Ⅸ SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES
類似粒子をいくつか含むシステム
54. Symmetrical and antisymmetrical states
Ir a system in atomic physics contains a number of particles of the
same kind, e.g. a number of electrons, the particles are absolutely
indistinguishable one from another. No observable change is made
when two of them are interchanged. This circumstance gives rise to
some curious phenomena in quantum mechanics having no analogue
in the classical theory, which arise from the fact that in quantum
mechanics a transition may occur resulting in merely the interchange
of two similar particles, which transition then could not be detected
by any observational means. A satisfactory theory ought, of course,
to count two observationally indistinguishable states as the same
state and to deny that any transition does occur when two similar
particles exchange places. We shail find that it is possible to reformu-
late the theory so that this is so.
Suppose we have a system containing x similar particles. We may
take as our dynamical variables a set of variables ¢, describing the
first particle, the corresponding set €, describing the second particle,
and so on up to the set £, describing the nth particle. We shall then
have the &,’s commuting with the €,’3 for r 4s. (We may require
certain extra variables, describing what the system consists of in
addition to the » similar particles, but it is not necessary to mention
these explicitly in the present chapter.) The Hamiltonian describing
the motion of the system will now be expressible as a function of the
£,,£5,-5€, The fact that the particles are similar requires that the
Hamiltonian shall be a symmetrical function of the &,, &o,...,€); Le. it
shall remain unchanged when the sets of variables é, are interchanged
or permuted in any way. This condition must hold, no matter what
perturbations are applied to the system. In fact, any quantity of
physical significance must be a symmetrical function of the é’s.
Let |a,>, |b,>,... be kets for the first particle considered as a dynami-
cal system by itself. There will be corresponding kets |a,), |b,>,... for
the second particle by itself, and so on. We can get a ket for the
assembly by taking the product of kets for each particle by itself,
for example
|@y>[bg>[€3>-+|Fn> = 1812 C3.+-In> (1)
208 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 54
say, according to the notation of (65) of § 20. The ket (1) corresponds
to a special kind of state for the assembly, which may be described
by saying that each particle is in its own state, corresponding to its
own factor on the left-hand side of (1). The general ket for the
assembly is of the form of a sum or integral of kets like (1), and
corresponds to a state for the assembly for which one cannot say that
each particle is in its own state, but only that each particle is partly
in several states, in a way which is correlated with the other particles
being partly in several states. If the kets |a,), |b,>,... are a set of
basic kets for the first particle by itself, the kets |a,>, |b,>,... will be
a set of basic kets for the second particle by itself, and so on, and the
kets (1) will be a set of basic kets for the assembly. We call the repre-
sentation provided by such basic kets for the assembly a symmetrical
representation, as it treats all the particles on the same footing.
In (1) we may interchange the kets for the first two particles and
get another ket for the assembly, namely
1by>|!4q>{63>.-19n> = [by dy g.--Gn>-
More generally, we may interchange the role of the first two particles
in any ket for the assembly and get another ket for the assembly.
The process of interchanging the first two particles is an operator
which can be applied to kets for the assembly, and is evidently a
linear operator, of the type dealt with in § 7. Similarly, the process
of interchanging any pair of particles is a linear operator, and by
repeated applications of such interchanges we get any permutation
of the particles appearing as a linear operator which can be applied
to kets for the assembly. A permutation is called an even permutation
or an odd permutation according to whether it can be built up from
an even or an odd number of interchanges.
A ket for the assembly |X is called symmetrical if it is unchanged
by any permutation, i.e. if
P\|X> = |X) (2)
for any permutation P. It is called antisymmetrical if it is unchanged
by any even permutation and has its sign changed by any odd
permutation, i.e. if P\|X) = +|X), (3)
the + or — sign being taken according to whether P is even or odd.
The state corresponding te a symmetrical ket is called a symmetrical
state, and the state corresponding to an antisymmetrical ket is called
an antisymmetrical state. In a symmetrical representation, the repre-
§ 54 SYMMETRICAL AND ANTISYMMETRICAL STATES 209
sentative of a symmetrical ket is a symmetrical function of the
variables referring to the various particles and the representative of
an antisymmetrical ket is an antisymmetrical function.
