We consider a dynamical system composed of w’ similar particles.
We set up a representation for one of the particles with discrete basic
kets |a®>, |x, jo >,.... Then, as explained in § 54, we get a sym-
metrical representation of the assembly of w’ particles by taking as
basic kets the products
|a#> [a&> lag... lo> == lad of af...0%-> (1)
in which there is one factor for each particle, the suffixes 1, 2, 3,..., w’
of the a’s being the labels of the particles and the indices a, b, ¢,...,9
denoting indices , ®, ®,... in the basic kets for one particle. Tf the
particles are bosons, so that only symmetrical states occur in nature,
then we need to work with only the symmetrical kets that can be
constructed from the kets (1). The states corresponding to these
symmetrical kets will form a complete set of states for the assembly
of bosons. We can build up a theory of them as follows.
We introduce the linear operator S defined by
S=w!lt> P, (2)
the sum being taken over all the wu’! permutations of the w’ particles.
Then S applied to any ket for the assembly gives a symmetrical ket.
We may therefore call S the symmetrizing operator. From (8) of § 55
it is real. Applied to the ket (1) it gives
wl Plot of af...08,> == Sla%a%a®...a%>, (3)
the labels of the particles being omitted on the right-hand side as
they are no longer relevant. The ket (3) corresponds to a state for
the assembly of u’ bosons with a definite distribution of the bosons
among the various boson states, without any particular boson being
assigned to any particular state. The distribution of bosons is specl-
fied if we specify how many bosons are in each boson state. Let
Ni, Ny, 24,... be the numbers of bosons in the states a, of), of)...
respectively with this distribution. The n’’s are defined algebraically
by the equation
attabtott..f-ed = ny of? 4 Ny) } Ny oe (4)
The sum of the ns is of course u’. The number of n”s is equal to
the number of basic kets [a ), which in most applications of the
226 THEORY OF RADIATION § 59
theory is very much greater than u’, so most of the n’’s will be zero.
If a, «®, of,..., a? are all different, i.e. if the n'’s are all 0 or 1, the
ket (3) is normalized, since in this case the terms on the left-hand
side of (3) are all orthogonal to one another and each contributes
u'!-1 to the squared length of the ket. However, if a4, 0, 0%)..., a7
are not all different, those terms on the left-hand side of (3) will
be equal which arise from permutations P which merely interchange
bosons in the same state. The number of equal terms will be
m,! ng! ng!..., so the squared length of the ket (3) will be
Cote? ol,..o9| 8? lataae...o% == ny! ng! ng!..- (5)
For dealing with a general state of the assembly we can introduce
the numbers ”,, M», %3,... of bosons in the states o, a, o®),..,
respectively and treat the n’s as dynamical variables or as observ-
ables. They have the eigenvalues 0, 1, 2,..., u’. The ket (3) is a
simultaneous eigenket of all the n’s, belonging to the eigenvalues
M4, Ny, Ng,.... The various kets (3) form a complete set for the
dynamical system consisting of u’ bosons, so the n’s all commute
(see the converse to the theorem of § 13). Further, there is only one
independent ket (3) belonging to any set of eigenvalues 7}, M9, %3,----
Hence the n’s form a complete set of commuting observables. If we
normalize the kets (3) and then label the -resulting kets by the
eigenvalues of the n’s to which they belong, ie. if we put
(m4! 25! 13!...) “FS fataba®...o8) = [ny ng ng...>, (6)
we get a set of kets |[n, ,73...>, with the n”s taking on all non-negative
integral values adding up to wu’, which kets will form the basic kets
of a representation with the n’s diagonal.
The n’s can be expressed as functions of the observables Oy, Og,
Q%g,.., &y Which define the basic kets of the individual bosons by
means of the equations
rg = >. 3a, ay? (7)
or the equations d a flo%) = ¥ f(a) (8)
holding for any function /.
Let us now suppose that the number of bosons in the assembly is
not given, but is variable. This number is then a dynamical variable
or observable u, with eigenvalues 0, 1, 2,..., and the ket (3) is an
eigenket of wu belonging to the eigenvalue u’. To get a complete
set of kets for our dynamical system we must now take all the
§ 59 AN ASSEMBLY OF BOSONS 227
symmetrical kets (3) for all values of u’. We may arrange them in
order thus ty Jaty, Slata®y, Satara”), ..., (9)
where first is written the ket, with no label, corresponding to the
state with no bosons present, then come the kets corresponding to
states with one boson present, then those corresponding to states
with two bosons, and so on. A general state corresponds to a ket
which is a sum of the various kets (9). The kets (9) are all orthogonal
to one another, two kets referring to the same number of bosons being
orthogonal as before, and two referring to different numbers of bosons
being orthogonal since they are eigenkets of wu belonging to different
eigenvalues. By normalizing all the kets (9), we get a set of kets like
(6) with no restriction on the 7’s (i.e. each n’ taking on all non-
negative integral values) and these kets form the basic kets of a
representation with the n’s diagonal for the dynamical system con-
sisting of a variable number of bosons.
