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Quantum Theory II Aggregates of Particles D. R. Bates
(Editor) Digital Instant Download
Author(s): D. R. Bates (editor)
ISBN(s): 9781483250663, 1483250660
Edition: Reprint
File Details: PDF, 48.67 MB
Year: 2013
Language: english

QUANTUM THEORY
77. Aggregates of Particles

PURE AND APPLIED PHYSICS
A SERIES OF MONOGRAPHS AND TEXTBOOKS
CONSULTING EDITOR
H. S. W. MASSEY
University College, London, England
Volume 1. F. H. FIELD and J. L. FRANKLIN, Electron Impact Phenomena and
the Properties of Gaseous Ions. 1957
Volume 2. H. KOPFERMANN, Nuclear Moments. English Version Prepared from
the Second German Edition by E. E. SCHNEIDER. 1958
Volume 3. WALTER E. THIRRING, Principles of Quantum Electrodynamics.
Translated from the German by J. BERNSTEIN. With Corrections
and Additions by WALTER E. THIRRING. 1958
Volume 4. U. FANO and G. RACAH, Irreducible Tensorial Sets. 1959
Volume 5. E. P. WIGNER, Group Theory and Its Application to the Quantum
Mechanics of Atomic Spectra. Expanded and Improved Edition.
Translated from the German by J. J. GRIFFIN. 1959
Volume 6. J. IRVING and N. MULLINEUX, Mathematics in Physics and Engineer
ing. 1959
Volume 7. KARL F. HERZFELD and THEODORE A. LITOVITZ, Absorption and Dis
persion of Ultrasonic Waves. 1959
Volume 8. LEON BRILLOUIN, Wave Propagation and Group Velocity. 1960
Volume 9. FAY AJZENBERG-SELOVE (ed.), Nuclear Spectroscopy. Parts A and B.
1960
Volume 10. D. R. BATES (ed.), Quantum Theory. In three volumes. 1961-62
Volume 11. D. J. THOULESS, The Quantum Mechanics of Many-Body Systems.
1961
Volume 12. W. S. C. WILLIAMS, An Introduction to Elementary Particles. 1961
ACADEMIC PRESS · New York and London

QUANTUM THEORY
Edited by
D. R. BATES
Department of Applied Mathematics
The Queen's University of Belfast
Belfast, North Ireland
Aggregates of Particles
1962
ACADEMIC PRESS New York and London

COPYRIGHT © 1962, BY ACADEMIC PRESS INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS,
WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS INC.
Ill FIFTH AVENUE
NEW YORK 3, N. Y.
United Kingdom Edition
Published by
ACADEMIC PRESS INC. (LONDON) LTD.
17 OLD QUEEN STREET, LONDON S.W. 1
Library of Congress Catalog Card Number 59-15762
PRINTED IN THE UNITED STATES OF AMERICA

Contributors to Volume 10-11
S. L. ALTMANN, Department of Metallurgy, University of Oxford,
Oxford, England
C. A. COULSON, Mathematical Institute, University of Oxford, Oxford,
England
D. M. DENNISON, Department of Physics, University of Michigan, Ann
Arbor, Michigan
K. T. HECHT, Harrison M. Randall Laboratory of Physics, University of
Michigan, Ann Arbor, Michigan
J. T. LEWIS, Brasenose College, University of Oxford, Oxford, England
M. J. SEATON, Department of Physics, University College, London, England
H. N. V. TEMPERLEY, Atomic Weapons Research Establishment, Alder-
maston, Berkshire, England
D. TER HAAR, The Clarendon Laboratory, University of Oxford, Oxford,
England
R. M. THOMSON, Department of Metallurgical Engineering, University of
Illinois, Urbana, Illinois
v

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Preface
Quantum Theory, comprising the three volumes, Elements, Aggregates,
and Radiation and High Energy Physics is intended as an advanced text
and reference on the fundamentals and applications of quantum theory.
It is primarily designed to meet the needs of postgraduate students.
The hope is that it will enable them to refresh and deepen their understanding
of the elementary parts of the subject, that it will provide them with surveys
of the more important areas of interest, and that it will guide them to the
main frontiers on which advances are being made. In addition, teachers
at universities and institutes of technology may find the compilation, which
is unusually wide in scope, useful when preparing lectures.
A knowledge is naturally assumed of classical mechanics, of electro
magnetic theory, of atomic physics, and (in Volume III) of the special theory
oi relativity. Familiarity with the ordinary techniques of mathematical
analysis is also assumed. However, the relevant properties of some of the
higher transcendental functions are summarized and accounts are given of
operator algebra and matrices (Volume I) and of group theory (Volume II).
In Volume I non-relativistic wave mechanics and matrix mechanics are
introduced; an extensive survey of the exactly soluble problems of the
point and of the continuous spectrum is presented ; the approximate methods
which are available for treating other stationary and time-dependent
problems are then developed in considerable detail ; and finally a very lengthy
chapter is devoted to scattering theory, the needs of both ionic and nuclear
physicists being met. To make this volume useful as a reference the treatment
of many of the topics is more comprehensive than is customary; in order
that it should nevertheless remain useful as a text the sections containing
the essentials are indicated at the beginning of certain of the chapters.
Volume II is concerned with the quantal treatment of systems of particles
— complex atoms, molecules, solids, and liquids. A chapter on quantum
statistics is included. It is hoped that theoretical chemists, as well as
theoretical physicists, will find the volume of value. Like the other volumes,
it is effectively complete in itself.
vii

viii PREFACE
In Volume III the compilation returns to the fundamentals of quantum
theory. The relativistic equations describing a single particle in an external
field of force are developed ; starting with the semiclassical theory a detailed
exposition is given of both the noncovariant and the covariant theory of
radiation ; the theory of the meson field is described ; and nuclear structure
is then discussed. The volume ends with a chapter on the question of whether
there are hidden variables underlying quantum theory. It is recognized that
the views expressed in this chapter are not widely accepted; but they are
undoubtedly stimulating.
A treatise by a group of authors is likely to have defects which would
be avoided in a text by a single author. The Editor believes that a compen
satory advantage is gained by having research workers writing on parts
of the subject in which they are particularly interested.
D. R. B.
The Queen's University of Belfast, Belfast, Northern Ireland
February, 1961

Contents
CONTRIBUTORS TO THIS VOLUME v
PREFACE vii
CONTENTS OF VOLUMES 10-1 AND 10-III xi
1. Complex Atoms 1
M. J. Seaton
1. Introductory Survey 3
2. The Central Field Model 9
3. Spectra of Alkali Atoms 14
4. The Exclusion Principle and the Periodic Table 23
5. Symmetry Properties of Atomic Wave Functions 25
6. Perturbation Procedures 28
7. The Quantum Theory of Angular Momentum 30
8. Coupling Schemes for Electron Wave Functions 37
9. The Helium Atom 40
10. Closed Shells and One Electron Outside of Closed Shells 48
11. Two Electrons Outside of Closed Shells 55
12. Three Electrons Outside of Closed Shells 65
13. The Calculation of Atomic Wave Functions 68
14. Radiative Transition Probabilities 75
References 84
2. Group Theory 87
S. L. Altmann
1. Symmetry Operators and their Representation 90
2. Elements of Group Theory 108
3. Representations 116
4. The Direct Product 134
5. Symmetry Properties of Functions 143
6. Some Important Groups 155
7. Tables of Characters for the Point Groups 171
References 182
ix

X CONTENTS
3. Chemical Binding 185
C. A. Coulson and J. T. Lewis
1. General Principles 186
2. Applications 205
References 246
4. Molecular Spectra 247
D. M. Dennison and K. T. Hecht
1. General Considerations 248
2. Diatomic Molecules 250
3. Polyatomic Molecules 269
References 322
5. Elements of Quantum Statistics 323
D. ter Haar
1. Introduction 323
2. Systems of Bosons 338
3. Systems of Fermions 346
References 350
6. Theory of Solids 351
R. M. Thomsorf
1. Free Electrons in a Box 354
2. The One-Electron Approximation 355
3. Band Approximation 364
4. Correlation Problems 381
5. Lattice Vibrations 388
6. Electron-Lattice Interactions 398
7. Imperfections 409
References 416
7. The Quantum Mechanics of Liquids 417
H. N. V. Temperley
1. de Boer's Extension of the Principle of Corresponding States 419
2. Some Simple Liquid Models » 423
3. The Properties of Liquid Helium 446
References 460
Author Index , 462
Subject Index 464

Contents of Volu
io-/
Elements
Preliminaries
H. M ARGENAU
Fundamental Principles of Quantum
Mechanics
H. M ARGENAU
Exactly Soluble Bound State
Problems
R. A. BUCKINGHAM
The Continuum
R. A. BUCKINGHAM
Stationary Perturbation Theory
A. DALGARNO
The Variational Method
B. L. MOISEIWITSCH
The Asymptotic Approximation
(AA) Method
BERTHA SWIRLES JEFFREYS
Transitions
D. R. BATES
Theory of Collisions
E. H. S. BURHOP
es 10-1 and 10-111
70-///
Radiation and High Energy
Physics
Relativistic Wave Equations
L. L. FOLDY
Noncovariant Quantum Theory of
Radiation
G. N. FOWLER
Covariant Theory of Radiation
G. N. FOWLER
Meson Theory and Nuclear Forces
J. C. GUNN
Nuclear Structure
K. A. BRUECKNER
Hidden Variables in the Quantum
Theory
DAVID BÖHM
xi

QUANTUM THEORY
A TREATISE IN THREE VOLUMES
/. Elements
//. Aggregates of Particles
///. Radiation and High Energy Physics