In the Schrédinger picture, the ket corresponding to a state of the
assembly will vary with time according to Schrédinger’s equation of
motion. If it is initially symmetrical it must always remain sym-
metrical, since, owing to the Hamiltonian being symmetrical, there
is nothing to disturb the symmetry. Similarly if the ket is initially
antisymmetrical it must always remain antisymmetrical. Thus a
state which is initially symmetrical always remains symmetrical and
a state which is initially antisymmetrical always remains antisym-
metrical. In consequence, it may be that for a particular kind of
particle only symmetrical states occur in nature, or only anti-
symmetrical states occur in nature. If either of these possibilities
held, it would lead to certain special phenomena for the particles in
question. ;
Let us suppose first that only antisymmetrical states occur in
nature. The ket (1) is not antisymmetrical and so does not corre-
spond to a state occurring in nature. From (1) we can in general form
an antisymmetrical ket by applying all possible permutations to it
and adding the results, with the coefficient —1 inserted before those
terms arising from an odd permutation, so as to get
» +P\a, bg C3---In>> (4)
the + or — sign being taken according to whether P is even or odd.
The ket (4) may be written as a determinant
lay> |@_> |@g> . . - (Gn?
[> [62> |e . - - 10.
I9> I92> IWsr> - - + IGn>
and its representative in a symmetrical representation is a determi-
nant. The ket (4) or (5) is not the general antisymmetrical ket, but
is a specially simple one. It corresponds to a state for the assembly
for which one can say that certain particle-states, namely the states
a,b,c,...,g, are occupied, but one cannot say which particle is in
which state, each particle being equally likely to be in any state. If
210 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 54
two of the particle-states a,b,c,...,g are the same, the ket (4) or (5)
vanishes and does not correspond to any state for the assembly.
Thus two particles cannot occupy the same state. More generally, the
occupied states must be all independent, otherwise (4) or (5) vanishes.
This is an important characteristic of particles for which only anti-
symmetrical states occur in nature. It leads to a special statistics,
which was first studied by Fermi, so we shall call particles for which
only antisymmetrical states occur in nature fermions.
Let us suppose now that only symmetrical states occur in nature.
The ket (1) is not symmetrical, except in the special case when all the
particle-states a,b,c,...,g are the same, but we can always obtain a
symmetrical ket from it by applying all possible permutations to it
and adding the results, so as to get
p Pla, by C3...9,>- (6)
The ket (6) is not the general symmetrical ket, but is a specially
simple one. It corresponds to a state for the assembly for which one
can say that certain particle-states are occupied, namely the states
a, 6,c,...,g, without being able to say which particle is in which state.
It is now possible for two or more of the states a,b,c,...,g to be the
same, so that two or more particles can be in the same state. In spite
of this, the statistics of the particles is not the same as the usual
statistics of the classical theory. The new statistics was first studied
by Bose, so we shall call particles for which only symmetrical states
occur in nature bosons.
We can see the difference of Bose statistics from the usual statistics
by considering a special case—that of only two particles and only two
independent states a and 6 for a particle. According to classical
mechanics, if the assembly of two particles is in thermodynamic
equilibrium at a high temperature, each particle will be equally likely
to be in either state. There is thus a probability 4 of both particles
being in state a, a probability } of both particles being in state b,
and a probability 4 of one particle being in each state. In the quan-
tum theory there are three independent symmetrical states for the
pair of particles, corresponding to the symmetrical kets |a,)|a.),
|5>|b,>, and |a,>[b,>+ |ag>|6,>, and describable as both particles in
state a, both particles in state 6, and one particle in each state
respectively. For thermodynamic equilibrium at a high temperature
these three states are equally probable, as was shown in § 33, so that
§ 54 SYMMETRICAL AND ANTISYMMETRICAL STATES 211
there is a probability 4 of both particles being in state a, a probability
4 of both particles being in state b, and a probability 4 of one particle
being in each state. Thus with Bose statistics the probability of two
particles being in the same state is greater than with classical statistics.
Bose statistics differ from classical statistics in the opposite direction
to Fermi statistics, for which the probability of two particles being
in the same state is zero.
In building up a theory of atoms on the lines mentioned at the
beginning of § 38, to get agreement with experiment one must assume
that two electrons are never in the same state. This rule is known as
Pauli’s exclusion principle. It shows us that electrons are fermions.
