Download The Quantization of Wave Fields

14
The Quantization of Wave Fields
The theory of quantum mechanics presented thus far in this book has
dealt with systems that, in the classical limit, consist of material particles.
We wish now to extend the theory so that it can be applied to the magnetic field and thus provide a consistent ba.9is for the quantum theory
of radiation .. ,The quantization of a wave field imparts to it some particle
properties; in the case of the electromagnetic field, a theory of light quanta
(photons) results. The field quantization technique can also be applied
to a 1/1 fIeld, such as that described by the nonrelativistic Schrodinger
equation (6.16) or by one of the relativistic equations (51.4) or (52.3).
As ,ye shall see in the nonrelativistic case (Sec. 55), it then converts a
one-particle theory into a many-particle theory, in a manner equivalent
to the transition from Eq. (6.16) to (16.1) or (40.7). Because of this
equivalence, it might seem that the quantization of 1/1 fields merely provides another formal approach to the many-particle problem. However,
the new formalism can deal as well with processes that involve the creation
or destruction of material particles (radioactive beta decay, mesonnucleon interaction),
490
THE QUANTIZATION OF WAVE FIELDS
ell
This chapter is intended to serve a.'l an introduction to 'l.uum"",
field theory.l We start in Sec. 54 ,vith a discussion of the classic:L1 lind
quantum equations of motion for a wave field, without specifying the
detailed nature of the field. The application to Eq. (6.16) is used as a
first example in Sec. 55. Several other particle wave equations (including
the relativistic Schrodinger and Dimc equations) have also been quantized
but are not discussed here. The electromagnetic field is considered in the
last two sections.
54DCLASSICAL AND QUANTUM FIELD EQUATIONS
A geneml procedure for the quantization of the equations of motion of
a classical system was obtained in Sec. 24. We start with the lagrangian
function for the system and verify that it gives the correct classical
equations. The momenta canonically conjugate to the coordinates of
the system are found from the lagrangian, and a hamiltonian function
is set up. The classical hamiltonian equations of motion are then converted into quantum equations by the substitution of commutator
brackets for Poisson brackets; this gives the change of the dynamical;!
variables with time in the Heisenberg picture. We now show how this
procedure can be applied in its entirety to a wave field ",(r,t), which we
assume for the present to be reaL 2
COORDINATES OF THE FIELD
A wave field is specified by its amplitudes at all points of space and the
dependence of these amplitudes on the time, in much the same W/l.y ns a
system of particles is specified by the positional coordinates qi aud their
dependence on the time. The field evidently hf.s an infinite n \lm ber of
degrees of freedom and is!tnalogous to a system that (1ousisi,H of lUI
infinite number of particles. It is natuml, t,hen, to lIKO tlw
",(r,t) at all points r as coordinates, in analogy wit.h the
nates qi(t) of Sec. 24.
It is not necessary, however, to pro(:()od in thiK wny. AK Itll alternative, W() can expand'" in some comnlcte orthonormal Ket of fUllctions
For further discussion, see P. A. M. "The Principles of Quantum l\iechanics,"
",I., chaps. X, XII (Oxford, New York, 1958); H. Goldstein, "Classical MeehanieR," "hnp. J 1 (Addison-Wesley, Reading, Mass., J 950) i J. D. Bjorkcn and S. D. Droll,
"Hd",\'iviHl.ic: Quantum Fields" (McGraw-Hili, New York, 1965); E. Henley and W.
Thin'illl(, "Elementary Quantum Ficld Theory" (McGraw-Hili, New York, 1962);
S. H. H<:liwc(,cr, "An Introduction to Relativistic QUtmtum Field Theory" (Harper &
Row, N.,w York, 1961); J. J. Sakurai, "Advanced Quantum Mcehanics" (AddisonW(,,,,,IllY, Mass., 1967).
• W. lIei",mhcl'l( 1\.11(1 W. Pauli, Z. Physik 66, 1 (1929); 69, 168 (11)30).
I
4t,1l
1
I
$
QUANTUM MECHANICS
492
Uk:
t/t(r,t) = Skak(l)uk(r)
The expansion coefficients ak in (54.1) call be regarded as the field coordinates, and the field equations can be expressed in termR of either t/t or the
ak. We shall use the wave amplitudes at all points as the field coordinates in this Rectioll. It will be convenient for some of the later work to
make use of the coefficients ak.