M. J. Seaton
1. Introductory Survey 3
1.1 Lines and Levels 3
1.2 Selection Rules 4
1.3 The Bohr Theory 4
1.4 Atomic Units 5
1.5 Relativistic Effects 5
1.6 Magnetic Moments and Spin-Orbit Energy 5
1.7 Nuclear Charge 7
1.8 Schrödinger Equation 7
1.9 Procedure and References 8
2. The Central Field Model 9
2.1 The Physical Idea 9
2.2 Self-Consistent Field 10
2.3 The Central Field Equation 10
2.4 Spherical Harmonics - 11
2.5 Spin Functions 12
2.6 One-Electron Orbitals . . . 12
2.7 Radial Functions 13
2.8 One-Electron Energies 13
2.9 Central Field Quantum Numbers 14
3. Spectra of Alkali Atoms 14
3.1 The Li Absorption Spectrum 15
3.2 The Li Emission Spectrum 15
3.3 Ground State Quantum Numbers in the Li Iso-Electronic Sequence . 17
3.4 Other Alkali Atoms 18
3.5 Alkali Fine Structure 18
3.6 The Zeeman Effect 20
3.7 The Paschen-Back Effect 22
4. The Exclusion Principle and the Periodic Table 23
4.1 The Pauli Exclusion Principle 23
4.2 The Periodic Table 24
1
1. Complex Atoms

2 M. J. SEATON
5. Symmetry Properties of Atomic Wave Functions 25
5.1 Exchange Symmetry 25
5.2 Antisymmetrical Functions for N Electrons 26
5.3 Parity 27
6. Perturbation Procedures 28
6.1 The Central Field Model and the Exact Electrostatic Hamiltonian . . 28
6.2 Commuting Operators 29
7. The Quantum Theory of Angular Momentum 30
7.1 Angular Momentum Operators 30
7.2 The Operator / _ 31
7.3 Angular Momentum Coupling 32
7.4 Calculation of Vector-Coupling Coefficients for jx = j2 = 1 33
7.5 Coupling Coefficients for jx = ;2 = £ 35
7.6 Properties of the Coupling Coefficients 35
7.7 Symmetry Properties of Coupled Eigenfunctions for jx = j2 36
7.8 Scalar Operators 36
8. Coupling Schemes for Electron Wave Functions 37
8.1 Coupling Schemes for One-Electron Problems 37
8.2 Many-Electron Systems: LS and ;'; Coupling 39
9. The Helium Atom 40
9.1 Separation of Space and Spin Coordinates 40
9.2 The He Ground State 41
9.3 He Excited States 41
9.4 The He Energy Level Diagram 43
9.5 Reduction of Integrals 44
9.6 Calculation of E(ls2p) 47
10. Closed Shells and One Electron Outside of Closed Shells 48
10.1 States Which May Be Represented in Terms of Single D Functions . . 48
10.2 The Normalization of D Functions 49
10.3 Energies of States Represented by Single D Functions 49
10.4 Interaction Between an Electron and a Closed Shell 51
10.5 Interaction Between Two Closed Shells 54
10.6 The Energy of a Closed Shell 54
10.7 Energy Expressions for Na+ and Na 54
11. Two Electrons Outside of Closed Shells 55
11.1 Allowed Values of SL 55
11.2 Interactions With Closed Shells 55
11.3 Two-Electron Wave Functions 56
11.4 The Electrostatic Energies: Theory 57
11.5 The Electrostatic Energies: Comparison with Experiment . . . . . 59
11.6 The Spin-Orbit Energy in LS Coupling: Theory 60
11.7 The Spin-Orbit Energy in LS Coupling: Comparison with Experiment . 62
11.8 Intermediate and ;';' Coupling 63
11.9 Configuration Interaction 65
12. Three Electrons Outside of Closed Shells 65
12.1 Parent Terms 65
12.2 Three Equivalent Electrons: Fractional Parentage Coefficients . . . . 66

1. COMPLEX ATOMS 3
13. The Calculation of Atomic Wave Functions 68
13.1 Variational Methods 68
13.2 Simple Analytic Functions for He 70
13.3 The Hartree-Fock Method: General Theory 71
13.4 The Hartree Self-Consistent Field 73
13.5 The Hartree-Fock Radial Equations 73
14. Radiative Transition Probabilities 75
14.1 Electric Dipole Radiation 75
14.2 The Dipole Moment 76
14.3 Electric Quadrupole Radiation and Magnetic Dipole Radiation . . . . 76
14.4 Spin, Parity, and Configuration Selection Rules 77
14.5 The One-Electron Electric Dipole Moment 78
14.6 Selection Rules for 5, L and / 79
14.7 Relative Intensities 79
14.8 The Matrix Elements of the Dipole Length and of the Dipole Momentum 81
14.9 The /-Sum Rule 81
14.10 The Calculation of Absolute Transition Probabilities 83
14.11 Forbidden Lines 83
References 84
1. Introductory Survey
1.1 Lines and Levels
The most detailed information concerning the structure of complex
atoms is obtained from spectroscopic observations. One may explain a
large number of lines in an atomic spectrum by postulating the existence
of a smaller number of energy levels. With energies Ea the line frequencies
are given by hv^ = Ea — Ea,. An atom with energy Eg, where Eg is the
smallest of the Eat is said to be in the ground state. In many spectra one
observes spectral series in which the separation of successive lines decreases
uniformly with increasing frequency and tends to zero at a definite spectral
limit corresponding to an energy E^ ; beyond this limit a continuous spectrum
may be observed.
The reality of energy levels is confirmed by electron impact studies;
with electrons of kinetic energy %mv2
incident on ground state atoms a line
of frequency v^ is observed only when mv%
is at least as great as (Ea — Eg).
As soon as the kinetic energy exceeds (E^ — Eg) it is found that collisional
ionization can occur; the spectral limit E^ therefore corresponds to an
ionization limit.

4 M. J. SEATON
1.2 Selection Rules
Once the energy level scheme has been obtained it is found that many
transitions between levels are not observed as spectral lines. One may
therefore suppose radiative transitions to be governed by certain selection
rules. The theory of atomic structure must explain, not only the existence
of energy levels, but also the selection rules governing transitions between
levels and the great variations of intensity of the observed lines.
1.3 The Bohr Theory
From a classical standpoint atomic spectra remained largely inexplicable ;
many of the problems presented are of an essentially quantum mechanical
nature. Bohr, in the first major theoretical advance, postulated that electron
angular momentum was quantized according to the rule L = nh with n
integer. He supposed a hydrogenic atom to be composed of a massive nucleus
of charge Ze and an electron of charge — e. Assuming the laws of non-
relativistic classical mechanics, with electrostatic interactions the only
forces between the particles, the energy levels were found to be given by
1 Z2
e2
2 n2
a0
and the radii of circular orbits by
n2
rn = —x a0. (2)
In place of energies it is often convenient to use term values defined by
Γ. — § . (3)
The Tn will then be positive numbers and, with En in cgs units, Tn will be
in units of cm-1
. From (1),
72
Tn=R-ï (4)
n2
with
R = ^-X — x 4 - = 109737 cm-1
. (5)
2 a0 he
Introducing the fine-structure constant a = e2
jc% we may put R = α/4π#0.

1. COMPLEX ATOMS 5
For the wave number (reciprocal of the wavelength) of a hydrogen line
n ->ri we obtain
The above formulas apply in the limit of an infinitely massive nucleus.
With a nucleus of mass M we must use the reduced mass mMj(m + M)
in place of the electron mass m. This means that R must be replaced by
7? - R
M
" 1 + (mjM) '
The Bohr theory, with Z = 1 and M equal to the proton mass, is then in
good agreement with observed hydrogen wave numbers so long as fine
structure, observed at very high resolution, is not considered.
1.4 Atomic Units
We define the atomic units of mass, length, and time on putting
m = e = h = 1 ; in these units En = — Z2
/2n2
and rn = n2
Z. The atomic
length unit is a0 = 0.5292 x 10~8
cm and the atomic energy unit is e2
/a0
which is equal to 27.20 eV or twice the ionization energy of the hydrogen
ground state. The fine structure constant, a = e2
jc%) is a dimensionless
number approximately equal to 1/137. In atomic units the velocity of light
is therefore c ^ 137.
1.5 Relativistic Effects
In atomic units the Bohr angular momentum condition for circular
orbits is L = vnrn= n. Therefore vn = Zjn and (vjc)2
^ (Zj'137n)2
. It
follows that (vjc)2
will be small for Z small and, in consequence, a non-
relativistic theory should provide a good approximation.
1.6 Magnetic Moments and Spin-Orbit Energy
A circulating current i produces a magnetic moment μ = iA/c where
A is the area of the current circuit described in the usual vector sense (Fig. 1).
For an electron of charge — e circulating with velocity v in an orbit of
radius r the mean current is i = — evfinr and the area is A = nr2
. The
magnetic moment is therefore μ = — ex X v/2c. Introducing the momentum
p = ms and the angular momentum L = r x p we obtain
^ = - 2 ^ · (6)

6 M. J. SEATON
This is the magnetic moment associated with an electron orbital angular
momentum L. An electron also has a spin angular momentum S and an
associated spin magnetic moment
FIG. 1.
Ps=-
2mc
28. (?)
The spin magnetic moment is said to be anomalous
because (/JS/S) is twice (μχ,/£).
The interaction energy between a magnet of moment
μ and a magnetic field H will be — μ · H. Consider a
magnet moving with momentum p in an electric field E.
In a frame of reference fixed relative to the magnet
there will be a magnetic field* H = ( E x p)/wc and the classical interaction
energy will be — μ · H. If the magnetic moment is due to electron spin
a relativistic treatment2
shows that the interaction energy is — μ5 · H.
Consider an electron moving in a central field with potential energy V(r).
The electrostatic potential per unit charge will be — Vje and the electric
field will be E = —
- grad (— Vje) from which we obtain
E = — —-r.
er dr
The magnetic field which interacts with the spin magnetic moment is there
fore
H = — E x p = — L,
mc emc r dr
L being r x p , and the interaction energy is
S L. (8)
This is known as the spin-orbit interaction. The expression (8) may be
obtainedf using Dirac's theory of the electron.
For hydrogenic ions
1 dV _ Ze2
r dr rz
* The transformation required to obtain this result is discussed by Heitler.
f See Condon and Shortley,3
p. 130.

