Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

342	Appendix J

z

2

ρ12

1

ρ1 ρ2

ã

y

x

Figure J.1 Distance between two particles 1 and 2 and their respective distances from the origin.

z

ρ12 2

1

ρ2

ρ1 è2

y

ϕ 2

x

Figure J.2 Rotation of the coordinate axes in Figure J.1 so that the z-axis lies along r1.

X		eÿ(r1 r2)		ð		ð	2ð		2ð
1
I l 0	… …	r.	slr12r22 dr1 dr2	…0	Pl(cos è2) sin è2 dè2	…0 sin è1 dè1	…0	dj1	…0	dj2

The integrals over è1, j1, and j2 are readily evaluated. Since P0(ì) 1, we may write the integral over è2 as

Evaluation of two-electron interaction interval				343
ð		1
…0	Pl(cos è2) sin è2 dè2 …ÿ1 Pl(ì)P0(ì) dì 2äl0
where equations (E.18) and (E.19) have been introduced. Thus, only the term with
l 0 in the summation does not vanish and we have
	I 16ð2… …	eÿ(r1 r2)	r12r22 dr1 dr2	(J:4)
		r.

In the second procedure, we substitute equation (J.3) directly into (J.1) and evaluate the integral over è2

ð				sin è2			1				ð
…0						dè2				s2 ÿ 2s cos è2)1=2	0
									(1
	(1		s2	ÿ	2s cos è2)1=2			s

1s [(1 s2 2s)1=2 ÿ (1 s2 ÿ 2s)1=2]

1s [(1 s) ÿ (1 ÿ s)] 2

The integrals over è1, j1, and j2 are the same as before and equation (J.4) is obtained. Since r. is the larger of r1 and r2, the integral I in equation (J.4) may be written in

the form

…

…

…

1

1

r1

1

I 16ð2

eÿr1 r12"

0

eÿr2 r22 dr2 r1 eÿr2 r2 dr2

# dr1

0

r1

1

16ð2

…0

eÿr1 r1f[2 ÿ (r12 2r1 2)eÿr1 ] r1(r1

1)eÿr1 g dr1 16ð2(85 85)

Accordingly, the ®nal result is

e

(r1 r2)

I … …

ÿ

dr1 dr2 20ð2

(J:5)

r12

Selected bibliography

Applied mathematical methods

Three widely used compendia of applied mathematics directed to the needs of chemistry and physics are the following.

G. B. Arfken and H. J. Weber (1995) Mathematical Methods for Physicists, 4th edition (Academic Press, San Diego).

M. L. Boas (1983) Mathematical Methods in the Physical Sciences, 2nd edition (John Wiley & Sons, New York).

K. F. Riley, M. P. Hobson, and S. J. Bence (1998) Mathematical Methods for Physics and Engineering (Cambridge University Press, Cambridge).

Undergraduate physical chemistry

The following undergraduate texts discuss the historical development of quantum concepts and introduce the elements of quantum mechanics.

R. A. Alberty and R. J. Silbey (1996) Physical Chemistry, 2nd edition (John Wiley & Sons, New York).

P.W. Atkins (1998) Physical Chemistry, 6th edition (Oxford University Press, Oxford; W. H. Freeman, New York).

D. A. McQuarrie and J. D. Simon (1997) Physical Chemistry: A Molecular Approach

(University Science Books, Sausalito, CA).

History and philosophy of quantum theory

J. Baggott (1992) The Meaning of Quantum Theory (Oxford University Press, Oxford). M. Jammer (1974) The Philosophy of Quantum Mechanics (John Wiley & Sons, New

York).

M. Jammer (1966)The Conceptual Development of Quantum Mechanics (McGrawHill, New York).

Some `classic' quantum mechanics texts

Emphasis on applications to chemistry.

H. Eyring, J. Walter, and G. E. Kimball (1944) Quantum Chemistry (John Wiley & Sons, New York).

344

Selected bibliography

345

L. Pauling and E. B. Wilson (1935) Introduction to Quantum Mechanics: With Applications to Chemistry (McGraw-Hill, New York; reprinted by Dover, New York, 1985).

Emphasis on applications to physics.