TIME DERIVATIVES
It is important to have clearly in mind the meaning of time derivatives
in classical and quantum field theories. In classical pl1rtici<" theory, both
total and partial time derivatives were defined in cOlllleetioll wifb :1
function F(qi,P."t) of the coordinates, momenta, and time; theBe derivatives are related through Eq. (24.22). Similarly, both dcrivatjv('B were
defined for a Heisenberg-picture operator and related to each ot,!lOr as
in Eq. (24.10). In classical field theory, t/t(r) is the analog of q" and
the only time derivative that can be defined is af/i)t; we refer to it a.N
f in analogy with qi in the particle case. Thus, in the claHsieal hamiltonian equations of motion of the field (54.19) beIO\'l], we illtcrpl'oI, ,f,
and also 'Ii', as partial time derivatives. However a functional P(t/t,1f,t)
can depend explicitly on the time as well as Oil the field, so that. it, i",
important to distinguish between dF /dt and aF/at in (54.20).
The same situation appears ill quantum field theory. No di:.;lill('.tion can be made between dift/dt and aift/at, and both are referred 1.0 as ,f.
On the other hand, a HeiRenberg-picture operator can depend
on the time, and the distinction between the two time derivative;; must
be made in Eq. (54.2:3).
CLASSICAL LAGRANGIAN EQUATiON
The lagrangian L(q;,rj;,t) used in Sec. 24 is a fUllction of the time and a
functional of the possible paths qi(t) of the system. The actual paths
are derived from the variational principle (24.17) :
o (t, L
it.
dl
0
=0
By analogy, we expect the field lagrangian to be a functional of the
lipid alllplitude f(r,!). It can usually be expressed as the integral over
all of a lagrangian L:
f,
JL(o/I,Vf,,f,t) dar
above, is at/t/at or dt/t/dt.' The appearance of Vt/t
of L is a consequence of the continuoUH dependence of t/t
Ill'! 1·('IIIi1.rlwd
HK HII ItrIl.UIll()II(.
(54.2)
•
t
[I
493
THE QUANTIZATION OF WAVE FIELDS
on r (continuously infinite number of of freedom); higher derivatives of ,y could also be present but, (10 tlo!, scem to ariHe in problems of
physical interest. The variational Umt eorrcHponds to (24.17)
is
L dt
=
Il
f' J
f
L dt d'r
where the infinitesimal variation
o,y or
il,y(r,t 1) = o,y(r,i 2) = 0
If L has the form indicated ill
aL
a,y o,y
oL
aL
+ L iJ{"iI,y/ il.t) f,
to the restrictions
ift
((lift)
iI.1'
(54.3)
«,iL) dlll'l' 0
(54.4)
it.s
variation can be written
(54.5)
ill"
I ,It/: of
xyz
where the summation over ;/:, 1/, Z 11111""';' sum of three terms with y
and z substituted for x. Now bt/: iI, II", dil'ft;rellce between the original
and varied'" and hence is 1,1)(' tilll!' d'·J'lv:d.ive of the variation of ,yo
This and the similar expressioll for .\{ .hI" ! (1,1'\ elm be written
. = -(o,y)
a
at
Oif;
f,
(ilf)
iI
iI.!' (bift)
fI,1'
Equation (54.3) then \lecollwH
raL
J La,y o,y + L
at
(o,y)
,I
,1.1' (llift)
clift {].
1
dt dar
=0
(54.0)
X'IIZ
The summation terms ill (I'd 1'0) t::ttl be integrated by parts with
respect to the space COortiill:l,I,('H, 1.111' ,'\lrrae(' terms vanish, either because
,y falls off rapidly enough ILl. illiillik diNI.:I,llee, or because,y obeys periodic
boundary conditions at Hw wnllH "fIt Inrge but finite box. The last term
of (54.6) can be integrated hy I'll.d:; wit.h respect to the time, and the
boundary terms vanish bee:J,\IHI' nr (£..1.,1). Equation (.')4.6) ean therefore
be written
(12
)11
J {aL04'
\' a r ilL il r \
1... ax
xyz
L5«'ift/ .
)
a
ill.