Planck’s law of radiation shows us that photons are bosons, as only the
Bose statistics for photons will lead to Planck’s law. Similarly, for
each of the other kinds of particle known in physics, there is experi-
mental evidence to show either that they are fermions, or that they
are bosons. Protons, neutrons, positrons are fermions, «-particles are
bosons. It appears that all particles occurring in nature are either
fermions or bosons, and thus only antisymmetrical or symmetrical
states for an assembly of similar particles are met with in practice.
Other more complicated kinds of symmetry are possible mathemati-
cally, but do not apply to any known particles. With a theory which
allows only antisymmetrical or only symmetrical states for a particu-
lar kind of particle, one cannot make a distinction between two states
which differ only through a permutation of the particles, so that the
transitions mentioned at the beginning of this section disappear.
55. Permutations as dynamical variables
We shall now build up a general theory for a system containing n
similar particles when states with any kind of symmetry properties
are allowed, i.e. when there is no restriction to only symmetrical or
only antisymmetrical states. The general state now will not be sym-
metrical or antisymmetrical, nor will it be expressible linearly in
terms of symmetrical and antisymmetrical states when n > 2. This
theory will not apply directly to any particles occurring in nature,
but all the same it is useful for setting up an approximate treatment
for an assembly of electrons, as will be shown in § 58.
We have seen that each permutation P of the n particles is a linear
operator which can be applied to any ket for the assembly. Hence
we can regard P as a dynamical variable in our system of » particles.
212 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 55
There are »! permutations, each of which can be regarded as a
dynamical variable. One of them, F, say, is the identical permutation,
which is equal to unity. The product of any two permutations is a
third permutation and hence any function of the permutations is
reducible to a linear function of them. Any permutation P has a
reciprocal P-1 satisfying
PP = PAP =P =1.,
A permutation P can be applied to a bra , the product
must be unchanged, since it is just a number, independent of any
order of the particles. Thus
(PCX|)P|¥> = and is thus equal to
would change it into B,.P,|X>, ie.
P,P,|X> = BPX.
Hence Pi = P,P, Pz, (13)
which expresses the condition for P, and FP, to be similar as an
algebraic equation. The existence of any P, satisfying (11) is suffi-
cient to show that P, and P, are similar.
56. Permutations as constants of the motion
Any symmetrical function V of the dynamical variables of all the
particles is unchanged by the application of any permutation P, so
P applied to the product V|X> affects only the factor |X), thus
PV|X> = VP|X).
Hence PV =VP, (12)
showing that a symmetrical function of the dynamical variables com-
mutes with every permutation. The Hamiltonian is a symmetrical
fanction of the dynamical variables and thus commutes with every
permutation. It follows that each permutation is a constant of the _
motion. This holds even if the Hamiltonian is not constant. If |Xt
is any solution of Schrédinger’s equation of motion, P|X¢) is another.
In dealing with any system in quantum mechanics, when we have
found a constant of the motion a, we know that if for any state of
motion, « initially has the numerical value «’, then it always has this
value, so that we can assign different numbers «’ to the different
states and so obtain a classification of the states. The procedure is
not so straightforward, however, when we have several constants of
the motion « which do not commute (as is the case with our permuta-
tions P), since we cannot in general assign numerical values for all
the a’s simultaneously to any state. Let us first take the case of a
system whose Hamiltonian does not involve the time explicitly. The
existence of constants of the motion « which do not commute is
then a sign that the system is degenerate. This is because, for a
3595.57 | P
214 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 56
non-degenerate system, the Hamiltonian H by itself forms a complete
set of commuting observables and hence, from Theorem 2 of § 19, each
of the a’s is a function of H and therefore commutes with any other a.
We must now look for a function f of the «’s which has one and
the same numerical value f’ for all those states belonging to one
energy-level H’, so that we can use f for classifying the energy-levels
of the system. We can express the condition for f by saying that it
must be a function of H and must therefore commute with every
dynamical variable that commutes with H, ie. with every constant
of the motion. If the a’s are the only constants of the motion, or if
they are a set that commute with all other independent constants of
the motion, our problem reduces to finding a function B of the a’s
which commutes with all the a’s. We can then assign a numerical
value ’ for 8 to each energy-level of the system. If we can find
several such functions 8, they must all commute with each other, so
that we can give them all numerical values simultaneously. We ob-
tain thus a classification of the energy-levels. When the Hamiltonian
involves the time explicitly one cannot talk about energy-levels, but
* the f’s will still give a useful classification of the states.