D. Bohm (1951) Quantum Theory (Prentice-Hall, New York).

P. A. M. Dirac (1947) The Principles of Quantum Mechanics, 3rd edition (Oxford University Press, Oxford) and 4th edition (Oxford University Press, Oxford, 1958). Except for the last chapter, these two editions are virtually identical.

H. A. Kramers (1957) Quantum Mechanics (North-Holland, Amsterdam).

L. D. Landau and E. M. Lifshitz (1958) Quantum Mechanics: Non-Relativistic Theory

(Pergamon, London; Addison-Wesley, Reading, MA).

Some recent quantum mechanics texts

Emphasis on applications to chemistry.

P. W. Atkins and R. S. Friedman (1997) Molecular Quantum Mechanics, 3rd edition (Oxford University Press, Oxford).

I. N. Levine (1991) Quantum Chemistry, 4th edition (Prentice-Hall, Englewood Cliffs, NJ).

F. L. Pilar (1990) Elementary Quantum Chemistry, 2nd edition (McGraw-Hill, New York).

J. Simons and J. Nichols (1997) Quantum Mechanics in Chemistry (Oxford University Press, New York).

Emphasis on applications to physics.

B.H. Bransden and C. J. Joachain (1989) Introduction to Quantum Mechanics (Addison Wesley Longman, Harlow, Essex).

C.Cohen-Tannoudji, B. Diu, and F. LaloeÈ (1977) Quantum Mechanics, volumes I and II (John Wiley & Sons, New York; Hermann, Paris).

D.Park (1992) Introduction to the Quantum Theory, 3rd edition (McGraw-Hill, New York).

J. J. Sakurai (1994) Modern Quantum Mechanics, revised edition (Addison-Wesley, Reading, MA).

Angular momentum

The following books develop the quantum theory of angular momentum in more detail than this text.

A. R. Edmonds (1960) Angular Momentum in Quantum Mechanics, 2nd edition (Princeton University Press, Princeton).

M. E. Rose (1957) Elementary Theory of Angular Momentum (John Wiley & Sons, New York; reprinted by Dover, New York, 1995).

R. N. Zare (1988) Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (John Wiley & Sons, New York).

346	Selected bibliography

Atoms and atomic spectra

H. A. Bethe and E. E. Salpeter (1957) Quantum Mechanics of Oneand Two-Electron Atoms (Springer, Berlin; Academic Press, New York; reprinted by Plenum, New York, 1977). A comprehensive non-relativistic and relativistic treatment of the hydrogen and helium atoms with and without external ®elds.

E. U. Condon and G. H. Shortley (1935) The Theory of Atomic Spectra (Cambridge University Press, Cambridge). A `classic' text on the application of quantum theory to atomic spectra.

More advanced applications of quantum mechanics

D. R. Bates (ed) (1961, 1962) Quantum Theory, volumes I, II, and III (Academic Press, New York and London). A compendium of articles covering the principles of quantum mechanics and a wide variety of applications.

S. Kim (1998) Group Theoretical Methods and Applications to Molecules and Crystals

(Cambridge University Press, Cambridge).

I.N. Levine (1975) Molecular Spectroscopy (John Wiley & Sons, New York). A survey of the theory of rotational, vibrational, and electronic spectroscopy of

diatomic and polyatomic molecules and of nuclear magnetic resonance spectroscopy.

N. F. Mott and H. S. W. Massey (1965) The Theory of Atomic Collisions, 3rd edition (Oxford University Press, Oxford). The standard reference for the quantummechanical treatment of collisions between atoms.

G. C. Schatz and M. A. Ratner (1993) Quantum Mechanics in Chemistry (PrenticeHall, Englewood Cliffs, NJ). An advanced text emphasizing molecular symmetry and rotations, time-dependent quantum mechanics, collisions and rate processes, correlation functions, and density matrices.

A. J. Stone (1991) The Theory of Intermolecular Forces (Oxford University Press, Oxford). An extensive survey of the applications of quantum mechanics to determine the forces between molecules.

Compilations of problems in quantum mechanics

I. I. Gol'dman and V. D. Krivchenkov (1961) Problems in Quantum Mechanics (Pergamon, London; Addison-Wesley, Reading, MA; reprinted by Dover, New York, 1993).