1. COMPLEX ATOMS 7
The component of 8 in any given direction may take the values ± %. To
obtain an order of magnitude estimate* of AEsp we put
With r = rn = n2
/Z we then obtain
ΔΕ ~ Z 4
^ i p
" 4(137) W
in atomic units. Since (vjc)2
^ (Z/137n)2
we may expect that, for small
Z, the spin-orbit energy will be of a magnitude comparable with that arising
from other magnetic and relativistic effects. For larger values of Z, however,
we may expect the spin-orbit energy to be the most important correction
to the nonrelativistic theory in which only electrostatic forces are considered.
For light atoms the spin-orbit energy will be small compared with the
electrostatic energy. One may therefore calculate the wave functions for
light atoms taking only electrostatic energies into account and then estimate
the spin-orbit energy using perturbation methods.
1.7 Nuclear Charge
The Bohr theory gives agreement with observed frequencies for the
following ions and assigned Z values:
Ion H He+ Li+2
Be+3
B+4
Z 1 2 3 4 5
This suggests that the atomic number, as determined by the sequence of
elements in the periodic table, is equal to the nuclear charge Z. Confirmation
is provided by other evidence, such as α-particle scattering and X-ray
spectra. The number of electrons will be denoted by N. For neutral atoms
N will be equal to Z, for positive ions N will be less than Z and for negative
ions N will be greater than Z.
1.8 Schrödinger Equation
For an iV-electron atom with nuclear charge Z the nonrelativistic
Schrödinger equation is
- ^ Σ tf + ν
(*χ,*2, ...,ΤΝ)-ΕΦ = 0. (9)
^ t = 1 J
* A more exact treatment will be given in § 3.5.

8 M. J. SEATON
Considering only electrostatic interactions the potential energy is
N N
V(tvrt,...,TN) = -Ze>2Jy + e* Σ 7" (10
)
i = l *,7 = 1
where r^ = r{ — r;-|. A coordinate system is used with the nucleus, assumed
to be of infinite mass, at the origin. We use the notation Σ^ to denote
a sum over all pairs (i, j), each pair being counted once only and i = / being
excluded: one may put
N N N-1
Σ = Σ Σ- (ID
i,j = 1 i = j' +1 ; = 1
The wave mechanical! theory for hydrogenic ions (Vol. I, Chapter 3,
§ 5.1.1) is obtained on putting N = 1 and V(r) = — Ze2
/r in (9). One obtains
the expression (1) for the energy levels but n is no longer interpreted as an
angular momentum quantum number.
1.9 Procedure and References
Exact solutions of the Schrödinger equation have not been obtained
for N ^ 2. For He and other light systems, however, elaborate variational
calculations give energies which, with relativistic and spin corrections, are
of an accuracy comparable to that of the best experimental determinations.
For heavier systems more drastic approximations are necessary. The
approach adopted is often suggested by physical as much as by purely
mathematical considerations.
We begin by considering the central field model and the interpretation
of alkali spectra. Electron spin variables are introduced at an early stage
for the reason that, even although the energy directly associated with electron
spin may be small, the spin variables are of fundamental importance in
connection with the symmetry properties of atomic wave functions. The
Pauli exclusion principle is advanced as a postulate which enables the
structure of the periodic table to be understood. It is then shown how the
essential content of the exclusion principle may be restated in a more general
form.
A clear understanding of the elements of perturbation theory is needed
for the study of complex atoms. The theory of angular momentum coupling,
discussed in § 7, is also essential. Complex atomic configurations are discussed
in § 8 to § 12. The last two sections deal with the calculation of atomic
wave functions and with the theory of radiative transition probabilities.

1. COMPLEX ATOMS 9
White's "Introduction to Atomic Spectra"4
and Herzberg's "Atomic
Spectra and Atomic Structure"5
are recommended for accounts of the
basic physics and more elementary accounts of the theory. A useful discussion
of the foundations of the theory is given by Eyring, Walter, and Kimball.6
The standard theoretical work is "The Theory of Atomic Spectra" by Condon
and Shortley.3
Since the appearance of this book the most important
theoretical advance is the introduction of tensor operator methods by
Racah.7
Valuable accounts of the quantum theory of angular momentum
and of tensor operator theory are contained in the recent books by Edmonds8
by Rose9
and by Fano and Racah.22
An excellent account of atomic
structure calculations is given in a recent book by Hartree.10
2. The Central Field Model
2.1 The Physical Idea
In order to obtain approximate solutions of the Schrödinger equation
(9) the first step is to choose a suitable simplified form for the wave function.
The "best" wave function consistent with this form may be determined
later using variational methods. In the central field model each electron
is considered to move in the field of the nucleus and a mean central field
due to the charge clouds of the other electrons. If electron* i moves in the
field VjHfù the total potential energy will be
N
=ΣVii«). (12)
On substituting this potential in (9) we obtain an equation in which the
variables are separable. A solution will be
where
and
φ = (pMtpifa). ..φΝ{τΝ)
_-■£* + *(*)-*
Ε = ΣΕ,
9>;(r<) = 0
(13)
(14)
(15)
* For the moment electrons are assumed to be distinguishable particles. Electron
i means the electron with coordinate r^.

10 M. J. SEATON
2.2 Self-Consistent Field
Let us suppose that solutions of the Eq. (14) have been obtained and
normalized to unity. The charge density due to electron i will be — ^|^(rt)|2
and the potential for electron / will be
(16)
where dx{ = dxi dy{ dz{. The requirement that Vj should be spherically
symmetric may be met by neglecting any nonspherically symmetric terms
on the right-hand side of (16).
It is seen that, given the potentials vit the wave functions φί may be
calculated and that, given the wave functions, the potentials may be cal
culated. By iterative numerical methods self-consistent solutions may be
obtained. This is the self-consistent field method of Hartree10
; it will be
discussed further in § 12.
1.0
r (atomic units)
2.0
FIG.2. The effective charge Zea(r). When r = 0, Zeff is equal to the nuclear charge
Z and when r —
► oo,Zeff tends to the residual charge z = (Z — N + 1). (The curve
given is that calculated by the Hartree-Fock method (§ 10.13.5) for the valence-electron
potential of Na.)
This equation is
2.3 The Central Field Equation
h2
2m
V2
+ v(r)-Enl<p(r)=0 (17)

1. COMPLEX ATOMS 11
The potential v(r) may be written
v(r) = -
e2
Zeft(r)
(18)
The behaviour to be expected for Ze{i(r), as deduced from (16), is shown
schematically in Fig. 2. For r = 0 we have Zefi(0) = Z and for large r we
have Zeff -> z where z = (Z — 2
V + 1) ; z will be termed the residual charge.
For neutral atoms z = 1.
2.4 Spherical Harmonics
Equation (17) permits a further separation of variables on introducing
polar coordinates. We put r = ^,θ,φ) = (r,r) where the unit vector ? = (r/r)
may be used in place of the polar angles θ,φ. The solutions of (17) may be
written
φ(ηΙιηιτ) = Y/W/(r) — Pw/(r)
where Ylm(r) is a normalized spherical harmonic defined by*
(19)
Yl[m, (r) = ( - 1)
|OT;I e x p {imi<p)
Y,_Kl(r) = +
(2π)!/2
exp (— i|»tj|9?)
(2/+l)(/-H)!
2(/+M)!
(2i+l)(/-|m,|)!
1/2
P]"*'1
(cos Θ)
2(l+m,)
1/2
(20)
^"■''(cosÖ).
It may be noted that Υ^ = 1/(4π)1/2
and that
y u = - {SßnfVJr sin θ, Υ10 = (3/4π)!/2
cos Θ,
y 1 _ 1 = (3/8^1
'2
e-'',
sin(9.
The orthonormality relations for the Ylm may be written
(21)
Yj'„;'(r)Y/m/(r)rfr = òwòmfn; (22)
where dr = sin Θ dd άφ. The spherical harmonics form a complete set of
functions in the ''space'' of the angular coordinates r = [θ,φ) hence the
angular dependence of any one-electron wave function may always be
represented by a linear combination of spherical harmonics. The inter-
* It should be noted that a definite phase choice is made in the definition of Y/w ;
this will be discussed in § 6.2.

