 Relativistic correction 189 Relativistic quantum theory 179 Relativity, theory of 69 Relaxation processes 211 Relaxation time of a spin -, longitudinal 233 -, transverse 232, 248 Renormalisation 193 Renormalisation theory 193 Residual gas analysis 32 Resolution, spectral, of closely spaced levels

265, 344ff. Resolving power - of a mass spectrometer 30f. - of a microscope 20 -, spectral 110, 265, 313, 344f., 350 Resonance, covalent-ionic 368f. Resonance fluorescence 92 Resonance light 350 Resonance line 169 Resonance - of the magnetic field with a spin frequency 228 - or the radiation field with electron transitions 245f.

Resonator 200 Rest energy of the electron 61 Rhodium 297 Ritz combination principle 95

Subject Index

Rotating wave approximation 30, 246 Russell-Saunders coupling 270ff. Rutherford's atomic model 3, 39, 44, 96 Rutherford scattering 37f., 40 -, anomalous 38, 44 -, experimental results 44 Rutherford scattering formula 39ff., 43 Rydberg atoms 112ff. Rydberg constant 94, 100, 313 Rydberg formula 95 Rydberg states 104

s series 169 S term 169, 271 Saturation spectroscopy, doppler-free 346ff. Scalar product of spin wavefunctions 217, 221 Scanning electron microscope 20 Scattering coefficient 12f., 36, 70 Scattering cross section II Scattering formula of Rutherford 43 Scattering of a particles 37 Schrodinger equation - for a free particle 121 - for a spin in a magnetic field 218, 227 - for a two-level atom in an radiation field 244 - for the harmonic oscillator 134 - for the many-electron problem 304, 307 -, for the radial part 147, 154 -, general I2Hf. - of the electron with spin-orbit coupling 223f. - , time independent 123 - with a central field 146 - with an electromagnetic field 215 Scintillation detector 22f., 38, 107 Screening effect 165 Screening number 283 Secondary electron multiplier (electron multiplier) 22f.

Secondary series 168 Secular determinant 241 Sedimentation equilibrium 7 Selection rules and symmetry 256f. -, for collisional excitation 109 -, for optical transitions 166, 184, 188, 205, 207,209,215,259,271,273,275,286,303,320 -, in multi pole radiation 259 Self energy 193 Separation ansatz for wavefunctions in a central

field 146 Separation tube 33 Series limit 94, 284, 286 Series-limit continuum 95,287 Sextet state 273 Shells, inner 165, 290 Shell structure 163ff., 296 - of atoms 293ff. - of the atomic energy levels 297 a transition 258 Silver, Auger effect 289 Single-mode laser 334

Subject Index

Singlet state 272f. Singlet system 267 f. Sodium 0 line 168f., 173, 187,285 -, anomalous Zeeman effect in 201, 206f. -, hyperfine splitting of 321 -, optical pumping 210 -, spin-orbit splitting 209 -, Stark effect 236 Sodium ion -, experiment showing the Zeeman effect 201 -, hyperfine structure of 320ff. -, resonance fluorescence 92 - , shell structure of 295 -, spectral series of 168 -, term diagram 168 Solid-state detector 107 Sommerfeld-Kossel displacement theorem 104 Sommerfeld fine structure constant 111, 190 Sommerfeld model 166 Sommerfeld's extension of the Bohr model 109ft. Sommerfeld's second quantum number 111 Space, atoms in 114 Space lattice interference 16 Spectral analysis 93 Spectral energy density 50 Spectral regions 91 Spectra of the alkali atoms 163ff. -, continuous 91 -, hydrogen-like 102 -, optical 3 Spectrograph 313 Spectrum of the hydrogen atom 93ff. Spherical harmonic functions 151 -, the simplest 153 Spin 150, 174ff., 178ff., 197, 216f., 267, 317, 362ff., 369

- of atomic nuclei 314ff. Spin echo 247ff. Spin expectation value 220ff. Spin flip 226, 229ff., 247f., 271, 273 Spin magnetism 173ff., 180, 183, 205ff. Spin matrices 216f. Spin motion, (in)coherent 232f. Spin operators 216ff. Spin-orbit coupling 183f., 207ff., 222ff., 268, 270,274

Spin-orbit coupling constant 186 Spin-orbit coupling energy 186, 189 Spin-orbit interaction 270, 284, 301, 317 Spin-orbit interaction energy 186, 223 Spin-orbit splitting 188 - in Bohr's atomic model 184ff. Spin precession 185, 220ff. Spin quantum number 179, 188, 216, 294 Spin resonance technique 230 Spinthariscope 21 f. Spin variable 304 Spin wavefunctions 216ff. Spontaneous emission in the laser 333f., 338 Square potential (particle in a box) 117 ff.