C. S. Johnson, Jr. and L. G. Pedersen (1974) Problems and Solutions in Quantum Chemistry and Physics (Addison-Wesley, Reading, MA; reprinted by Dover, New York, 1987).

G. L. Squires (1995) Problems in Quantum Mechanics: with Solutions (Cambridge University Press, Cambridge).

Index

absorption of radiation 187

adiabatic approximation 268±9, 272±3 allowed transitions 192

amplitude

of vibration, de®ned 107 of a wave, de®ned 2

see also harmonic oscillator, wave motion angular frequency

of vibration, de®ned 107 of a wave, de®ned 3

see also harmonic oscillator, wave motion angular momentum

addition of 204±6

conservation of 131, 192, 204±5 general theory of 132±7, 155

see also orbital angular momentum, spin angular momentum

anharmonic oscillator 246±8, 278±80 antisymmetric wave function, see wave function atomic orbitals 175±82, 254±6

average values, see expectation values azimuthal quantum number, de®ned 175

Balmer series 156, 188±9, 193 Biot and Savart law 202

Bohr, N. 43, 151, 156±7

Bohr correspondence principle 43±5, 120 Bohr magneton, de®ned 152

Bohr radius 175, 184 Boltzmann constant 228±9 Born, M. 34, 38, 41, 265 Born±Huang treatment 266±9

Born±Oppenheimer approximation 264±6, 268±9, 273

Bose±Einstein condensation 229±30 boson 197±8, 217±18, 221±3, 229±30 bra vector, de®ned 81

see also eigenfunctions Brackett series 188±9

center-of-mass coordinates 149, 157±8, 269±71 classical mechanics

of harmonic oscillator 106±9, 128

of magnetic moment 151±4 of orbital motion 130±1

of rigid rotor 148±9

closure property, see completeness relation commutators

de®ned 66

for generalized angular momentum 133, 155 for orbital angular momentum 132, 155, 161,

203±4

for position and momentum 104 properties 66, 70, 77, 100±1, 104±5

for spin angular momentum 197, 199, 203±4 complete set of functions, de®ned 75±7 completeness relation 76±7, 84, 118, 218±20, 300,

309, 317 Compton, A. 19

Copenhagen interpretation 34

correspondence principle, see Bohr correspondence principle

Coulomb's law 151, 160, 226 creation operator, see ladder operators

Davison±Germer experiment 19 de Broglie, L. 19

de Broglie wavelength 19, 49±50, 58 degeneracy 72±4, 78±9, 85, 92, 96

de®ned 62±3, 67

for harmonic oscillator 113±14, 129 for hydrogen-like atom 176, 191±2 for particle in a box 63

and perturbation theory 248±56 for rigid rotor 151

delta function, see Dirac delta function destruction operator, see ladder operators determinants, properties of 334±5 deuterium 190, 193

diatomic molecule 269±80 Dirac, P. A. M. 195

Dirac delta function 15±16, 22, 76±7, 291±5 Dirac equation and electron spin 195±6 Dirac notation 69, 71, 80±3, 86

dispersion law, de®ned 4 see also wave motion

347

348

Index

Ehrenfest theorems 43±5, 59±60, 99, 105 eigenfunctions 67±75, 80±3, 85, 87, 96, 104

completeness of 75±7

expansions in 75±7, 84, 88±91, 94, 198 for generalized angular momentum 133±4 normalization 38±41, 47, 51, 58, 69, 86, 98

orthogonality 51, 69, 71±4, 80, 211, 215, 217 scalar product of 68±9

simultaneous 77±9

for spin angular momentum 197±9 well-behaved, de®ned 68, 162

see also wave function

eigenket, see eigenfunctions, wave function eigenvalues 67±8, 71, 85, 87±90, 104

for generalized angular momentum 133±4, 137 for orbital angular momentum 138, 140

for spin angular momentum 197, 199 see also degeneracy, energy levels

eigenvector, see eigenfunctions, wave function Einstein, A. 18±19, 229

electron spin 29, 32±4, 85, 190, 194±6, 201±6, 223±4

electron volt, de®ned 168 emission of radiation 187 energy levels

for diatomic molecule 276±9

for harmonic oscillator 113±14, 127±9, 323 for hydrogen-like atom 168, 175±6, 191±2,