12 M. J. SEATON
pretation of /, ml as angular momentum quantum numbers will be discussed
further in § 6.
2.5 Spin Functions
It is often necessary to include spin variables in the one-electron functions
(cf. Vol. I, Chapter 2, § 4.4). We postulate the existence of a spin angular
momentum which may have the value in any given direction, conventionally
chosen to be the Oz axis, of msh with ms = ± J. We introduce a spin co
ordinate a which may be equal to ± £ and the spin functions ô(msa). These
are delta functions in the sense that ô(msa) is zero for ms Φ a and unity for
ms = a. The orthonormality relation for these functions is
]£ ô(msa)ô(msa) = ômsms'- (23)
The functions ô(msa) form a complete orthonormal set in the "spin space''
o any spin state of an electron may be represented in terms of a linear
combination of the functions ô(msa).
2.6 One-Electron Orbitals
When spin is included the one-electron functions may be written
φ{ηΙιηίηι8) = d(tnsa) Yimfi) — Pm(r)· (24)
For the coordinates we use the single symbol x = (τ,σ).
More generally we may consider one-electron functions <p(oc|x) where a
might stand for the set of quantum numbers (nlm^) or might stand for
some other convenient set of quantum numbers. The <p(a|x) are termed
one-electron orbitate. Any function 9?(oc|x) may be expressed as a linear
combination of functions of type (24).
For any operator P, one electron matrix elements may be written
(α|Ρ|α') = <ρ*(α|χ)Ρ<ρ(α'|χ) dx (25)
where j . . .dx means an integration over the spatial coordinates r and a
sum over the spin coordinates a. In particular
(α|α') = I <p*(a|x)<p(a'|x) dx.
An orbital will be normalized if (α|α) = 1.

1. COMPLEX ATOMS 13
2.7 Radial Functions
The radial functions Pnl(r) in (24) will be solutions of
d2
1(1+1)
dr2 U{r) — Eni Pm(r) = 0 (26)
where
2m . v _ 2m ^
u(r) = - p - v(r) and Fnl=-— Enl. (27)
Since Enl is negative for bound states, εη1 will be positive. Numerical methods
must be used to obtain solutions of (26) which are everywhere bounded
and continuous. Such solutions will exist only for certain discrete eigenvalues
of eM/; these must be found by trial and error (practical procedures are
discussed by Hartree10
). The quantum number n is defined to be such that
the number of nodes in the radial function, excluding the origin and infinity, is
equal to (n — I — 1). This is consistent with the usage of n for hydrogen.
For given / the function with no nodes has n = (I + 1) ; this is the smallest
n and corresponds to the largest εη1.
2.8 One-Electron Energies
Having calculated Enl we define Znl by
Comparison with the hydrogenic formula (1) and with the form of Ze{l(r)
shown in Fig. 2 leads us to expect that Znl will be smaller than Z but greater
than the residual charge z = (Z — N + 1).
It is important to consider the dependence of Znl on / and on n. Considering
the /-dependence first we may expect that Znl will be smaller than ZnV if
/ is greater than /'. The physical reason is that an electron with large angular
momentum is usually found at large radial distances where Zef[(r) is small
whereas an electron with small angular momentum will be more likely to
penetrate into the region of large Zeff. The mathematical reason is obtained
on considering (26) for / large. The equation will be dominated by the term
— /(/ + l)/r2
for r small and will be sensitive to u(r) only for large values
of r where Zeff is small.
The dependence of Znl on n may be obtained in a similar way. For large
n the electron is most likely to be at large radial distances and therefore
Znl will be small. Hence Znl will decrease with increasing n.

14 M. J. SEATON
A formula in many ways more useful than (28) is the Ritz formula
1 z2
e2
which defines the quantum defect μη1. The effective quantum number is
n*t = n — μη1. It may be shown10
that the quantum defect tends to a
finite limit as n -► oo and it is usually found that μη1 varies slowly with n.
Comparing (28) with (29) and recalling that Znl decreases with increasing /,
we may expect μη1 to decrease with increasing /.
2.9 Central Field Quantum Numbers
The one-electron central field functions are specified by the quantum
numbers n^m^^ the energy depending only on ni and l{. For N electrons
the total energy is
i
The quantum numbers
ηλ11}η212ί... ηΝΙχ
are referred to as the configuration. Two or more electrons which have the
same quantum numbers n}{ are said to be equivalent. A configuration of
q equivalent nl electrons is denoted by nlq
.
3. Spectra of Alkali Atoms
A characteristic of the alkali atoms Li, Na, K, Rb, and Cs is that a single
outer electron is readily detached. This is termed the valence electron. The
evidence for this, other than spectroscopic, includes the low ionization
potentials* (5.39 ev for Li compared with 24.58 ev for He, and 5.14 ev for Na
compared with 21.56 evforNe), the fact that in the solid state the alkalis
are metals (the outer electron explaining the high electrical conductivity),
and the tendency of the alkalis to form ionic compounds. We seek to explain
the general features of alkali spectra by assuming the model of a single
electron moving in a central field. We begin by considering the identification
of observed energy levels1
in Li.
* Numerical values for atomic energies are quoted from the tables of Charlotte
Moore.12
t Our treatment is similar to that given by Herzberg.5

1. COMPLEX ATOMS 15
3.1 The Li Absorption Spectrum
This is obtained on passing light through atomic lithium vapour. With
moderate resolution we observe a series of dark absorption lines converging
to a spectral limit. This is the principal series. The wave numbers will
be denoted by σηρ, and the wave number of the limit by T^p). One obtains
T{p) = 43487 cm-1
. Assuming the ground state to be the initial state for
the absorption process the energy levels required to obtain the principal
series are shown in Fig. 3. This assumption requires that T{p) = 43487 cm- 1
be the term value of the ground
state. This agrees with the
ionization limit as determined
from electron impact studies.
Putting anp = T{p) - Tnp it
is found that a fair approxima
tion is obtained using the simple
hydrogenic formula
A better approximation is
Tn = R
np
[n - μηρ)2
with μηρ = 0.0471 and almost perfect agreement is obtained with
μηρ = 0.0471 - 0.0241/n2
.
3.2 The Li Emission Spectrum
In addition to the principal series at least three other series may be
observed in emission; the sharp and the diffuse series (recognized by the
appearance of the lines) and the fundamental series with wave numbers
close to those of the hydrogen Paschen series (n' -*w = 3). The wave
numbers may be represented by the following approximate formulas:
Sharp series
a„s = Tw - π , J w = 28583 cm-1
, μΜ = 0.40,
(n (Àns)
« = 2,3,4,...
Continuum
^Ζ
Ground state
FIG. 3. Energy level diagram for the Li
absorption spectrum.

16 M. J. SEATON
Principal series
Onp = T{p) —
Diffuse series
Ond = T(d)
Fundamental series
R
(n- μηρ)2
'
R
(n — μηά)2
'
Onf = T{f) —
R
(n- μ„/)2
'
T{p) = 43487 cm-1
, μηρ ^ 0.047,
n = 2,3,4,.
T{d) = 28583 cm-1
, μηά = 0.001,
n = 3,4,5,...
T(f) = 12204 cm"1
, μηί = 0.000,
n = 4,5,6,...
2Q 2p Hydrogen
n = œ
n = 2
FIG. 4. The Li energy level diagram. The energy levels of H are shown on the right.
The diagram shows the first three transitions of the principle series (np —
► 2s), the sharp
series (ns —
► 2p), the diffuse series (nd —
► 2p) and the fundamental series (nf —> Sd).
The corresponding energy level diagram is shown in Fig. 4. The diagram
and the observed wave numbers are consistent with the relations
T(p) = T2s, T{s) = T(d) = T2p, Τφ = Tw
in Fig. 4 we have four series of levels with spectroscopic labels s, p, d,
and /. This classification of the observed levels might be expected to be a
classification in terms of the quantum numbers /. Using the fact that the
quantum defect μη1 decreases with increasing /, we obtain the following
relation between spectroscopic labels and /-quantum numbers:
1=0 1 2 3
s p d f.

1. COMPLEX ATOMS 17
The notation is continued with
/ = 4 5 6 7
g h i k.
The quantum numbers nl are usually written nsfnpfnd,... for / = 0,1,2,... .
From the diagram (Fig. 4) we obtain the selection rule that all observed
lines correspond to a change of ± 1 in I.
Our assignment of principal quantum numbers n agrees with theoretical
expectation for / = 1,2,3 the lowest p state being 2p, the lowest d state 3d
and the lowest / state 4/. There is, however, an apparent anomaly in that
the lowest s state is 2s and not Is. We say that Is is an excluded state for
the valence electron.
I 2 3 4 567
π 1 1 1—m 7
/
y
s
Li
j i I
1.0 0.5 0
1/z
FIG. 5. The effective quantum number (n — μ) for the ground states of the Li iso-
electronic sequence. The residual change z is 1 for Li, 2 for Be+, 3 for B+2
,. . . . The
quantum defects are plotted on a linear scale of z. It is seen that (n — μ) tends to
2 as z tends to infinity.
3.3 Ground State Quantum Numbers in the Li Iso-Electronic Sequence
The assignment of principal quantum number 2 for the lowest s state
in Liisconfirmed by considering the iso-electronic sequence Li, Be+
, B+ 2
, C+ 3
..
of ions with the same number of electrons but increasing nuclear charge. The
corresponding spectra are denoted by Li I, Be II, B III, C IV,..., the Roman
numerical being equal to the residual charge z. For the ground state we put
L.J
=1
I
c:
1.5
(η-μ)2

18 M. J. SEATON
and plot the effective quantum number (n — μ) as a function of jz (Fig. 5).
As the nuclear charge increases the field approximates more closely to the
Coulomb form and the quantum defect decreases. It is seen that the effective
quantum number tends to 2 as z goes to infinity.
3.4 Other Alkali Atoms
Similar analyses may be made for the other alkalis.* The selection
rules Al = ± lis found to hold but very weak lines are occasionally observed
which violate the rule. Table I gives approximate quantum defects μη1
TABLE I
QUANTUM DEFECTS μηί FOR THE ALKALIS
Series
Atom s p d f g
Li
Na
K
Rb
Cs
0.40 (2)
1.35 (3)
2.19 (4)
3.13 (5)
4.06 (6)
0.04 (2)
0.85 (3)
1.71 (4)
2.66 (5)
3.59 (6)
0.00 (3)
0.01 (3)
0.25 (3)
1.34 (4)
2.46 (5)
0.00 (4)
0.00 (4)
0.00 (4)
0.01 (4)
0.02 (4)
-
0.00 (5)
0.00 (5)
0.00 (5)
0.00 (5)
(assumed constant in each nl series) for the alkali atoms. It is seen that
μη1 increases with Z and decreases with I. Following each quantum defect
we give in brackets the value of n for the lowest level in each series.
3.5 Alkali Fine Structure
All alkali levels except s levels are found to be split into two components,
the splitting increasing rapidly with Z. We seek to explain this in terms
of spin-orbit interaction. From (8) we have
Jïip = f(r)8-L (30)
for the spin-orbit Hamiltonian, ξ(r) being
*
W = ^ ? T ! ·
(31)
* Unambiguous identification of the various series depends on consideration of
fine structure and of the Zeeman effect. For the assignment of principal quantum
numbers of the heavier alkalis more reliance must be placed on theory.