(?!:;)}
o,y di d r = 0
il,y
3
(54.7)
Since (54.3) is valid for an arhitrnry VII,!'I;I.uon o,y at each point in space,
Eq. (54.7) is equivalent to the dilTt·r"lll.i;d equation
(54.8)
al:.
ao/;
f:. ax o(o,y
lax)
ii
ill
(ilL)
at/:
Equation (54.8) is the classical iie\d
density L(4',V4',.j;,t).
0
,·/lII:I.LlIlI!
derived from the lagrangian
QUANTUM MECHANICS
494
FUNCTIONAL DERIVATIVE
In order to pursue further ·the analogy with particle mechanics, it is
deOlirable to rewrite Eq. (54.8) in terms of L rather than L. Since the
aggregate of values of y; and J; at all points is analogous to the qi and Ii,
of particle theory, we require derivatives of L with respect to y; and J;
at particular points. These are called functional derivatives and are
denoted by i!LjiN and i!LjilJ;. Expressions for them can be obtained
by dividing up all space into small eells and replacing volume integrals
by summations over these cells. The average values of quantities such
as y;, Vy;, and J; in the ith cell are denoted by subscripts i, and the volume
of that cell by OT,. Then
L(Y;i, (Vy;)" .p" tJ
OTt
appronelwx Ii in the limit in which all the OT, approach zero.
In ximilar ftu:lhioIl, the t integrand in Eq. (54.6) or (54.7) can be
r<lllltwed
f -
a
LOY;i
Or.
f
+ of, OT,
where the variation inL is now produeed by independent variations in
the Y;i and the .pi. Suppose now that all the OY;i and ofi are zero except
for a particular oY;j. It is natural to relate the functional derivative of
L with respeet to Y; for a point in the jth cell to the ratio of oL to oY;j;
we therefore define
ilL
ily;
r
oL
... oY;j OTj
=
aL
"a [
aL
ay; - L. aX iJ(oy;jox)
]
(54.9)
xV'
Similarly, the functional derivative of L with respeet to f is defined by
setting all the OY;i and of, equal to zero except for a particular ofi:
ilL
iJf
lim. oL
.Irj->O of j Orj
iJL
(54.lO)
'af
Here again the point r at which the functional derivative is evaluated is
Substitution of 0';4.9) and O';4.lO) into (54.8) gives
in the jth celL
o iJI,
at iJ.p
aL
iJy;
= 0
(54.11)
whieh dosely resembles the lagrangian equations (24.18) for a system of
partieltlH.
I
411
THE QUANTIZATION OF WAVE FIELDS
CLASSICAL HAMILT.ONIAN EQUATIONS
The momentum canonically conjugate to 1/;j can be defined as in particle
mechanics to be the of oL to the infinitesimal change o,h when all
the other 0"" and all the OY;i are zero. We thus obtain
p.
= Or
o1/;j
J
J
iJ'"
12)
j
It follows from (54.11) and (154.12) that
Pj
Or;
(.14.13)
The analogy with Eq. (24.19) then gives for the hamiltonian
LPi"',
H =
L =
i
L iJ1/;.
"'i Or, i
(54.14)
L
We 'write H as the volume integral of a hamiltonian density Hand
assume that the cells are small enough so that the difference between a
volume integral and the corresponding cell summation can be ignored;
we then have
J Hd
H =
3
r
H=
iJL
71'==iJ'"
L
ilL
il",
The approximate hamiltonian (54.14), with the relations (.14.12) and
(.14.13), can be manipulated in precisely the same way as the hamiltonian
i
I
for a system of particles. Instead of showing this explicitly, we now work
with the true field hamiltonian H given in (54.15), which is a functional
of 1/; and 71' from which", has been eliminated. The classical hamiltonian
equations of motion will be derived without further recourse to the cell
approximation. The variation of L produced by variations of 1/; and '"
can be written, with the help of (54.11) and (.14.15),
oL =
J
01/; + 0"') d r
3
= J[0(71'''')
=
oH
r
'\
:\
J(ir01/; + 71'0",) dar
The variation of H produced
can be written
oH
=
+ iro1/; -
+ aL +
"'07rJ dar
f(iro1/; - ",071') dar
(54.16)
the corresponding variations of 1/; and 71'
J(iJH
01/; + iJH 071') dar
iJ1/;
iJ7r
(54.17)
496
QUANTUM MECHANICS
It follow" from
{,W
ilt/;
illl
,111
(,11.1'11('" discussion of functional derivatives that
i)//
ilt/;
LilaH
u;r; li( aNax)
rut,
./11
'-' ax
illf
XUt
Eqs.