254±6, 261±2, 328

for particle in a box 50, 62±3 for rigid rotor 150

see also perturbation theory, variation method Estermann, I. 19

exchange forces 223

exchange operators 210±11, 213±16, 230 exclusion principle, see Pauli exclusion principle expectation values 41±5, 47±8, 59, 85, 87±91,

104±5

for harmonic oscillator 128

for hydrogen-like atom 181, 184, 186±7, 192±3, 329±30

for particle in a box 104 time dependence 97±8

Fermi, E. 19

Fermi energy in free-electron model 227±9 fermion 197±8, 217±18, 221±9, 231

®ne structure constant 203

force constant for vibration 106, 273 Fourier series 4, 285±9

Fourier transform 8, 10, 14±15, 22, 37, 40, 42, 58, 102, 289±91, 294

free-electron gas 226±9, 231 frequency

angular, de®ned 3

of vibration, de®ned 107 of a wave, de®ned 2

see also harmonic oscillator, wave motion Frisch, R. 19

Frobenius method, see series solution method

gaussian distribution 10±11, 15, 17±18, 21, 34±5, 102

Gerlach, W. 26

see also Stern±Gerlach experiment Goudsmit, S. 194±6

group velocity, see wave packet gyromagnetic ratio 196

Hamiltonian operator 85, 87, 93, 95±6 de®ned 47

for harmonic oscillator 109 for helium atom 224, 256

for hydrogen-like atom 159, 161, 191, 260 for a molecule 263±4

for multi-particle system 60

for non-interacting particles 220 perturbed 240, 246, 248

for rigid rotor 150

for spin±orbit coupling 202±3, 206 symmetry of 209, 212, 216 three-dimensional 59

harmonic oscillator 106±29 classical 106±9, 128

energy levels 113±14, 127±9, 323 Hamiltonian operator 109 isotropic 127±9

ladder operators 110, 128 linear 109±21

matrix elements 121±5, 129 applied to nuclear motion 275±6 perturbed 246±8

SchroÈdinger equation 109, 126 three-dimensional 125±9

and variation method 235±6

wave functions 114±21, 127, 320±3 zero-point energy 113

see also anharmonic oscillator Heisenberg, W. 34

Heisenberg uncertainty principle 21±3, 29, 45±6, 99±105, 155

for harmonic oscillator 125

for time and energy 22, 31, 103±4 and zero-point energy 50

see also wave packet ± uncertainty relation helium

atom 217±18, 223±5, 256±60 ion 193

liquid 218, 230

Hellmann±Feynman theorem 96±7, 186, 192 Hermite polynomials 117±18, 296±300, 323 hermitian operator, see operator

Hilbert space 80 Hooke's law 107 Humphreys series 188±9

hydrogen-like atom 156±7, 160±93 atomic orbitals 175±82, 254±6 Bohr model 156±7, 168, 175 Bohr radius 175, 184

in electric ®eld 254±6, 260±1 energy continuum 174±5

Index

349

energy levels 168, 175±6, 191±2, 254±6, 261±2, 328

expectation values 181, 184, 186±7, 192±3, 329±30

Hamiltonian operator 159, 161, 191, 260 ionization potential 169

ladder operators 163

in a magnetic ®eld 190±2 perturbed 206, 254±6, 262 quantum numbers 175±6

radial distribution function 181, 184±6, 192 radial functions 161±75, 181, 183, 185, 326±8 Rydberg constant 156±7, 188, 190, 193 SchroÈdinger equation 159±61

spectra 156±7, 187±90, 192±3 spin±orbit coupling 201±6, 262 wave functions 161, 175

identical particles 85, 208±17, 221 de®nition 208

non-interacting 220±3, 226±30

and symmetry of wave functions 209±11, 230 see also boson, exchange operators, fermion,

permutation operators ionization potential

for hydrogen 169 for lithium ion 193

ket vector, de®ned 80 see also eigenfunctions

ladder operators 106

for generalized angular momentum 134±7, 155 for harmonic oscillator 110, 128

for hydrogen-like atom 163

for orbital angular momentum 140±1 for spin angular momentum 197, 199

Laguerre polynomials 310±12 associated 171±4, 192, 312±17, 328

Larmor frequency 153±4 Legendre polynomials 301±4

associated 147, 304±9, 325±6 linear variation function 237±9, 261 linear vector space 337±9