1. COMPLEX ATOMS 19
The first step in the calculation of the spin-orbit energy is to introduce the
total angular momentum
J = S + L.
From J2
= (S + L)2
we obtain
S.L = £ ( J 2
- S 2
- L 2
) .
Anticipating results to be obtained in § 7 we take the allowed values
of S · L to be
^ [ / ( / + l ) - s ( s + l ) - / ( / + l ) ]
with s = £ and / = / ± £· The second step in the calculation of the spin-
orbit energy is to replace ξ(r) by the mean value JP2
j ξ(r) dr. Combining
these results the expression for the spin-orbit energy is
AE(nlj) = i[/(/ + 1) - i - /(/ + 1)]?(«/) (32)
with
00
ζ(ηΙ) = ηΛρ2
Ηΐ(ήξ(ήατ. (33)
0
The theory therefore predicts a doublet fine structure with separation
ÔE(nl) = AE(nlj = I + J) - AE(nlj = I - ) = {2l + 1)ζ{ηΙ). (34)
A difficulty arises in the case of s states. With / = 0, the only allowed value
of / is + . The expression for 8 · L is zero but the integral (33) is divergent.*
A more exact treatment, using the Dirac relativistic theory (Condon and
Shortley,3
p. 130) shows that the definition (31) of ξ(r) should be modified
for small r and that the integral for £(ws) then converges. There is therefore
no splitting of s states but, according to the relativistic theory, there is a
shift of s state energy levels.
The predictions of theory are in agreement with observations. Observed
radiative transitions between fine structure levels are found to satisfy the
selection rules
Aj = 0,±1 but not (/ = 0) -*(/ = 0).
* It may be recalled that Pni(r) behaves as rl + 1
for small r and that £(r), defined
by (31), behaves as f~8
; for / = 0 the integrand of (33) therefore behaves as r_1
.

20 M. J. SEATON
Taking Pnl to be a hydrogenic function for nuclear charge z and v = — ze2
/r
one obtains (Condon and Shortley,3
p. 117)
„ „ = _«W *4
^n
' mVal nH(2l+l)(l+l)-
This gives for the term-value difference
ÒTnl=
hT =
nWTT)' (35)
This agrees with experiment in that it predicts that ôTnl should decrease
with n and with I and should increase with increasing z.
A quantitative check on the theory may be made5
for the 2p splitting
in the Li isoelectronic sequence. We replace z in (35) by Z2p defined by
T2p = RZlpl4:. This gives calculated values of ôT2p which are compared
with observed values in Table II.
TABLE II
F I N E STRUCTURE OF 2p STATES IN THE Li ISOELECTRONIC SEQUENCE
o^tcm-1
)
Li
Be+
B+2
C+3
N+4
0+5
It should be noted that (35) does not give agreement with observed fine
structure in hydrogen. This is because other magnetic and relativistic
corrections are of the same order as the spin-orbit energy.
3.6 The Zeeman Effect
For an alkali atom we continue to consider the model of a single electron
moving in a central field. By (6) and (7) the magnetic moment of the atom
will be
* = - 2 ^ L + 28
> = - 2 ^ , + S
>
Observed
0.338
6.61
34.4
107.4
259.1
533.8
Calculated
0.395
6.39
32.1
100.4
243.1
500.8

1. COMPLEX ATOMS 21
since J = L + S. If placed in a uniform magnetic field the atom will have
an additional energy,
J£(magO = ^ - H = + - | 3 - - ( J + S ) . (37)
Choosing a coordinate system such that H is in the Oz direction,
where |H| = Jf. From a strict quantum mechanical standpoint we should
regard (37) as an additional operator to be included in the Hamiltonian.
A perturbation theory treatment of the contribution will be discussed in
§ 9.1. For the present we consider a more physical approach.
//
/ /
FIG. 6. Coupling of S and L to give a resultant J. In the absence of an external magnetic
field J is a constant vector and S and L precess around J due to spin-orbit coupling.
For the Zeeman effect (weak magnetic field) one considers only the components S and
L of S and L which are parallel to J.
J will be a constant vector in the absence of an external field and may
still be treated as a constant vector in the limit of a weak field. If the spin
magnetic moment were not anomalous (§ 1.6.) the calculation would be
very simple. We would have ΔΕ = [eJ^ßmc)Jx = {e3^l2mc)Umjf assuming
the allowed values of Jz to be Km^ with w;- = /,(/ — 1),... (— /). This result
is not in agreement with experiment; the anomalous magnetic moment
must be taken into account. We consider that J = S + L is a constant
vector, that 8 and L are constant in magnitude but that, due to spin-orbit
coupling, S and L rotate around J (see Fig. 6a). We consider S to be the
sum of two components, a component S parallel to J and a component 5 X
perpendicular to J (Fig. 6b). The interaction between S1
- and H gives a
rapidly fluctuating contribution to the energy which is neglected (it being

22 M. J. SEATON
assumed that S 1
rotates many times around / during the course of an
observation to determine the energy). The parallel component of 8 is
S =
(S · J) J
P
Considering only this component (37) gives
S J
ΔΕ =
2mc 1+-
P
J,
Since L2
= (S - J)2
= S2
+ J2
- 2S · J we have
J£(mag.) =
etf
2mc
1 +
(S2 + p _ L2)
2/2 Λ
and replacing S2
J2
,L2
by fc2
£(£ + l),Ä2
/(/ + l),h2
l(l + 1) and / , by Ηηιρ
zd£(mag.) = 2 ^ r g ( ^ w ,
where
g(i/) = 1 +
2/(/ + 1)
(38)
(39)
is the Lande g-factor. These results agree with experiment for fields so weak
that AE (mag.) is small compared with the spin-orbit energy AE(nlj).
H I
FIG. 7. In the Paschen-Back effect it is assumed that the spin-orbit coupling between
S and L has been broken down by a very large magnetic field. The vectors S and L
precess independently about the direction Oz of the applied field. Only the components
Sz and Lz axe taken into account when calculating spin-orbit interactions.
3.7 The Paschen-Back Effect
We consider an external magnetic field so large that ΔΕ (mag.) is much
greater than the spin-orbit energy. The coupling of 8 and L to give J is
then completely broken down and / does not enter into the theory. Only

1. COMPLEX ATOMS 23
the components Sz and Lz of S and L will be constant, since the vectors
S and L rotate around the magnetic field direction (Fig. 7). Replacing
Sz and Lz by %ms and Hml one obtains
Zl£(mag.) = - — (mt + 2ms)%.
For the spin-orbit energy we consider the operator f(r)S· L (§ 3.5.). With
the rotating vectors of Fig. 7 only SZLZ need be considered, the mean values
of SXLX and SyLy being zero. In the Paschen-Back limit the spin-orbit
energy is therefore ζ(ηΐ)ηι5ψηι.
4. The Exclusion Principle and the Periodic Table
It has been seen that many properties of the alkali atomsmay be understood
in terms of a model of a single electron moving in a central field. We may
now ask : why is it that the model proves so satisfactory for these particular
elements? We consider more generally how the entire structure, of the
periodic table may be understood. In addition to the idea of the central
field model we require the postulate of the Pauli principle.
4.1 The Pauli Exclusion Principle
This states that no two electrons can be in the same quantum state. We
shall usually consider the quantum states
9>(η&»,*»,|χ) = ô(msa)Yimi(T)(llr)Pnl(r).
For a given nl there are 2(21 + 1) such states, corresponding to ml = I,
(I — 1)... (— I) and ms = ± . It should be emphasized that, so far as
the exclusion principle is concerned, there is no necessity to choose the
particular set of states* (pfolm^). The important point is that the functions
ç>(tt/wjWs) form a complete set in the sense that any other set of functions
for the central field energy level nl could be expressed as a linear combination
of the functions (pfolm^) and that the number of linearly independent
functions would always be 2(21 + 1).
A closed shell is said to exist when 2(2/ + 1) equivalent electrons occupy
all the quantum states of a level nl.
* We could, for example, choose functions with angular momenta quantized in
the Ox direction instead of in the Oz direction.