the 16) and (.54.17) for arbitrary variations at/; and
field equations in hamiltonian form:
('Ollllllll'i:;"11 or
r.1l
1.111'11
&//
if;
i-l1r
(54.18)
aH
ilH
11-
(.54.19)
at/;
The hamiltonian equation for the time rate of change of a functional
'" of t/; alld 7f can now be found. We express F as the volume integral of
II,,· functional densij,y F(t/; ,11", t) , which for simplicity is
not to depend explicitly on the time or on the gradientR of t/;
The foregoing analysis can be used to show that
nNsullwd
or
7f.
dF =
+ J + aF + J(iJF aH _ ilF ilH) d3
at
at/;
= aF
+ {F,H}
iJ11" iJt/;
r
(54.20)
This equation also serves t.o define the Poisson bracket expression for two
functionals of the field variables. The right side of Eq. (.54.20) is not
changed if F also depends on Vt/; or Vrr
Prob. 2), It is apparent
from (54.20) that H is a constant of the motion if it does not depend
explicitly on the time; in this case, H is the total energy of the field.
QUANTUM EQUATIONS FOR THE FIELD
The analogy between particle coordinates and momenta qi, Pi and the cell
averages t/;i, Pi suggests that we choose as quantum conditions for the
lipid
[t/;"t/;,l = [Pi,Pj ] = 0
=
ihOij
(54.21)
that we have converted the wave field from a real numerical
lH'rmitian operator in the Heisenberg picture.
W(, IIOW that the cell volumcs are very small. Then Eqs.
I) 1.11.11 Ion with the help of (54.12) and
in terms of
1111.11.1114
I'lIlIdioli 10
:I
THE QUANTIZATIQN OF WAVE FIELDS
417
1/1 and 7r:
[1/1 (r,t),I/I(r'
'"' ['IT(r,t) ,7r(r'
[I/I(r,t),7r(r'
= t'lio(r,r')
=0
where o(r,r') = 1/0T' ir r nnd r' are in the same cell and zero otherwise.
The function o(r,r') haM the property that ff(r)o(r,r') d 3r is equal to the
average value of J fOl' the cell in which r' is situated. Thus, in the limit
in which the cell volumes approach zero, Il(r,r') can bc replaced by the
three-dimellsional Dirac 0 function 1l 3 (r
r'). The Quantum conditions.
for tho canonical field variables then become
[I/I(r,t),I/I(r',t»)
[1/1 (r,t) ,7r(r'
=
[7r(r,t),7r(r',t)]
=
ihll
0
3 (r
(54.22)
The equation of motion for any quantum dynamical variable F is
obtained from Eq.
10) or by replacing the Poisson bracket in Eq.
the commutator bracket divided by ih.
dF
dt
aF
at
+
1 [F H)
'
(54.23)
The commutator bracket can be evaluated with the help of (54.22) when
explicit expressions for F and H in terms of 1/1 and 7r are
Thus Eqs.
and (54.23) completely describe the behavior of the quantized
field that is specified by the hamiltonian H.
FIELDS WITH MORE THAN ONE COMPONENT
Thus far in this section we have dealt with fields that can be described
a single real amplitude. If the field has more than one component
1/11, 1/12, . . . , the lagrangian density has the form L(I/Il, 4.1/11, ,Itt, 1/13, 4.1/12,
';'2, . . . ,t). Then if each of the field components iH vlLried inlillpcnd.
ently, the variational equation (54.3) leads to 1m eqllll,tloll of UIIl form
(54.8) or (54.11) for ench of 1/11, 1/12, . . .. A mOnl(lllt,lIlll
conjugatc to each 1/1, can be defilled lUI ill I';q. (M.
The hamiltonian -density lU11l the form
H
I
"I
=
L7r,';', -
L
(54.24)
and the hamiltonian equations COlllliRt. of It pail' like (M.19) for each
Equation (54.23) is unchanged, and tho commutation relations
are replaced
[I/I.(r,t),1/1",(r',t)] = [7r.(r,t),7r•• (r',t)J = 0
[I/I.(r,t),7r,.(r',t))
=
ihll..,o3(r - r')
8.