If there is no interaction between the bosons and if the basic kets
lo), |x®,... correspond to stationary states of a boson, the kets (9)
will correspond to stationary states for the assembly of bosons. The
number w of bosons is now constant in time, but it need not be a
specified number, i.e. the general state is a superposition of states
with various values for uw. If the energy of one boson is H(a), the
energy of the assembly will be
> (os) = 2 gH" (10)
from (8), H* being short for the number H(a*). This gives the
Hamiltonian for the assembly as a function of the dynamical
variables n.
60. The connexion between bosons and oscillators
In § 34 we studied the harmonic oscillator, a dynamical system of
one degree of freedom describable in terms of a canonical q and p,
such that the Hamiltonian is a sum of squares of g and p, with
numerical coefficients. We define a general oscillator mathematically
as a system of one degree of freedom describable in terms of a
canonical g and p, such that the Hamiltonian is a power series in ¥
and p, and remains so if the system is perturbed in any way. We
shall now study a dynamical system composed of several of these
oscillators. We can describe each oscillator in terms of, instead of
q and p, a complex dynamical variable 7, like the y of § 34, and its
228 THEORY OF RADIATION § 60
conjugate complex 7, satisfying the commutation relation (7) of
§ 34. We attach labels 1, 2, 3,... to the different oscillators, so that
the whole set of oscillators is describable in terms of the dynamical
variables 41,° 2, Nase Fi Te a--- Satisfying the commutation
relations
Na oN Na = 9;
Nate Ta = 9, (11)
Fa Meo Ha = Sab-
Put Nata = Me (12)
so that Fate = Mtl. (13)
The n’s are observables which commute with one another and the
work of § 34 shows that each of them has as eigenvalues all non-
negative integers. For the ath oscillator there is a standard ket for
the Fock representation, |0,> say, which is a normalized eigenket of n,
belonging to the eigenvalue zero. By multiplying all these standard
kets together we get a standard ket for the Fock representation for
the set of oscillators, 10, }O,) 10) (14)
which is a simultaneous eigenket of all the n’s belonging to the
eigenvalues zero. We shall denote it simply by |0>. From (13) of §34
7jql0> = 0 (15)
for any a. The work of §34 also shows that, if m,, mg, ”5,... are any
non-negative integers, ott nt nB...|0> (16)
is a simultaneous eigenket of all the n’s belonging to the eigenvalues
Ny, Ng, My,... respectively. The various kets (16) obtained by taking
different n’’s form a complete set of kets all orthogonal to one another
and the square of the length of one of them is, from (16) of § 34,
n,!ng!ng!.... From this we see, bearing in mind the result (5), that
the kets (16) have just the same properties as the kets (9), so that
we can equate each ket (16) to the ket (9) referring to the same n’
values without getting any inconsistency. This involves putting
S focterPod’....0.7 = NaN Ter Iq|0>- (17)
The standard ket |0> becomes equal to the first of the kets (9), corre-
sponding to no bosons present.
The effect of equation (17) is to identify the states of an assembly
of bosons with the states of a set’ of oscillators. This means that the
§60 CONNEXION BETWEEN BOSONS AND OSCILLATORS = 229
dynamical system consisting of an assembly of similar bosons 18 equiva-
lent to the dynamical system consisting of a set of oscillators—the two
systems are just the same system looked at from two different points of
view, There is one oscillator associated with each independent boson
state. We have here one of the most fundamental results of quantum
mechanics, which enables a unification of the wave and corpuscular
theories of light to be effected.
Our work in the preceding section was built up on a discrete set
of basic kets |x) for a boson. We could pass to a different discrete
set of basic kets, [84> say, and build up a similar theory on them.
The basic kets for the assembly would then be, instead of (9),
I>, IB4>, SIB4B™>, SIB4B3B°, .... (18)
The first of the kets (18), referring to no bosons present, is the same
as the first of the kets (9). Those kets (18) referring to one boson
present are linear functions of those kets (9) referring to one boson
present, namely iB = Ss joa? = [p'D>sp. (56)
The connexion between |p’> and |p’D) is like the connexion between
the basic kets when one changes the weight function of the representa-
tion, as shown by (38) of § 16.
With nj, photons in each discrete photon state p’, the Gibbs
density p for the assembly of photons is, according to (68) of § 33,
p => [p’D>n, n, , according to (73)
of § 33. From (57) this equals
(x'|plx'> = | =
é
h(2zv')!