24 M. J. SEATON
4.2 The Periodic Table
We require the ideas on binding order developed in § 2.8 : that, for example,
an atomic Is electron will be the most tightly bound and that a 2s electron
will be more tightly bound than a 2p electron.
For the first 36 elements of the periodic table we are led to the assignment
of ground configurations shown in Table III. We start with the H Is ground
TABLE III
T H E PERIODIC TABLE
H Is Al -3s2
3/> Mn -3d5
4s2
He Is2
Si 3£2
Fe -3de
4s2
Li ls2
2s P 3£3
Co -3rf7
4s2
Be - 2 s 2
S 3/>4
Ni -3tf8
4s2
B -2s2
2/> Cl 3£5
Cu -3rf10
4s
C 2p2
A 3p* Zn -3d1 0
4s2
N 2£3
K 3pHs Ga 4s2
4/>
O 2/>4
Ca 4s2
Ge 4/>2
F 2£δ
Se - 3 d 4s2
As 4/>3
Ne 2£β
Ti -3d2
4s2
Se 4/>4
Na 2/>«3s V -3d3
4s2
Br 4£5
Mg 3s2
Cr -MHs Kr 4/>«
state. In He we add a second Is electron and complete the Is2
closed shell.
The added electron in Li is then 2s. This explains immediately why the
Is state is excluded for the Li valence electron. With Be we complete the
2s2
shell and start adding 2p electrons from B to Ne, the closed 2p* shell
being reached at Ne. With Na we have again an alkali atom, with a 3s
electron outside of closed shells. We continue in the same way building up
to argon with configuration ls2
2s2
2£e
3s2
3£e
. Many of the regularities of
the periodic table, originally deduced from chemical properties of the
elements, are immediately apparent. Thus the rare gases He, Ne and A
have all their electrons in closed shells, and the halogens F and Cl have
outer npb
configurations.
To proceed with the periodic table beyond argon more appeal to ex
perimental evidence is necessary. Thus one might wonder whether a 4s or
a 3d electron would be most tightly bound in K. Experiment shows that
in fact it is 4s. Further complications arise in the building up of the 3d shell.
Thus vanadium, with outer configuration 3<23
4s2
, is followed by chromium
which has configuration 3dHs and not 3dHs2
.

1. COMPLEX ATOMS 25
5. Symmetry Properties of Atomic Wave Functions
5.1 Exchange Symmetry
The Pauli principle was stated in terms of the approximation, based
on the central field model, of assigning quantum numbers to each individual
electron. We now show that the essential content of the exclusion principle
may be restated in a different form which does not depend on the use of
approximate models. This form concerns the symmetry of the wave functions
for interchange of electron coordinates.
Any function f(xvx2) is said to be symmetric if f(xvx2) = + /(x
2>x
i)
and to be antisymmetric if /(x!,x2) = — /(χ
2>χ
ι)· Let φ(χνχ2) be a solution
of the Schrödinger equation
(H - Ε)φ(χνχ2) = 0 (40)
for a two-electron atom. We shall show that φ(χνχ2) must be either symmetric
or antisymmetric. The proof depends on the fact that the Hamiltonian
H(xltx2) is symmetric; this follows* from the fact that electrons are in
distinguishable particles. We first consider an energy E = E0 such that
(40) has nondegenerate solutions φ0. This means that if φ0 and φ0' are any
two solutions of (H — Ε0)φ = 0 then φ0' = αφ0 where a is a constant.
It will be shown later that the ground state of the He atom is an example
of such a nondegenerate state. H being symmetric, ψ0(χ2,χ1) will be a solution
of (H — Ε0)φ = 0 if φ0(χνχ2) is asolution, and therefore ^(x^x^ = αφ0(χνχ2).
But interchange of coordinates does not change the normalization of the
function and therefore a2
= 1 and a — e10
with ô real. The effect of inter
changing coordinates is thus to multiply the function by a phase factor,
φ0(χ2>χ1) = βίο
φ0(χνχ2).
Interchanging twice we have
φ0{Χ2,Χ1) = βίδ
φ0(χνΧ2) = ^(/r0(X
2>X
l)
so that e2tô
= 1 and etô
= ± 1. This proves that 0o(x2,x1) = ± φ0(χνχ2)
for nondegenerate states. Now let us suppose that the atom is brought
into some other state, which may belong to a degenerate energy level, by an
external perturbation. Before the perturbation is turned on the complete
time-dependent wave function will be φ0(χνχ2,ή = φ0(χνχ2) exp (— iE^tß)
* The symmetry of H is obvious for the electrostatic Hamiltonian in (9)

26 M. J. SEATON
and during the transition the complete wave function will be a solution of the
time-dependent Schrodinger equation
Η(χ1}χ2>ήφ(χνχ2,ή = ίη, — φ(χνχ2,ή.
ot
But since H(xvx2>t) is always a symmetric function of x1}x2 any function
φ which is originally symmetric will always remain symmetric, and any
function which is originally antisymmetric will always remain antisymmetric.
We may therefore conclude that all wave functions for many-electron systems
must be either symmetric or antisymmetric for interchange of the coordinates of
any pair of electrons. It is readily shown that the physical properties associ
ated with symmetric wave functions are different from those associated
with antisymmetric wave functions, and since we do not observe different
types of pairs of negative electrons it may be concluded that only symmetric,
or antisymmetric, electron states occur in nature. The Pauli principle
allows us to choose between these two possibilities. The requirement that
H be symmetric imposes the condition that the central field potentials
vvv2 be identical. A symmetrical solution for the two-electron central field
problem is
ψΜ^ιίψΜ^) + <Ρ(αι|χ
2Μα
2|χ
ι)
and an antisymmetric solution is
?(αι|χ
ιΜα2|χ
2) - ?(αι|χ
2Μα2|χ
ι)· (37)
The antisymmetric solution vanishes if ax = a2. We therefore conclude
that only antisymmetric electron states occur in nature. The Pauli principle
is then satisfied automatically.
Henceforth, following Condon and Shortley, we shall adopt the convention
of using capital Ψ only for antisymmetric functions.
5.2 Antisymmetrical Functions for N Electrons
Suppose that we are given a set of one-electron orbitals ^(^Ιχχ),
<p(*2x
2)>' · ·Ψ(Λ
ΝΧ
Ν)' We may then obtain the functions
|<Ρ(α
ι|χ
ι) <P{*2*I)---<P(*NXI)
9?(αι|χ2) <p(a2|x2)...<p(aN|x2)
D(AX) =
]/N
Ψ((Χ.1ΧΝ) < P ( < X 2| X N ) - - .Ç>(OCJV|XJV)
(41)
where A is written for the set of quantum numbers (α^ο^,... α^) and X
for the set of coordinates (xx,x2,.. .χΛτ). Functions of the type (41) will be

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year, he shall be prohibited coming to the church, and when dead be
refused ecclesiastical burial.”
All, rich and poor, noble and simple, on coming to the Sacrament of
Penance, were treated alike. An old fifteenth-century book of
Instructions says—
“Every body that shall be confessed, be he never so hye
degree or estate, ought to shew loweness in herte,
lowenes in speche and lowenes in body for that tyme to
hym that shall hear hym; and or he begynne to shew what
lyeth in hys conscience, fyrste at hys beginnyng he shall
say, Benedicite: and afturwards hys confessor hath
answered Dominus. Sume than, whych be lettered, seyn
here Confiteor til they come to Mea culpa: sume seyn no
ferthere, but to Quia peccavi nimis; some seyn no
Confiteor in latin till at the last end. Of these maner
begynnings it is lytyl charge, for the substance of
Confession is in opyn declaration and schewyng of ye
synnes, in whyche a mannus conscience demyth hym
gulty agenst God. In thys declaration be manye formes of
shewyng, for some scheme and divyde here confession in
thought, speche and dede, and in thys forme sume can
specyfye here synnes, and namely in cotydian confession,
as when a man is confessed ofte; oythes as every day or
every othur day or onus in sevene nyght. Also sume
schewe and here confession by declaration of ye fyve
wyttes, and all may be well as in such cotydyan
confession. Also sume, and the most parte lettyred and
unletteryd, schewe openly her synnes be confession of ye
sevene dedly synnes, and thane they schewe what they
have offendyd God agenste Hys precepts, and then in
mysdyspendyng of here fyve wyttes, and thanne in not
fulfyllyng ye seven dedus of mercy. And so, whanne they
have specyfyed what comyth to here mynde, then yn ye
ende, they yelde them cowpable generally to God and
putte hem in Hys mercy, askyng lowly penaunce for her

synnnes and absolution of here confessor in the name of
holy church.”
The instructions, given by the Canons of the English Church, as to
the method to be followed by priests in hearing confessions, are
simple and to the point. They are to remember that they are doctors
for the cure of spiritual evils, and to be ever ready “to pour oil and
wine” into the wounds of their penitents. They are to bear in mind the
proverb, that “what may cure the eye need not cure the heel,” and
are to apply the proper remedy fitting to each disease. They are to
be patient, and “to hear what any one may have to say, bearing with
them in the spirit of mildness, and not exasperating them by word or
look.” They are “not to let their eyes wander hither and thither, but
keep them cast downwards, not looking into the face of the penitent,”
unless it be to gauge the sincerity of his sorrow, which is often
reflected most of all in the countenance. Women are to be confessed
in the open church, and outside the (lenten) veil, not so as to be
heard by others but to be seen by them.
The place where confessions might be heard was settled in the
Constitutions of Archbishop Walter Reynold, in 1322.
“Let the priest,” it is said, “choose for himself a common
place for hearing confessions, where he may be seen
generally by all in the church; and do not let him hear any
one, and especially any woman, in a private place, except
in great necessity and because of some infirmity of the
penitent.”
Myrc, in his Instructions, says that in Confession the priest is to
“Teche hym to knele downe on hys kne,
Pore other ryche, whether he be,
Then over thyn yen pulle thyn hod,
And here hys schryfte wyth mylde mod.”
The place usually chosen by the priest to hear the confessions of his
people was apparently at the opening of the chancel, or at a bench
end near that part of the nave. In some of the churchwardens’