(54.25)
QUANTUM MECHANICS
498
COMPLEX FIELD
Thus far we have dealt with fields that are real numerical functions in
the classical case I1nd hermitian operators in the Heisenberg picture in
the quantum case. A different situation that is of immediate interest for
the nonrelativu.,t,j(l Hehrodinger equation is a single ifi field that is complex
or nonhermhil1ll.
___c____c_____
.::·c". In t.he e1mll'lienJ case we can express ifi in terms of real fields
ifi2 as
'"
e':"!
ifi*
+iifi2)
2-1(ifi1 - iifi2)
"'1 and
(54.26)
Wo HImI\' Iir'HI, tim!, the lagrangian equations of the form (54.8) obtained
hy ilI«iopmllJolII, variation of ifi and ifi* are equivalent to those obtained
vnr'ial.ioll 01' ifil ILnd ifi2' It follows from (54.26) that
;/
ilifi
::ll
&ifi1
- i
&ifi2
)
&
= 2-t &ifi1
+ i &ifi2
'1'11111'1 LIIl! ifi, ifi* equations are obtained by adding and subtracting the ifi1l
ifi.
III I-limilaf fashion, the classical momenta canonically conjugate to
ifi II.lld ifi"" arc seen to be
11'
::l-
1(11'1 -
i1l'2)
if = 2-'(11'1
+ ill'Z)
(54.27)
'1'111' Il0(lOlld momentum is written as if rather than 11'* in order to emphasize
UIL\ f,l.d 01111. it. is defined as being canonically conjugate to ifi* and is not
1I1'III'Illlll.l'ily I,he complex conjugate of 11'. Indeed, as we shall see in the
twx!. Hl'l'lcioll, if is identically zero for the nonrelativistic Schrodinger
('l1uul.ioll. Iinwever, whenever the lagrangian is real, 11'1 and 11'2 are inde11"111 h1ll I, or (llIdl other and if = 11'*. In this case 11'1"'1 + 11'2"'2 = + 11'*"'*,
unci 1.111' IlItlnil1.onian is unchanged.
'1'111' (',(IfT()HpOnding quantum case is obtained from the commutation
(fd.::lii) with 8 = 1, 2. If 11'1 and 1l'2 are independent, then all
\mlt',. of vHl'illhles except the following commute:
lifi(r,I),IT(r',l)]
[ifit(r,t),1ft(r',t)] = ih Q3(r
r')
(54.28)
51111QUANTIZATION OF THE NONRELATIVISTIC
SCHRt)DINGER EQUATION
It tir'l-Il. mmrnple of the application of the field-quantization technique
dllvplopOll ill Lhe preceding section, we consider here the quantization of
I.Iw lIolIl'olnt.iviHLie Rchrodinger equation (6.16). The application implies
!,Imt. Wil 11,1'0 Lroai.itlfJ: Eq. (6.16) as though it were a classical equation that
dOH(ll'ihlll-l tlw
llloti()1l
of some kind of material fluid.
As we shall see, the
..
THE QUANTIZATION OF WAVE FIELDS
4..
resulting quantized field theory ie; equivalent to a many-particle Schl'ildinger equation, (W.I) 01' (·10.7). For this reason, field quantization
is often called second (tluwli;:lllion; this term implies that the transition
from classical
quantization.
CLASSICAL LAGRANGIAN AND HAMILTONIAN EQUATIONS
The lagrangian dmlHiLy may he taken to be
/t,2
L
?'hljl"'"
,
V.j;* . vljI
2m
V(r,t)ljI*ljI
(55,1)
As shown ai, til(, plld of the preceding section, ljI and ljI* can be varied
separatdy 1,0 ubl.aill the lagrangian equations of motion. The equation
of tho f(ll'll! ([,.1.1") Umt results from variation of ljI is
11.
ill"," ,
2
,
\i 2lj1*
2m
+ V(r,t)ljI*
whiel! iH Ul(' wmplex conjugate of Eq. (6.16).
Eq. (Ii.· .. ·
{.2
I.