Similarly, the matrix element of Hy referring to the absorption of a
photon from the continuous state p% with the atom jumping from
state «° to state a’ is
=
¢P'lo’|Hg|0®> =
Coe" |, |X%9>- (75)
é
A(2av%)t
and the matrix element referring to the scattering of a photon from
the continuous state p°l® to the continuous state p’l’ with the atom
jumping from state «” to state a’ is
, (76)
nee e ,
= Orhimly't (LT) by-aas (77)
there being two terms in (74) which contribute to it. These matrix
elements will be used in the next section. The matrix elements
referring to the simultaneous absorption or emission of two photons
may be written down in the same way, but they lead to physical
effects too small to be of practical importance.
64. Emission, absorption, and scattering of radiation
We can now determine directly the coefficients of emission, absorp-
tion, and scattering of radiation by substituting in the formulas of
§ 64 EMISSION, ABSORPTION, AND SCATTERING OF RADIATION 245
Chapter VIII the values for the matrix elements given by (75), (76),
and (77).
For determining the emission probability we can use formula
(56) of §53. This shows that for an atom in a state a? the proba-
bility per unit time per unit solid angle of its spontaneously emitting
a photon and dropping to a state a’ of lower energy is
2 2
Now the energy and momentum of a photon of frequency v are
W = hy, P = hy/e.
Again, from the Heisenberg law (20) of § 29,
Cot! |e == — 2rrtv(a?a')
of the formula (44) of § 51
(k|V |p = = = =
and
E' —E, = b+ Hp(«°)—Hp(2")—h hv’ = —hfp' + (00°) ],
there being now two photons, of frequencies v® and v’, in existence
for the intermediate state. Substituting in (44) of § 51 the values of
the matrix elements given by (75), (76), and (77), we get for the
sates coefficient
i aI OD) Barae +
+> fe faa ble ola SIP can
If we write (81) in terms of x instead of 2, we get
rey a (l 1) 8rg0 — > voxel” )o( 0” (= ria ae aad
eel) oy
We can simplify (82) with the help of the quantum conditions.
We have
Ly Lyo—LyoXy = 0,
248 THEORY OF RADIATION § 64
which gives
> {ox" [ty] of the interaction
energy are of the second order of smallness compared with the
ones, at any rate when the scattered particles are photons.
2
poy’3 . (85)
a”
65. An assembly of fermions
An assembly of fermions can be treated by a method similar to
that used in §§ 59 and 60 for bosons. With the kets (1) we may use
the antisymmetrizing operator A defined by
A=w!4D4P, (2')
summed over all permutations P, the + or — sign being taken
according to whether P is eyen or odd. Applied to the ket (1) it gives
WIE S + Plot ob o§...09> = AlatoPa®...0%>, (3')
a ket corresponding to a state for an assembly of u’ fermions. The
§ 65 AN ASSEMBLY OF FERMIONS 249
ket (3’) is normalized provided the individual fermion kets |a”, |a”,...
are all different, otherwise it is zero. In this respect the ket (3’) is
simpler than the ket (3). However, (3’) is more complicated than (3)
in that (3') depends on the order in which a”, 2, o”,... occur in it,
being subject to a change of sign if an odd permutation is applied
to this order.
We can, as before, introduce the numbers 7), No, 3,... of fermions
in the states ao, o, «,... and treat them as dynamical variables or
observables. They each have as eigenvalues only 0 and 1. They form
a complete set of commuting observables for the assembly of fermions.
The basic kets of a representation with the n’s diagonal may be taken
to be connected with the kets (3’) by the equation
Alotaboe...a%> = |, ngng...> (6’)
corresponding to (6), the n’s being connected with the variables
at, a, of... by equation (4). The -- sign is needed in (6’) since, for
given n’”’s, the occupied states «%, «”, a°,... are fixed but not their
order, so that the sign of the left-hand side of (6’) is not fixed. To
set up a rule which determines the sign in (6’), we must arrange all
the states « for a fermion arbitrarily in some standard order. The
a's oceurring in the left-hand side of (6’) form a certain selection from
all the a’s and the standard order for all the a’s will give a standard
order for this selection. We now make the rule that the + sign should
occur in (6’) if the «’s on the left-hand side can be brought into their
standard order by an even permutation and the — sign if an odd
permutation is required. Owing to the complexity of this rule,
the representation with the basic kets |n,ngn3...> is not a very
useful one.
If the number of fermions in the assembly is variable, we can set
up the complete set of kets
Id, Ja, Ajatx®>, Alatarae>, (9’)
corresponding to (9). A general ket is now expressible as a sum of
the various kets (9’).