Stark effect 237f. -, linear 237, 241ff. -, quadratic 237, 238ff. State, bound 358 States, non-bound 357 Statistical interpretation of quantum mechanics 81ff.

Stefan-Boltzmann law 50f

393

Stern-Gerlach experiment 173, 178f., 181ff., 325 Stimulated emission in the laser 334 Superposition - of spin wavefunctions 217, 220, 227 - of wavefunctions for different atomic levels 260,345

- , see also Linear combination Symmetries and selection rules 251ff. Symmetry transformation 252ff. Synchrotron radiation 47f.

Temperature determination, spectroscopic 98 Term analysis 206, 209f. Term diagram, complete 302ff. Terms, nomenclature 273 Thermal conductivity 15 Thermal diffusion 33 Thermal radiation 48f., 49 Thermionic emission 107 Thomas factor 185 Thomson's parabola method 68 Time-of-flight mass spectrometer 32 Total angular momentum 270, 273, 317 Total angular momentum quantum number 183f,

188, 209 Total orbital angular momentum 183f., 186, 205,

209, 270, 273 Total spin 270 Total wave function 301 Transformation behaviour of wavefunctions - on reflection 252ff. - on rotation 254 ff. Transition elements 297 Transitions -, allowed and forbidden 109 - , see also Selection rules Triplet state 272f. Triplet system 267f. Tunnel effect 359ff. Turbine drive, for gyroscope 198 Two-electron system 303 Two-level system -, atomic 244, 247ff. -, spin 216, 247ff. Two-photon absorption, doppler-free 348f.

Ultraviolet catastrophe 51 Uncertainty relation - for energy and time 85 -, Heisenberg's 83ff. Unit cell 8 Uranium isotopes 33

394

Valence, chemical 294, 297 Valence electron 163,270,297, 370 Valence electron (radiant electron) 111 -, screening 165 Van der Waals equation for real gases 15 Variational principle, equivalence to the Schrodinger equation 308f., 361

Vector model for hyper fine structure 317 Vector model for spin-orbit coupling 183ff. -, quantum mechanical basis 222ff. Vector potential 213, 260 Velocity dependence of mass 69 Velocity focussing (Aston's method) 30 Virtual processes 193 Viscosity 14, 18 Volume effect 313 Volume of an atom 8

Wave character of light 47f. Wave, electromagnetic 47 Wave functions, radial part - in a central field 153ff. - in a Coulomb field 155ff. - , see also Eigenfunctions Wavelength standard 92 Wavelength, unit of 92 Wavenumber 93 Wave packet 77ff. - , flowing apart of 80 Wien's displacement law 51 Wigner-Weisskopf theory 345 Work function 58, 281

X band 200 X-radiation 48, 58 -, characteristic 282ff. -, from outer shells 279

-, wavelength 279f. -, -, measurement of 9 X-ray absorption coefficient 287 X-ray absorption edges 283 X-ray absorption spectra 286ff.

Subject Index

X-ray bremsstrahlung spectrum 280ff. X-ray detectors 279 X-ray diffraction by crystals 8, 15 -, Laue method 15 X-ray fluorescence 282 X-ray interference 9 X-ray line spectrum 282ff. X-ray quantum 279f. X-ray spectra 279ff. -, characteristic 279, 280 - fine structure in 284f. -, white 281

Zeeman components 204 -, polarization of 21 Off. Zeeman effect 200ff., 236 -, anomalous 178, 20lf., 206f., 222ff., 276 -, classical electron theory of 202 - in hyperfine structure 321 f. -, ordinary 201f., 204ff., 210, 213ff., 276 -, vector diagram 322 -, vector model 204ff. Zeeman energy

of a proton 325 - of an electron 325 - of the nucleus 324 Zeeman spectroscopy 197 Zeeman triplet, ordinary 205 Zero point energy of the harmonic oscillator 87, 138

Zero point oscillations of atoms 88