'l'm
\i 2lj1
Variation of ljI* gives
+ V(r,t)ljI
(55.2)
'1'111, IIlOlllontum canonically conjugate to ljI is
?r
ilL
<"I",
thljl*
(55.3)
How(w(,1' "'. dOI'H not appear in the lagrangian density, so that i ie;
identil,ally 1,"1'0. It therefore impossible to satisfy the of tho
conuHutllliOIl rdatjoflS (54.28) (or the corresponding classical I'O[:-;SOIIbradwL n'ild,ioll), so that ljI*, i caIUlOt be regarded as a pail' of
conj lI!l;lI.to They can easily be eliminated from the hamilLonian
since?r" Ill'vt'r and Eq. (55.3) gives ljI* in terms of ?r.1
Tho Imllliitollinll density is
H
L
itt
i
-V?r'VljI VlI'l/t
2m
II,
I Tlw "()tWIIlHi"" IJlltl " ""11 identified with >/;. is related to the appearance of only
th" firH!. .. nl.·, 1.111'" d..... vlltive in the wave equation (55.2), since in this case'" can be
expr""A"d III 1."IIIIiI of >I, nlld ;!,s space derivatives through the wave equation. If the
wave "'IlIltl.lo" IH of H",,,,,,,I order in the time derivative, >/; and", are independent;
then 'If ill ..1'111,1,,'<1 I.., J" I'Id.lll'f thau to >/;., and both !/I, 1<' and >/;., ii' are pairs of canonical
variabl"H. '1'1". lIollrl'l",l.;v;HI,;e Hchrodinger equation and the Dirac equation are of
the fOl'lIIer !,.VI"', witor.,,,,,, th" relativistic Schrodinger equation is of the latter type.
SOD
QUANTUM MECHANICS'
The hamiltonian cqUlttjOllS of motion obtained from (54.19), with the
of (54.18). arc!
i
-
.
i Viii, '12
fI, '11"
2m 11"
Il' "'"
ft,
\
£h
+ 2m
'12ljt
if;
Vljt
I
1&
or l.linHC (jquations is the same as (55.2), and the second equation,
t,oll;ol.hm· wil.h (55.a), is the complex conjugate of (55.2). We have thus
Tlw
AhoWIl, 1'1'O1l1
il.
Itl'(l ill
the point of view of classical field theory, that the lagrangian
(11!).1) and the canonical variables and hamiltonian derived from
agreement with the wave equation (6.16) or (SS.2).
QUANTUM EQUATIONS
as the hamiltonian, (54.23) as the equation of motion, and
linolt. of
(54.28) as the quantum condition on the wave field. Since
ljt is now a Heisenberg-picture operator rather than a numerical function,
ljt* is replaced by ljtt, which is the hermitian adjoint of ljt rather than its
complex conjugate. Further, as remarked above, the Heisenberg-picture
operators ljt, ljtt have no explicit dependence on the time, so that their
equations of motion are given by (54.23) or (24.10), with the first term
on the right side omitted and #/dt on the left side identified with "'.
The hamiltonian is conveniently written with replacement of 7r by ihljtt
and becomes
H
f
Vljtt. vljt
+ Vljtt.p) d·r
,
"
\
(55.5)
_
and (22.16) then shows that H is hermitian.
hamiltonian given in (55.5) is the operator that represents
the total energy of the field; it is not to be confused with the operator
(23.2), which is the energy operator for a single particle that is described
by the wave equation (6.16). We have as
given no explicit representation for the new operators ljt and H and therefore cannot say on
what they might operate. The choice of a particular representation is
!lot necessary so far as the Heisenberg equations of motion are concerned
hut is for the physical interpretation of the formalism that we
lI;i ve irtter in this section.
ThH commu.tation relations are
Ul;t,Hlol""U
=0
r/)
I
j
..
lot
THE QUANTIZATI.QN OF WAVE FIELDS
The omission of t from ['IH' tLl'l(llIIlOli t or the field varhthl{Jf; implies
hoth fields in a com III II [,ttt,OI' 1I1'IWI(llI, 1'1'1'01' t,o Lhe same time, In accordance with the earlinr (liJolflltl'!l'!ioll, I,lio oqul\l,ioIl of motion for f is
rf,H)
=
[f, J '(1'f '" V'f' d I If, J V'ft'f'
vnrinble r' has been substituted
evaluated with the
where primes indi<mj,o I.Iml, /1.11 ill
for r. The second t.m'lII 011 LlH'
help of (.55.6) to gi vo
fV'(fft'f'
(5.5.7)
il /,'
fl'V/f)
Itnl"
dar'
.IT'(IPf "
JV'f'
- r') (Pr'
(55.8)
f eommutes with V, ",hill!! iii
II, IIIlllwI·jmtl function.