We follow this method in dealing with our permutations P. We
must find a function y of the P’s such that PyP-! = x for every P.
It is evident that a possible y is > P,, the sum of all the permutations
in a certain class c, i.e. the sum of a set of similar permutations, since
> PF, P- must consist of the same permutations summed in a differ-
ent order. There will be one such y for each class. Further, there can
be no other independent y, since an arbitrary function of the P’s can
be expressed as a linear function of them with numerical coefficients,
and it will not then commute with every P unless the coefficients of
similar P’s are always the same. We thus obtain all the x’s that can
be used for classifying the states. It is convenient to define each x as
an average instead of a sum, thus
Xe = NDP,
' where 7, is the number of P’s in the class c. An alternative expression
for x, is Xe = nt Y PP, P-l, (13)
P
the sum being extended over all the n! permutations P, it being easy
to verify that this sum contains each member of the class ¢ the same
number of times. For each permutation P there is one x, x(P) say,
§56 PERMUTATIONS AS CONSTANTS OF THE MOTION 235
equal to the average of ail permutations similar to P. One of the
xs is x(R) = 1.
The constants of the motion y,, X2,---, ¥_ Obtained in this way will
each have a definite numerical value for every stationary state of the
system, in the case when the Hamiltonian does not involve the time
explicitly, and also in the general case can be used for classifying
the states, there being one set of states for every permissible set of
numerical values yj, %2)---, X¥m for the y’s. Since the y’s are always
constants of the motion, these sets of states will be exclusive, i.e.
transitions will never take place from a state in one set to a state in
another.
The permissible sets of values x’ that one can give to the y’s are
limited by the fact that there exist algebraic relations between the
x’s. The product of any two x’s, x, X,. is of course expressible as
a linear function of the P’s, and since it commutes with every P it
must be expressible as a linear function of the y’s, thus
{ i} [ A)
Xp Xq = 8% Xi 8 Xo Tn Xm: (14)
where the a’s are numbers. Any numerical values x’ that one gives
to the y’s must be eigenvalues of the y’s and must satisfy these same
algebraic equations. For every solution x’ of these equations there
is one exclusive set of states. One solution is evidently y,, = 1 for
every Xp, giving the set of symmetrical states. A second obvious
solution, giving the set of antisymmetrical states, is y, = --1, the
+ or — sign being taken according to whether the permutations in
the class p are even or odd. The other solutions may be worked out
in any special case by ordinary algebraic methods, as the coefficients
@ in (14) may be obtained directly by a consideration of the types
of permutation to which the y’s concerned refer. Any solution is,
apart from a certain factor, what is called in group theory a character
of tke group of permutations. The y’s are all real dynamical variables,
since each P and its conjugate complex P-! are similar and will occur
added together in the definition of any x, so that the x’’s must be all
real numbers.
The number of possible solutions of the equations (14) may easily
be determined, since it must equal the number of different eigen-
values of an arbitrary function B of the y’s. We can express B as
a linear function of the y’s with the help of equations (14); thus
B == by yy tbe xo+--- tlm Xm (i
on
\
i
216 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 56
Similarly, we can express each of the quantities B®, B°,..., B™ as a
linear function of the y’s. From ‘the m equations thus obtained,
‘together with the equation x(P,) = 1, we can eliminate the m un-
knowns X14, Xo.» Xm» Obtaining as result an algebraic equation of
degree m for B,
Bre, Bt +-¢, Bre, == 0,
The m solutions of this equation give the m possible eigenvalues
for B, each of which will, according to (15), be a linear function of 6,,
b,..., b, whose coefficients are a permissible set of values x1, Xos--63 Xm
The sets of values x’ thus obtained must be all different, since if
there were fewer than m different permissible sets of values y’ for the
x’s, there would exist a linear function of the y’s every one of whose
eigenvalues vanishes, which would mean that the linear function itself
vanishes and the y’s are not linearly independent. Thus the number of
permissible sets of numerical values for the y’s is just equal tom, which
is the number of classes of permutations or the number of partitions
of x. This number is therefore the number of exclusive sets of states.