lithium atom 230 ion 193

lowering operators, see ladder operators Lyman series 156, 188±9

magnetic moment

of electron spin 26±9, 32±3, 190, 196, 201±2, 204

of hydrogen atom 190±2 orbital 151±5, 190, 201 of particles with spin 196

magnetic interactions

with atoms 153±5, 190±2 within an atom 201±2

see also Stern±Gerlach experiment magnetic quantum number, de®ned 176, 191

magneton, Bohr, de®ned 152 Marshall, L. 19

matrices

properties of 331±40 spin 200±1, 207

matrix elements 81

of harmonic oscillator 121±5, 129 metallic crystal 226, 228±9, 231 molecular spectra 279

molecular structure 263±80

adiabatic approximation 268±9, 272±3 Born±Huang treatment 266±9

Born±Oppenheimer approximation 264±6, 268±9, 273

diatomic molecule 269±80

nuclear wave function 265±8, 273±4, 276 moment of inertia 149, 278

momentum operator 43±4, 58, 70, 86

momentum space wave function, see wave function

± in momentum space Morse potential 279±80

normalized functions 38±41, 47, 51, 58, 69, 86, 98 nuclear motion, see molecular structure

operator

adjoint 82±3, 163

exchange 210±11, 213±16, 230

for generalized angular momentum 132±4, 155 hermitian 69±75, 77±87, 104

linear 65±8, 85, 104

for momentum 43±4, 58, 70, 86

for orbital angular momentum 131±2, 139, 149±50, 155, 160±1, 202±6

parity 94±6, 253 permutation 212±16, 219±21 projection 83±4

self-adjoint, see operator ± hermitian

for spin angular momentum 196±7, 199, 202±6 see also Hamiltonian operator, ladder operators

Oppenheimer, J. R. 265

orbital angular momentum 130±2, 138±46, 155 classical 130±1

eigenvalues 138, 140 ladder operators 140±1

and magnetic moment 151±5, 190, 201 operators 131±2, 139, 149±50, 155, 160±1,

202±6

operators in spherical polar coordinates 138±40 spin±orbit coupling 201±6, 262

wave functions 138±47, 323±6 orbitals, atomic, see atomic orbitals

orthogonal functions 51, 69, 71±4, 80, 211, 215, 217

expansions in terms of 75±7, 84, 88±91, 94, 198 orthonormal functions, de®ned 51±2, 69 oscillator, see harmonic oscillator

overlap integral 237

parity 95, 192 operator 94±6, 253

350

Index

Parseval's theorem 10, 18, 35, 41, 288±9, 291 particle in a box 48±52, 64, 91, 104±5, 230

energy levels 50, 62±3 perturbed 261±2 three-dimensional 61±3, 226 and variation method 234±5 wave functions 50±2, 62

Paschen series 156, 188±9 Pauli, W. 195±6, 221

Pauli exclusion principle 221±2, 225, 227 Pauli spin matrices 200±1, 207 permutation operators 212±16, 219±21 perturbation theory 239±58, 261±2

degenerate 248±56

®rst-order 240±3, 245, 250±4, 257±8, 261 applied to harmonic oscillator 246±8 applied to helium atom 257±8

applied to hydrogen atom 262

applied to hydrogen atom in electric ®eld 254±6 applied to a molecule 265±6, 276±9 non-degenerate 239±45

related to variation method 245 second-order 240, 243±5, 261 applied to spin±orbit coupling 262

Pfund series 188±9

phase velocity, see wave packet photon 1, 18±19, 24±6, 30±2, 187

spin of 217 Planck, M. 18

Planck relation 18, 157, 187 plane wave 2±9, 11, 22, 40

postulates of quantum mechanics 85±94, 196, 217 principle quantum number, de®ned 175 probability density, see wave function ± and

probabilities

radial distribution function 181, 184±6, 192 radiation, absorption and emission of 187 raising operators, see ladder operators Rayleigh±SchroÈdinger perturbation theory, see

perturbation theory

reduced mass 149, 158, 175, 188, 270±1 re¯ection coef®cient, see tunneling

rigid rotor 148±51, 274±6, 278 rotational constant 150, 275

Rydberg constant 156±7, 188, 190, 193 Rydberg potential 279±80

Schmidt orthogonalization 72±3, 104 SchroÈdinger, E. 1, 20, 37 SchroÈdinger equation

for harmonic oscillator 109, 126 for hydrogen-like atom 159±61 for a molecule 264±5

for particle in a box 48, 61 time-dependent 37, 59, 85, 92±4 time-independent 47, 59, 93, 96±7