Evaluation of the
first term Oil i.ho right JoIid •• of (MI.7) iH Hilllplified by performing a partial
integration on f'(1fl' • Vf' dil" 1,0 ohLni II
f f t 'V'2f' dJr'; the surface terms
vanish bccaui-)() f Ilit.lw!' vlI,lIilllll'H /Lt, infinity or obeys periodic houndary
conditions. W (l UtilI'! oill-Itill
[f,JV'f t ' • '(1'f (til,,11 = - tinl"
'f,N"V'2f'
Jc'V'2f') /)3(r - r')
d 3r'
-V2f (55,9)
Substitutioll or (oIio.H) ILIlIi (MI.IJ) illl,!) (55.7) yields Eq.
so that
the eqmd,jullK ubl,ldunli 1'1'0111 (,I".NNionl nnd quantum field theories agree.
A similu.r ealOlllntioll JoIhOWH 1,llId, UII' oquati'OIl ihJ;t = [ft,H) yields the
hermitian adjoin!, of II)q. it mm also be seen directly that this
equation is tho hormil,iltll Itlijoillt of the equation [f,H) so long as
H is hermitian.
If V is inUep()lIdollt, of t, /I ImH 110 explicit dependence on the
and Eq. (54.2:J) HhoWH 1.11101, 1/ ill 11, of the motion. Thus the
energy in the field iii UIIIlHI,UIlL AlIoLller interesting operator is
N
=
N t fd 8r
"
(55.
The commutator of N wit,h i./Ill V pnrl. of II can be written as
JfV'(ftfft'f' - ft'f'f'f) d,:lnPr'
•
502
QUANTUM MECHANICS
With the. help of (.55Jj) the parenthesis in the integrand is
+ (P(r - r')]",' _ ",t'",'",t",
",t'",t",'", + ",t",'oJ(r - r') _ ",t'",'",t",
+ ",t",'o3(r r')
";1",,,,t',,,' - ",t'",'",t", = ",t[",t'",
_ ",t'",'",t",
=0
since tho Il i'lilldion vanishes unless r
calculation shows that
1",1""
v'",t' . V'",']
[",tv'",'
r'.
A similar but slightly more
(v'",t')",] . V'/lJ(r - r')
TJUI dOllhle integral of this over rand r' is zero. Thus Eq. (55.10) shows
Lhal, N iH !1 constant of the motion.
1(, e:w also be shown that the commutator brackets in (55.6) are
COIIH\'ItIlt,H of the motion, so that these equations are always valid if they
ILI'O nt, n particular time.
1
,
THE N REPRESENTATION
We now specialize to a representation in which the operator N is diagonal.
Since N is hermitian, its eigenvalues are real. A convenient and general
way of specifying this representation is by me!1nS of an expansion like
(54.1) in terms of some complete orthonormal set of functions Uk(r) ,
which we assume for definiteness to be discrete. We put
",(r,t)
=
2:k ak(t)uk(r)
"'t(r,t)
2:k akt(t)u:(r)
(55.11)
where the Uk are numerical functions of the space coordinates and the ak
are Heisenberg-picture operators that depend on the time. Equations
11) can be solved for the ak:
Ju: (r)",(r,t) d
ak(t)
3r
akt(t)
JUk(r)",t(r,t) dar
I
,
Thus, if we multiply the last of the commutation relations (55.6) by
u:(r)ul(r') on both sides and integrate over rand r', we obtain
[ak(t),a/(t)]
JJu;(r)ul(r') 83 (r - r') d 3rd 3r' =
Ilkl
(55.12)
of the orthonormality of the Uk.
In similar fashion, it is apparent
I.hnl. ltk and al commute and that akt and alt commute, for all k and 1.