All dynamical variables of physical importance and all observable
quantities are symmetrical between the particles and thus commute
with all the P’s. Thus the only functions of the P’s of physical
importance are the y’s. The states corresponding to |y’> and to
S(P)\x’>, where |y’> is any eigenket of the y’s belonging to the eigen-
values x’ and f(P) is any function of the P’s such that f(P)|y> < 9,
are observationally indistinguishable and are thus physically equiva-
lent. There is a definite number, n(x’) say, of independent kets which
can be formed by multiplying |y’> by functions of the P’s, which
number depends only on the y”s. It is the number of rows and
columns in a matrix representation of the P’s in which each x is
equal to x’. If |x’> corresponds to a stationary state, n(x‘) will be
its degree of degeneracy (so far as concerns degeneracy caused by the
symmetry between the particles). This degeneracy cannot beremoved
by any perturbation that is symmetrical between the particles.
57. Determination of the energy-levels
Let us apply the perturbation method of § 43 and make a first-order
calculation of the energy-levels in the case when the Hamiltonian
does not involve the time explicitly. We suppose that for our unper-
turbed stationary states of the assembly each of the similar particles
has its own individual state. With n particles, we shall have n of
§ 57 DETERMINATION OF THE ENERGY-LEVELS 217
these states, corresponding to kets Jat, |a*>,..., |a”> say, which we
assume for the present to be all orthogonal. The ket for the assembly
is then [XD = foed> odd... fo®, (16)
like (1) with a, «?,... instead of a, b,.... If we apply any permutation
P to it we get another ket
P|X> = [ap> 05>... lay (17)
say, 7, 8... 2 being some permutation of the numbers 1, 2,..., ”,
corresponding to another stationary state of the assembly with the
same energy. There are thus altogether m! unperturbed states with
this energy, if we assume there are no other causes of degeneracy.
According to the method of § 43 when the unperturbed system is
degenerate, we must consider those elements of the matrix represent-
ing the perturbing energy V that refer to two states with the same
energy, i.e. those of the type (X|P, VB,|X>. These will form a matrix
with 2! rows and columns, whose eigenvalues are the first-order
corrections in the energy-levels.
We must now introduce another kind of permutation operator
which can be applied to kets of the form (17), namely a permutation
which acts on the indices of the «’s. We denote such a permutation
operator by P*. The essential difference between the P’s and the
P’s may be seen in the following way. Let us consider a permutation
in the general sense, say that consisting of the interchange of 2 and 3.
This may be interpreted either as the interchange of the objects 2 and
3 or as the interchange of the objects in the places 2 and 3, these two
operations producing in general quite different results. The first of
these interpretations is the one that gives the operators P, the objects
concerned being the similar particles. A permutation P can be
applied to an arbitrary ket for the assembly. A permutation with the
second interpretation has a meaning, however, only when applied
to a ket of the form (17), for which each of the particles is in a ‘place’
specified by an «, or to a sum of kets of the form (17). A permutation
P may be considered as an ordinary dynamical variable. A permuta-
tion P* may be considered as a dynamical variable in a restricted
sense, valid when one is dealing only with states obtainable by super-
position of the various states (17). This is the case for our present
perturbation problem.
We can form algebraic functions of the P* which will be other
operators applicable to kets of the form (17). In particular we can
218 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 57
form x(P2%), the average of all P*’s in a certain class c. This must
equal y(P,), the average of the permutation operators P in the same
class, since the total set of all permutations in a given class must
evidently be the same whether the permutations are applied to the
particles or to the places the particles arein. Any P commutes with
any P%, ie. P, Px = P2P,, (18)
By labelling the «’s by the same numbers 1, 2, 3,..., » which label
the particles, we set up a one-one correspondence between the «’s and
the particles, so that given any permutation PF, applying to the par-
ticles, we can give a meaning to the same permutation P% applying
to the a’s. This meaning is such that, for the ket |X given by (16),
PGL|X> = |X). (19)
Since the various kets |a1>, |o*),... are orthogonal, |X> and P|X) are
orthogonal unless P = 1. It follows that, for any coefficients cp,
D2 6p = Cp, (20)
provided |X> is normalized, the summation being over all the n!
permutations P or P*, with P, fixed. Now define VY by .