To continue with the development we introduce a set of linear
operators n, 7, one pair 7,, 7, corresponding to each fermion state «*,
satisfying the commutation relations
Na Not No Na = 9
Na ot No fa = 0, (11’)
Dao Qa = Sap:
250 THEORY OF RADIATION § 65
These relations are like (11) with a -+ sign instead of a — on the left-
hand side. They show that, for a + 6, 7, and 7, anticommute with
n» and 7, while, putting b = a, they give
m= 9, = 0, FaMat Nata = 1. (11’)
To verify that the relations (11’) are consistent, we note that linear
operators y, 7 satisfying the conditions (11’) can be constructed in
the following way. For each state a we take a set of linear operators
Fxq> Sy Fcq like the o,, o,, o, introduced in § 37 to describe the spin
of an electron and such that o,,, Gyq; 6,4 commute with 9,,, Op, %»
for 6 a. We also take an independent set of linear operators €,,
one for each state «*, which all anticommute with one another and
have their squares unity, and commute with all the o variables.
Then, putting
Na = 2$a(Fra—tya)> Na = HalOrattya),
we have all the conditions (11’) satisfied.
From (11")
("4 ja)” = Naa Na Ie = Nall—e Na) a = Na Na
This is an algebraic equation for 7,7,, showing that »,7, is an
observable with the eigenvalues 0 and 1. Also y, 7, commutes with
1, i, for 6 4a. These results allow us to put
Nata = Na (12')
the same as (12). From (11”) we get now
Na Ve = I—n,, (13)
the equation corresponding to (13),
Let us write the normalized ket which is an eigenket of all the n’s
belonging to the eigenvalues zero as |0>. Then
Ng\O> = 0,
so from (12’) CO| M4 Fg|O> = 0.
Hence FqlO> = 9, . (15')
like (15). Again
showing that 7,,|0> is normalized, and
showing that »,|0> is an eigenket of n, belonging to the eigenvalue
unity. It is an eigenket of the other n’s belonging to the eigenvalues
zero, since the other n’s commute with »,. By generalizing the
§ 65 AN ASSEMBLY OF FERMIONS 251
argument we see that 4_ 7p Ne---4y|07 18 normalized and is a simul-
taneous eigenket of all the n’s, belonging to the eigenvalues unity
for Mg Nyy Ney, R, and zero for the other n’s. This enables us to put
Alatoal...07> = 14 Ne Mg|O> (17)
both sides being antisymmetrical in the labels a, 6, c,...,g. We have
here the analogue of (17).
If we pass over to a different set of basic kets [8+> for a fermion,
we can introduce a new set of linear operators 74 correspor.ding to
them. We then find, by the same argument as in the case of bosons,
that the new 7’s are connected with the original ones by (21). This
shows that there is a procedure of second quantization for fermions,
similar to that for bosons, with the only difference that the commu-
tation relations (11’) must be employed for fermions to replace the
commutation relations (11) for bosons.
A symmetrical linear operator Up of the form (22) can be expressed
in terms of the y, 7 variables by a similar method to that used for
bosons. Equation (24) still holds, and so does (25) with S replaced
by A. Instead of (26) we now have
Oy Ney Marg (0 = pa 2 (—)9 1 Nes Ney |0>
= 2 Na 2 (—)’-*n;,* Nay Nag? |O>85x,<4| U\b>, (26’)
nz,i meaning that the factor y,, must be cancelled out, without its
position among the other 7,’s being changed before the cancellation.
Instead of (27) we have
Fe Nay Nag LOD =D (ng? Ney Nae |9 Spe, (27')
so (28) holds unchanged and thus (29) holds unchanged. We have
the same final form (29) for Up in the fermion case as in the boson
case. Similarly, a symmetrical linear operator Vp of the form (30) can
be expressed as v= Se ny (ab |V led) ua Fes (35"}
auc
the same as one of the ways of writing (35).
The foregoing work shows that there is a deep-seated analogy
between the theory of fermions and that of bosons, only slight
changes having to be made in the general equations of the formalism
when one passes from one to the other.
There is, however, a development of the theory of fermions that
has no analogue for bosons. For fermions there are only the two
252 THEORY OF RADIATION § 65
alternatives of a state being occupied or unoccupied and there is
symmetry between these two alternatives. One can demonstrate the
symmetry mathematically by making a transformation which inter-
changes the concepts of ‘occupied’ and ‘unoccupied’, namely
Na = Nas Na = Nar
The creation operators of the unstarred variables are the annihilation
operators of the starred variables, and vice versa. Thestarred variables
are now seen to satisfy the same quantum conditions and to have all
the same properties as the unstarred ones..
If there are only a few unoccupied states, a convenient standard
ket to work with would be the one for which every state is occupied,
namely |0*> satisfying
Mq|0*> = [O*).
It thus satisfies nilo*> = 0,
or ial0*) = 0.
Other states for the assembly will now be represented by
Ta 1 Ne-|0*),
in which variables appear referring to the unoccupied fermion states
a,b,c.... We may look upon these unoccupied fermion states as holes
among the occupied ones and the 7* variables as the operators of
creation of such holes. The holes are just as much physical things
as the original particles and are also fermions.