Schwarz's inequality 46, 284

secular determinant 78, 239, 251±2, 255 selection rules 192

series solution method 318±20

for harmonic oscillator 110, 147±8, 162, 320±3 for orbital angular momentum 323±6

for radial equation 326±8 Slater determinant 221±2 Sommerfeld, A. 226

spherical harmonics 139±47, 161, 175, 177, 192, 274

spin angular momentum 85, 194±207

of bosons 197±8, 217±18, 221±3, 229±30 discovery of 194±6

eigenfunctions 197±9 eigenvalues 197, 199

of electron 29, 32±4, 85, 190, 194±6, 201±6, 223±4

of fermions 197±8, 217±18, 221±9, 231 gyromagnetic ratio 196

ladder operators 197, 199 operators 196±7, 199, 202±6 singlet and triplet states 224 spin one-half 198±201

see also Pauli spin matrices spinor 199

spin±orbit coupling 201±6, 262 spring constant 107

square pulse distribution 12±13, 15, 35

Stark effect in atomic hydrogen 254±6, 260±1 state function 30, 38, 40, 85±6

see also wave function stationary state 47±8, 52, 59, 93±4 Stern, O. 19, 26

Stern±Gerlach experiment 26±9, 32±4, 195 symmetric wave function, see wave function

Thomas precession 202 Thomson, G. P. 19

transmission coef®cient, see tunneling tunneling 53±7, 64

Uhlenbeck, G. E. 194±6

uncertainty principle, see Heisenberg uncertainty principle

uncertainty relation, see wave packet ± uncertainty relation

Urey, H. 190

variation method 232±9, 260±1 excited state energies 236±7 ground state eigenfunctions 234 ground state energy 232±4

applied to harmonic oscillator 235±6 applied to helium atom 259±60

applied to hydrogen atom in electric ®eld 260±1 linear variation function 237±9, 261

applied to particle in a box 234±5 related to perturbation theory 245

variation theorem 232±3, 236 vibration

of molecular bonds 106 see also harmonic oscillator

virial theorem 187, 192

Index

351

wave

frequency of, de®ned 2±3 nodes of 2

see also plane wave, wave motion, wave packet wave equation, see SchroÈdinger equation

wave function

collapse of 30±4, 85, 92

for free particle 20±3, 29±34

for harmonic oscillator 114±21, 127, 320±3 for hydrogen-like atoms 161, 175 interpretation of 29±34, 86

many-particle 60±1, 209±25

in momentum space 40±1, 58, 128±9 normalization 38±41, 47, 51, 58, 69, 86, 98 nuclear 265±8, 273±4, 276

for orbital angular momentum 138±47, 323±6 for particle in a box 50±2, 62

and probabilities 30, 38±41, 47, 58, 60, 86, 91± 2, 118±21, 198, 209, 222±3

properties of 29±34, 48±9 for rigid rotor 150

symmetry of 209±11, 214±21, 223±4, 230 time-dependent 37

time-independent 47 well-behaved, de®ned 68, 162

see also eigenfunctions, state function, stationary state

wave motion 2±18 composite wave 4±9 standing wave 7

see also plane wave, wave packet wave number

of plane wave, de®ned 3 of spectral line 187±8, 190

wave packet 8±18, 20±2, 29, 34, 36±7, 85 dispersion of 4±5, 15±18, 20±1, 29, 34 group velocity of 7, 20±1, 23, 34

of minimum uncertainty 102±3 phase velocity of 4±5, 20±1, 34 uncertainty relation 12, 14±15, 18

see also gaussian distribution, square pulse distribution

wavelength

de Broglie 19, 49±50, 58 of a wave, de®ned 2

see also wave motion

weighting function 69, 72, 74, 77, 162, 164±5

Young, T. 24

Young's double-slit experiment 23±6, 29±32

Zeeman effect 190±2

Zinn, W. H. 19