Vp = (X|VP|X). (21)
We then have, for any two permutations P, and P,,
(AP, VEX = = Ve. p,
= 2 Vp (X| Ph, Py|X>
with the help of (20). From (18) this gives
(XP, VE,|X> = & Ye KE, P*B,|X>. (22)
We may write this result as
VS Pe, (23)
B
where the sign means an equation in a restricted sense, the
operators on the two sides being equal so long as they are used only
with kets of the form P|X> and their conjugate imaginary bras.
The formula (23) shows that the perturbing energy V is equal, in
the restricted sense, to a linear function of the permutation operators
P* with coefficients Vp given by (21). The restricted sense is adequate
for the calculation of the first-order correction in the energy-levels,
as this calculation involves only those matrix elements of V given by
§ 57 DETERMINATION OF THE ENERGY-LEVELS 219
(22). The formula (23) is a very convenient one because the expression
on its right-hand side is easily handled.
As an example of an application of (23) we shall determine the
average energy of all those states, arising from the unperturbed state
(16), that belong to one exclusive set. This requires us to calculate
the average eigenvalue of V for those states (17) for which the y’s
have specified numerical values x’. Now the average eigenvalue of
P* for any of these states equals that of P*P%(P%)-1 for arbitrary
P” and thus equals that of ni > P¢p% P%)-1, which is y‘(P%) or
x'(P,). Hence the average eigenvalue of V is } Vpx'(P). A similar
P
method could be used for calculating the average eigenvalue of any
function of V, it being necessary only to replace each P* by x’(P) to
perform the averaging.
The number of energy-levels in an exclusive set x = x’ that arise
from a given state of the unperturbed system is equal to the number
of eigenvalues of the right-hand side of (23) that are consistent with
the equations y = x’. This number is the number n(x’) introduced
at the end of the preceding section, and is thus just the degree of
degeneracy of the states in this set.
We have assumed that the individual kets |o!), {a”),... which deter-
mine the unperturbed state according to (16) are all orthogonal. The
theory can easily be extended to the case when some of these kets are
equal, any two that are not equal being still restricted to be orthogonal.
We now have some permutations P* such that P*|X> = |X),
namely those permutations which involve only interchanges of
equal «’s. Equation (20) will now hold if the summation is extended
only over those P’s which make-P*|X> different. With this change
in the meaning of 2 all the previous equations still hold, including
the result (23). For the present |X) there will be restrictions on the
possible numerical values of the y’s, e.g. they cannot have those
values corresponding to |X) being antisymmetrical.
58. Application to electrons
Let us consider the case when the similar particles are electrons.
This requires, according to Pauli’s exclusion principle discussed in
§ 54, that we take into account only the antisymmetrical states. It
is now necessary to make explicit reference to the fact that electrons
have spins, which show themselves through an angular momentum
220 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 58
and a magnetic moment. The effect of the spin on the motion of
an electron in an electromagnetic field is not very great. There
are additional forces on the electron due to its magnetic moment,
requiring additional terms in the Hamiltonian. The spin angular
momentum does not have any direct action on the motion, butit comes
into play when there are forces tending to rotate the magnetic moment,
since the magnetic moment and angular momentum are constrained
to be always in the same direction. In the absence of a strong
magnetic field these effects are all small, of the same order of magni-
tude as the corrections required by relativistic mechanics, and there
would be no point in taking them into account in a non-relativistic
theory. Theimportance of the spin lies not in these small effects on the
motion of the electron, but in the fact that it gives two internal states
to the electron, corresponding to the two possible values of the spin
component in any assigned direction, which causes a doubling in the
number of independent states of an electron. This fact has far-reaching
consequences when combined with Pauli’s exclusion principle.