accounts there is mention of a special seat or bench, called the
“shryving stool,” “the shriving pew,” “the shriving place;” whilst at St.
Mary the Great, Cambridge, there appears to have been a special
erection for Lent time, as there is an entry of expense for “six irons
pertaining to the shryving stole for lenton,” which suggests that these
iron rods were to support some sort of a screen round about the
place of confession. Perhaps, however, it may have been for an
extra confessor, since, as already related, in one place it is said that
the parish paid for three extra priests “to shreve” in Holy Week.
The Holy Eucharist.—All adults of every parish were bound to
receive the Holy Communion at least once a year under pain of
being considered outside the benefits and privileges of Holy Church
and of being refused Christian burial, if they were to die without
having made their peace. Besides the Easter precept, all were
strongly urged to approach the Holy Eucharist more frequently, and
especially at Christmas and Easter, and, as has been already
pointed out, there is some evidence to show that, in point of fact, lay
people did communicate more frequently, and especially on the
Sundays of Lent.
At Easter and other times of general Communion the laity, after their
reception of the Sacrament, were given a drink of wine and water
from a chalice. The clergy were, however, directed to explain
carefully to the people that this was not part of the Sacrament. They
were to impress upon them the fact that they really received the
Body and Blood of our Lord under the one form of bread, and that
this cup of wine and water was given merely to enable them to
swallow the host more securely and easily after their fast.
Extreme Unction.—
“This Sacrament,” says the Synod of Exeter, “is to be
considered as health giving to both body and soul ...
wherefore it is not the least of the Sacraments, and parish
priests, when required, should show themselves ever
ready to visit the sick, and to administer it to such as ask,
without asking or expecting any payment or reward.

“We further order that, avoiding all negligence, parish
priests shall be watchful and careful in the care committed
to them, and that without reasonable cause they never
sleep out of their parishes. And further that in case they do
ever so, they procure some fitting substitute, who knows
how to do everything which the cure of souls requires.”
If by the fault, negligence, or absence of his priest any one, old or
young, shall die without Baptism, Confession, Holy Communion, or
Extreme Unction, the priest convicted of this is to be forthwith
suspended from the exercise of his ecclesiastical functions, and this
suspension is not to be relaxed until he has done fitting penance “for
so grave a crime.”

SACRAMENT OF EXTREME UNCTION
Visitation of the Sick.—The subject of Extreme Unction, “the
Sacrament of the sick,” to be given in danger of death through
sickness, raises the question of the visitation of the sick in a
mediæval parish. The order that all parish priests should visit the
sick of their district every Sunday has already been noticed. It was,
moreover, a positive law of the Church, that every priest should go at
once on being called to a sick person, no matter what time of the day
or night the summons might come. Priests were ordered also to
impress upon all doctors the need of urging sick people and their
friends to send immediately for the priest in all cases of serious
illnesses. Priests, however, were not to wait to be called, but directly
they heard that any of their people were unwell they were warned to
go at once to them.
A chance story, used to enliven a fifteenth-century sermon, illustrates
the readiness of priests to go to the sick whenever they were
summoned.
“I read,” says the preacher, “in Devonshire, besides
Axbridge dwelt a holy vicar, and had in his parish a sick
woman that lay all at the death, half a myle from him in a
town. The which woman at midnight sent after this vicar to
come and give her her rites. Then this vicar with all haste
that he might he rose and rode to the church and took
God’s body in a box of ivory,” etc.
Archbishop Peckham legislated for the mode of carrying the Blessed
Sacrament to the sick, or rather he codified and made obligatory the
usual practice. The parish priest was to be vested in surplice and
stole, and accompanied by another priest, or at least by a clerk. He
was to carry the Blessed Sacrament in both hands before his breast,
covered by a veil, and was to be preceded by a server carrying a
light in a lantern, and ringing a hand bell, to give notice to the people
that “the King of Glory under the veil of bread” was being borne
through their midst, in order that they might kneel or otherwise adore
Him.

If the case was so urgent, that there was no time for the priest to
secure a clerk to carry the light and bell, Lyndwood notes that the
practice was for the priest to hang the lamp and bell upon one of his
arms. This he would also do in large parishes, where sick people
had to be visited at a distance and on horseback. In this case the
lamp and bell would be hung round the horse’s neck.
On the return to the church, should the Blessed Sacrament have
been consumed, the light was to be extinguished and the bell
silenced, so that the people might understand, and not, in this case,
kneel as the priest passed along. Lyndwood adds that the people
should be told to follow the Sacrament with “bowed head, devotion of
heart, and uplifted hands.” They were to be taught also to use a set
form of prayer as the priest passed along, such as the following:
“Hail! Light of the world, Word of the Father, true Victim, Living Flesh,
true God and true Man. Hail flesh of Christ, which has suffered for
me! Oh, flesh of Christ, let Thy blood wash my soul!” The great
canonist says that he himself on these occasions was accustomed to
make use of the well-known “Ave verum Corpus, natum ex Maria
Virgine,” etc.

HEARSE AND PALL, FIFTEENTH CENTURY. CANTORS AT
LECTERN
The bell and light, or lights, for the visitation of the sick, were to be
found by the parish, and the churchwardens’ accounts consistently
record expenses to procure and maintain these lights. In some
places, apparently, the people found two such lanterns instead of the
one which the law obliged them to furnish. In the Archdeacon’s
visitations, also, there were set inquiries to see that the parish did its
duty in this matter. In one such examination there are references to
the necessary “cyphus pro infirmis,” which is stated to be good, bad,
or wanting altogether. What this may have been is not quite clear;
but probably it was the dish in which the priest purified his fingers,
after having communicated the sick person. Myrc gives a rhyming
summary of what a priest should know about visiting the sick. He is
to go fast when called; he is to take a clean surplice and a stole,
“and pul thy hod over thy syght;” in case of death being imminent, he

is not to make the sick man confess all his sins, but merely charge
him to ask God’s mercy with humble heart. If the sick man cannot
speak, but shows by signs that he wishes for the Sacraments
—“Nertheless thou schalt hym Soyle, and give hym hosul and holy
oyle.”
The bishops watched carefully to see that no laxity should creep into
the mode of giving the Viaticum to the sick. Bishop Grandisson, in
1335, issued a special mandate to the priests of his diocese on the
matter, as he had heard that some carelessness had been noticed.
He reminds them that the Provincial Constitutions were clear in their
prescriptions that all were to wear a surplice and stole, unless the
weather were bad, and then these might be carried and put on
before the room of the sick man was entered. They must always
have the light borne before them, however, and the bell was to be
rung to call the attention of the people generally to the passing of the
Sacrament, and thus enable them to make their adoration.
According to most books of instruction on the duties of priests,
before the sick man was anointed or received the holy Viaticum, the
parson was to put to him what were known as “the seven
interrogations.” He was to be asked: (1) if he believed the articles of
the faith and the Holy Scriptures; (2) whether he recognized that he
had offended God Almighty; (3) whether he was sorry for his sins; (4)
whether he desired to amend, and if God gave him more time, by His
grace he would do so; (5) whether he forgave all his enemies; (6)
whether he would make all satisfaction; (7) “Belevest thowe fully that
Criste dyed for the, and that thow may never be saved but by the
merite of Cristes passione, and thonne thonkest therof God with
thyne harte as moche as thowe mayest? He answerethe, Yee.”
“Thanne let the curat desire the sick persone to saye In
manus tuas &cetera with a good stedfast mynde and yf
that he canne. And yef he cannot, let the curate saye it for
hym. And who so ever may verely of very good
conscience and trowthe without any faynyng, answere
‘yee,’ to all the articles and poyntes afore rehersed, he
shalle live ever in hevyne with Alle myghtie God and with

his holy cumpany, wherunto Ihesus brynge bothe youe
and me. Amen.”
Marriage.—So far in this chapter the Sacraments which every
parishioner had to receive at one time or other have been briefly
treated. It remains to speak of the Sacrament of Matrimony, which,
though not absolutely general, yet commonly affected most people in
every parish. “Marriage,” says Bishop Quevil, in the Synod of Exeter
—“marriage should be celebrated with great discretion and
reverence, in proper places and at proper times, with all modesty
and mature consideration; it should be celebrated not in taverns nor
during feastings and drinkings, nor in secret and suspect places.”
That a matter of this importance should be rightly done, the Synod
lays down the law of the Catholic Church on the point; no espousal
or marriage was to be held valid unless the contract was made in the
presence of the parish priest and three witnesses. For, although the
contract of the parties was the essential factor in marriage, still,
“without the authority of the Church, by the judgment of which the
contract had to be approved, marriages are not to be contracted.”