In dealing with an assembly of electrons we have two kinds of
dynamical variables. The first kind, which we may call the orbital
variables, consists of the coordinates x, y, z of all the electrons and
their conjugate momenta p,, p,, p,. The second kind consists of the
spin variables, the variables o,, o,, o,, a8 introduced in § 37, for all
the electrons. These two kinds of variables belong to different degrees
of freedom. According to §§ 20 and 21, a ket fixing the state of the
whole system may be of the form |.4>|B>, where |A) is a ket referring
to the orbital variables alone and {B) is a ket referring to the spin
variables alone, and the general ket fixing a state of the whole system
is a sum or integral of kets of this form. This way of looking at things
enables us to introduce two kinds of permutation operators, the first
kind, P* say, applying to the orbital variables only and operating
only on the factor |A> and the second kind, P* say, applying only
to the spin variables and operating only on the factor |B>. The P2’s
and P?’s can each be applied to any ket for the whole system, not
merely to certain special kets, like the P®’s of the preceding section.
The permutations P that we have had up to the present apply to all
the dynamical variables of the particles concerned, so for electrons
they will apply to both the orbital and the spin variables. This means
that each P, equals the product
P, = P= Ps, (24)
§ 58 APPLICATION TO ELECTRONS 221
We can now see the need for taking the spin variables into account
when applying Pauli’s exclusion principle, even if we neglect the spin
forces in the Hamiltonian. For any state occurring in nature each
P, must have the value +1, according to whether it is an even or
an odd permutation, so from (24)
PtPo = -l. (25)
The theory of the three preceding sections would become trivial if
applied directly to electrons, for which each P, = +1. We may,
however, apply it to the P* permutations of electrons. The P®’s are
constants of the motion if we neglect the terms in the Hamiltonian
that arise from the spin forces, since this neglect results in the
Hamiltonian not involving the spin dynamical variables o at all. The
P’s must then also be constants of the motion. We can now intro-
duce new x’s, equal to the average of all of the P*’s in each class, and
assert that for any permissible set of numerical values y’ for these y’s
there will be one exclusive set of states. Thus there exist exclusive sets
of states for systems containing many electrons even when we restrict
ourselves to a consideration of only those states that satisfy Pauli’s
principle. The exclusiveness of the sets of states is now, of course,
only approximate, since the y’s are constants only so long as we
neglect the spin forces. There will actually be a small probability for
a transition from a state in one set to a state in another.
Equation (25) gives us a simple connexion between the P*’s and
P°’s, which means that instead of studying the dynamical variables
P* we can get all the results we want, e.g. the characters y’, by
studying the dynamical variables P’. The P°’s are much easier to
study on account of there being only two independent states of spin
for each electron. This fact results in there being fewer characters x’
for the group of permutations of the o-variables than for the group
of general permutations, since it prevents a ket in the spin variables
from being antisymmetrical in more than two of them.
The study of the P?’s is made specially easy by the fact that we
can express them as algebraic functions of the dynamical variables ¢.
Consider the quantity
Org = HI O91 Fag Oy1 Fyn tn Set = ELL + (Gy, Fa)}-
With the help of equations (50) and (51) of § 37 we find readily that
(6, G2)? = (Oy Fng+Fya Cyat Fn Fea)” = 3—2(Gy, Gg), (26)
and hence that
O48 = U1+2(6,, 03) -+(6y 9)"} = 1. (27)
222 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICLES § 58
Again, we find
1 . L:
O12 Ory = NOx + Op. 1024 Oyg Tt yy G9},
1 Le .
Ox2 Oy, = HOn2 + n14 WO y1 Fzg— UG zy Oyo}
and hence O12 On1 = Fzq Oy.
Similar relations hold for o,, and o,, so that we have
O12 5, = 5, Oj,
or Oy. 0, OF = a.
From this we can obtain with the help of (27)
O19 6, O7G' = 94.
These commutation relations for O,, with o, and o, are precisely the
same as those for P?,, the permutation consisting of the interchange
of the spin variables of electrons 1 and 2. Thus we can put
—_ Co
Or = CPi,
where c is a number. Equation (27) shows that c = +1. To deter-
mine which of these values for c is the correct one, we observe that
the eigenvalues of P?, are 1, 1, 1, —1, corresponding to the fact that
there exist three independent symmetrical and one antisymmetrical
state in the spin variables of two electrons, namely, with the notation
of § 37, the states represented by the three symmetrical functions
Folia) Sa(Fz2)s Selo) fp (22), Sod Oe) fp(C22) theo) fo(oz2), and the one
antisymmetrical function f,(o24) fg(o22)—fe(G1) fa(ze)- Thus the mean
of the eigenvalues of Pf, is 4. Now the mean of the eigenvalues of
(o,, 6.) is evidently zero and hence the mean of the eigenvalues of O,,
is 3. Thus we must have c = +1, and so we can put
Po, = {14+ (,, 52)}. (28)
In this way any permutation P? consisting simply of an interchange
can be expressed as an algebraic function of the o’s. Any other per-
mutation P? can be éxpressed as a product of interchanges and can
therefore also be expressed as a function of the o’s. With the help of
(25) we can now express the P?’s as algebraic functions of the o’s and
eliminate the P°’s from the discussion. We have, since the — sign
must be taken in (25) when the permutations are interchanges and
since the square of an interchange is unity,
Phy = —3{1+(6,, 69)}. (29)
The formula (29) may conveniently be used for the evaluation of
§ 58 APPLICATION TO ELECTRONS 223
the characters y’ which define the exclusive sets of states. We have,
for example, for the permutations consisting of interchanges,
Xue = X(Pe) = 5(? ae (3,, @)
a
If we introduce the dynamical variable s to describe the magnitude of
the total spin angular momentum, 4 2 s, in units of #, through the
formula
s(s-+1) = (3 9,4 > @;),
r
in ayreement with (39) of § 36, we have
23 (6,0) = (Yop ¥o)— S(6
= 48(s-+1)—3
1 4s(s+1)—3n n(n—4)-+4s(s+1)
Xia = 5(! : n(n—1) ~~ 2n(n—1) - (9)
Thus yj. is expressible as a function of the dynamical variable s and
of n the number of electrons. Any of the other y’s could be evaluated
on similar lines and would have to be a function of s and only, since
there are no other symmetrical functions of all the o dynamical
variables which could be involved. There is therefore one set of
numerical values x’ for the y’s, and thus one exclusive set of states,
for each eigenvalue s’ of s. The eigenvalues of s are
Hence
1 1 1
gn, in—lI, én—2, see
the series terminating with 0 or §.
We see in this way that each of the stationary states of a system
with several electrons is an eigenstate of s, the magnitude in units of
hs of the total spin angular momentum } > a,, belonging to a definite
r
eigenvalue s’. For any given s’ there will be 2s’+1 possible values
for a component of the total spin vector in any direction and these
will correspond to 2s" 1 independent stationary states with the same
energy. When we do not neglect the forces due to the spin magnetic
moments these 2s’+-1 states will in general be split up into 2s’+1
states with slightly different energies, and will thus form a multiplet
of multiplicity 2s’+-1. Transitions in which s’ changes, i.e. transitions
from one multiplicity to another, cannot occur when the spin forces
are neglected and will have only a small probability of occurrence
when the spin forces are not neglected.
224 SYSTEMS CONTAINING SEVERAL SIMILAR PARTICES § 58
We can determine the energy-levels of a system with several
electrons to the first approximation by applying the theory of the
preceding section with the kets |a’> referring only to the orbital
variables and using formula (23). If we consider only the Coulomb
forces between the electrons, then the interaction energy V will
consist of a sum of parts each referring to only two electrons, which
will result in all the matrix elements Vy) vanishing except those for
which P* is the identical permutation or is simply an interchange of
two electrons. Thus (23) will reduce to
Vaht DV. Pte (31)
rcs
¥,, being the matrix element referring to the interchange of electrons
ry and s. Since the P*’s have the same properties as the P*’s, any
function of the P°’s will have the same eigenvalues as the corre-
sponding function of the P?’s, so that the right-hand side of (31)
will have the same eigenvalues as
Wt 2d Veo Pres
r V,{1+(e,,6,)}} . (32)
r in the orbital variables for two electrons
may be the same. Suppose |x and |«?> are the same. Then we must
take only those eigenvalues of (31) that are consistent with P% = 1,
or those eigenvalues of (32) that are consistent with P%, = 1 or
P%, = —1. From (28) this condition gives (¢,,¢,) = —3, so that
(o,+¢,)? = 0. Thus the resultant of the two spins «, and a, is zero,
which may be interpreted as the spins o, and «, being antiparallel.
Thus we may say that two electrons in the same orbital state have
their spins antiparaliel. More than two electrons cannot be in the
same orbital state.