SACRAMENT OF MATRIMONY
The first matter to be attended to in arranging for a marriage in any
parochial church was, as now, the publication of the banns in the
church on three successive Sundays or feast days. This was to
secure the proof of the freedom of the parties to marry. In a book of
instructions for parish priests, written about 1426, some interesting
information is given as to marriage.
“The seventh Sacrament is wedlock,” it says, “before the
which Sacrament the banes in holy church shal be thryes
asked on thre solempne dayes—a werk day or two
between, at the lest: eche day on this maner: N. of V. has
spoken with N. of P. to have hir to his wife, and to ryght
lyve in forme of holy chyrche. If any mon knowe any
lettyng qwy they may not come togedyr say now or never
on payne of cursyng.”
On the day appointed for the marriage, at the door of the church, the
priest shall interrogate the parties as follows:—
“N. Hast thu wille to have this wommon to thi wedded wif.
R. Ye syr. My thu wel fynde at thi best to love hur and hold
ye to hur and to no other to thi lives end. R. Ye syr. Then
take her by yor hande and say after me: I N. take the N. in
forme of holy chyrche to my wedded wyfe, forsakyng alle
other, holdyng me hollych to the, in sekenes and in hele,
in ryches and in poverte, in well and in wo, tyl deth us
departe, and there to I plyght ye my trowthe.”
Then the woman repeated the form as above.
It was this “Marriage at the church door” which had to be
established, according to Bracton, in any question as to the legality
or non-legality of the contract. After this “taking to wife at the church
door,” the parties entered the church and completed the rite in the
church itself. As in the case of baptisms, churchings, and funerals,
the fee for marriages was fixed at 1d., but apparently all who could
afford it, gave more.

“Three ornaments,” says the author of Dives and Pauper
—“three ornaments (at marriage) belonged principally to
the wyfe: a rynge on her finger, a broche on hyr breste,
and a garlande on hir head. The rynge betokeneth true
love; the broche betokeneth clenness of herte and chastity
that she ought to have; and the garland betokeneth the
gladness and the dignity of the sacrament of wedlock.”
Some of the ornaments for the bride at marriage the parish provided.
The nuptial veil was one of the things which the churchwardens were
supposed to find, and frequent inquiries were made concerning it in
the parochial visitations. In one parish the wardens possessed “one
standing mazer to serve for brides at their wedding;” and in another,
a set of jewels was left in trust for the use of brides on their wedding
day. If lent outside the parish, they were to be paid for, and the
receipt was to go to the common purposes of the church to which
they belonged.

CHAPTER X
THE PARISH PULPIT
The influence on parochial life of the Sunday sermon and what went
with it can hardly be exaggerated. It was not only that it was at this
time that the priest instructed his people in their faith and in the
practice of their religion; but the pulpit was the means, and in those
days the sole means, by which the official or quasi-official business
of the place was announced to the inhabitants of a district. The great
variety of matters that had necessarily to be brought to the notice of
the parishioners would have all tended to make the pulpit utterances
on the Sunday, in a pre-Reformation parish, both interesting and
instructive. In this chapter it is proposed to illustrate some of the
many features presented at the time of the Sunday sermon; and first
as to the regular religious teaching of faith and morals.
The first duty of the Church, after seeing to the administration of the
Sacraments and the offering of the Sacrifice of the Altar, was
obviously to teach and direct its children in all matters of belief and
practice. This was done from the pulpit, which was in all probability
an unpretentious wooden erection, perhaps in the screen, or at the
chancel arch. In one case there is given the cost of the erection of a
pulpit of wood; another churchwardens’ account speaks of “clasps
for” the pulpit (?), possibly hinges for the door; a third tells of “a
green silk veil for the pulpit”; and a fourth of “cloth and a pillow” for it.
The chief interest, however, is not in the thing itself, but in its use.

PULPIT, 1475, ST. PAUL’S, TRURO
It is impossible to think that Chaucer’s typical priest was a mere
creation of his imagination. The picture must have had its
counterpart in numberless parishes in England in the fourteenth
century. This is how the poet’s priest is described:—
“A good man was ther of religioun,
And was a poure parsoun of a town;
But riche he was of holy thought and werk.
He was also a lerned man, a clerk,
That Christe’s Gospel trewely wolde preche,
His parischens devoutly wolde he teche.

But Christe’s lore and His Apostles twelve
He taughte, but first he folwede it himselve.”
It will be remembered, too, that the story Chaucer makes his priest
contribute to the Canterbury Tales is nothing else than an excellent
and complete tract, almost certainly a translation of a Latin
theological treatise, upon the Sacrament of Penance.
As a sample, however, of what is popularly believed on this subject
at the present day, it is well to take the opinion of by no means an
extreme party writer, Bishop Hobhouse. “Preaching,” he says, “was
not a regular part of the Sunday observances as now. It was rare,
but we must not conclude from the silence of our MSS. (i.e.
churchwardens’ accounts) that it was never practised.” In another
place he states, upon what he thinks sufficient evidence, “that there
was a total absence of any system of clerical training, and that the
cultivation of the conscience as the directing power of man’s soul,
and the implanting of holy affections in the heart seem to have been
no part of the Church’s system of guidance.” That this is certainly not
a correct view as to the way in which the pastors of the parochial
churches in pre-Reformation days discharged—or rather neglected
—their duties, in view of the facts, appears to be certain. The
grounds for this opinion are the following: for practical purposes we
may divide the religious teaching, given by the clergy, into the two
classes of sermons and instructions. The distinction is obvious. By
the first are meant those set discourses to prove some definite
theme, or expound some definite passage of Holy Scripture, or
deduce the lessons to be learnt from the life of some saint. In other
words, putting aside the controversial aspect, which, of course, was
rare in those days, a sermon in mediæval times was much what a
sermon is to-day. There was this difference, however, that in pre-
Reformation days the sermon was not probably so frequent as in
these modern times. Now, whatever instruction is given to the people
at large is conveyed to them almost entirely in the form of set
sermons, which, however admirable in themselves, seldom convey
to their hearers consecutive and systematic, dogmatic and moral
teaching. Mediæval methods of imparting religious knowledge were

different. For the most part the priest fulfilled the duty of instructing
his flock by plain, unadorned, and familiar instructions upon matters
of faith and practice. These must have much more resembled our
present catechetical instructions than our modern pulpit discourses.
To the subject of set sermons I shall have occasion to return
presently, but as vastly more important, at any rate in the opinion of
our Catholic forefathers, let us first consider the question of familiar
instructions. For the sake of clearness we will confine our attention to
the two centuries (the fourteenth and fifteenth) previous to the great
religious revolution under Henry VIII.
Before the close of the thirteenth century, namely, in a.d. 1281,
Archbishop Peckham issued the celebrated Constitutions of the
Synod of Oxford which are called by his name. There we find the
instruction of the people legislated for minutely.
“We order,” runs the Constitution, “that every priest having
the charge of a flock do, four times in each year (that is,
once each quarter), on one or more solemn feast days,
either himself or by some one else, instruct the people in
the vulgar language, simply and without any fantastical
admixture of subtle distinctions, in the articles of the
Creed, the Ten Commandments, the Evangelical
Precepts, the seven works of mercy, the seven deadly sins
with their offshoots, the seven principal virtues, and the
Seven Sacraments.”
The Synod then proceeded to set out in considerable detail each of
the points upon which the people must be instructed. Now, it is
obvious that if four times a year this law was complied with in the
spirit in which it was given, the people were very thoroughly
instructed indeed in their faith. But was this law faithfully carried out
by the clergy, and rigorously enforced by the bishops in the
succeeding centuries? That is the real question. I think that there is
ample evidence that it was. In the first place, the Constitutions of
Peckham are referred to constantly in the fourteenth and fifteenth
centuries as the foundation of the existing practices in the English
Church. Thus, to take a few specific instances in the middle of the
fourteenth century, the decree of a diocesan Synod orders—

STONE PULPIT BRACKET,
WALPOLE ST. ANDREW,
NORFOLK
“That all rectors, vicars, or chaplains holding ecclesiastical
offices shall expound clearly and plainly to their people, on
all Sundays and feast days, the Word of God and the
Catholic faith of the Apostles; and that they shall diligently
instruct their subjects in the articles of faith, and teach
them in their native language the Apostles’ Creed, and
urge them to expound it and teach the same faith to their
children.”
Again, in a.d. 1357, Archbishop
Thoresby, of York, anxious for the
better instruction of his people,
commissioned a monk of St.
Mary’s, York, named Gatryke, to
draw out in English an exposition
of the Creed, the
Commandments, the seven
deadly sins, etc. This tract the
archbishop, as he says in his
preface, through the counsel of
his clergy, sent to all his priests—
“So that each and every one,
who under him had the charge
of souls, do openly in English,
upon Sundays teach and
preach them, that they have
cure of the law and the way to
know God Almighty. And he
commands and bids, in all that
he may, that all who have
keeping or cure under him,
enjoin their parishioners and
their subjects, that they hear
and learn all these things, and
oft, either rehearse them till
they know them, and so teach
them to their children, if they any have, when they are old

enough to learn them; and that parsons and vicars and all
parish priests inquire diligently of their subjects at Lent-
time, when they come to shrift, whether they know these
things, and if it be found that they know them not, that they
enjoin them upon his behalf, and on pain of penance, to
know them. And so there be none to excuse themselves
through ignorance of them, our father, the Archbishop, of
his goodness has ordained and bidden that they be
showed openly in English amongst the flock.”
ARCHIDIACONAL VISITATION

SACRAMENT OF MATRIMONY
To take another example: the Acts of the Synod, held by Simon
Langham at Ely in a.d. 1364, order that every parish priest frequently
preach and expound the Ten Commandments, etc., in English (in
idiomate communi), and all priests are urged to devote themselves
to the study of the Sacred Scriptures, so as to be ready “to give an
account of the hope and faith” that are in them. Further, they are to
see that the children are taught their prayers; and even adults, when
coming to confession, are to be examined as to their religious
knowledge.
Even when the rise of the Lollard heretics rendered it important that
some check should be given to general and unauthorized preaching,
this did not interfere with the ordinary work of instruction. The orders

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