HuhHtiLution of (55.11) into the expression for N shows that
N='2,
where
Nk
. (55.13)
k
11. iM nu,.4ily thnt each Nk commutes vrith aJl the others, so that they
call ho dilt/l:ollnlizmi Himultaneollsly.
t
101
THE QUANTIZATION' OF WAVE FIELDS
CREATION, DESTRlI.CTION, AND NUMBER OPERATORS
The commutatioll relaj,iollH for the operators ak and ak woro
solved in Sec. 21) ill oOlllllidion the harmonic oscillator. There it
was found that tlw 8olut.ion of (25.10), in the representation in which
ata is diagonal, eO!lHiHj,s of the matrices (2.'5.12). It follows that the
states of the qUltflLized field, in the representation in which each Nk is
T
diagonal, are the kett!
(55.14)
inl,n2, . . . nk, . . . )
where each nk is an eigenvalue of Nk and must be a positive integer or
zero. We also have the relations
aklnl,
nk,
) = nk!l n l,
)
(n.
1, . . . )
. . . nk
1, . . . ) (55.15)
. . . nk -
+ 1)*\n!,
+
nk,
a.tlnt,
Thus ak t and ak are called creation and destruction operator8 for the state
k of the field.
The number operator Nk need not be a constant of the motion,
although we have seen from Eq. (55.10) that N = :zl"h is a constant.
The rate of change of Nk is given hy
ihNk
[akta.,II]
where H is obtained from (55.5) and (55.11):
a/al J H
=
..
Vui • VUl
a/al JU;'" ( -
+
:;. \7 2 +
VU;UI) dar
v)
11.1
dar
(55.16)
It is not difficult to show from (fiIU2) that a particular Nk is constant if
and only if all the volume intogml,; in (55.16) arc zero for which either
j or l is equal to k. These int,ogmlH are just the matrix elements of the
one-particle hamiltonian (23.2), 1'40 I.h!tt the necessary and sufficient condition that Nk be a constant of !;lw motion is that all such off-diagonal
elements that involve the state Uk be zero.!
The case in which the Uk are eigenfunctions of (23.2) with eigenvalues Ek is of particular illterl!iiL The integrals in (55.16) are then
E10 jh and the field hamiltonian IW(:OU1CH
(55.17)
H
aktakEk
ivkEk
L
k
L
k
This particular N representation
ill t,}10
one in which H is also diagonal;
ThiH for the quantized field is dOR"ly related to the corresponding result,
containml in Eq. (35.5), for the one-partido prol)llbility amplitude.
1
504
the kef,
QUANTUM MECHANICS
In" .
OPOI':!'I.OI'
fl.
I i / " . , , ) has j,he eigenvalue 'J:,nkE k for the toj.al energy
I t, it-! npl'llH'1I t. t.hat all the
are constant ill thi" case.
CONNECTION WITH THE SCHROOINGER EQUATION
<111/1.11 Li ,/,.,01 li.·ld UH,or,Y is closely related to the many-particle SehrcidiltfJ:4\t· cIll'litLioll in Sec, 40. If the Uk are eigenfunctions of the
Ol!t'-I'II.I·Lled.. IlHluill,olliall (23.2), the field theory shows that
Thn
which the number of particles n, in the kth state is 11
I'mii Li VI' i lIi-eger or zero, and the energy is 'J:,nkE k • Each solution
(,all 1,,- .h'H(\I'il.pd by ket
. . . nk, . , . these kets form a complete
Ol'l.hollOI'lIIII,1 HnC, alld there is just one solution for each set of number"
III,
()1I the other hand, a stationary many-particle wave function
Iii", 1111' .p ill I';q. (-iO.1) can be written a.s a product of olle-particle wave
fUlidiollH if there is no interaction hetween the
'1'114' linolLI' combination of such products that is symmetric with
of any
of pa.rticle coordinates can be specified uniquely
the number of particles in each state. Again, the number of
in eaeh state is a positive integer or zero, and the energy is the
Hum of alt the particle energies.
We see then that the quantized field theory developed thus far in
this section is equivalent to the Schrodinger equation for several noninteracting particles, provided that only the symmetric solutions are
retained in the latter case. We are thm; led to It theory of
that
Einstein-Bose statistics, It can be shown that the two theories are
completely equivalent even if interactions between narticles are taken
into account. l
It is natural to see if there is some way in which the quantized-field
formalism can be modified to yield a theory of particles that obey FermiDirac statistics. As discus;,;ed in Sec. 40, a system of such particles can be
described by a many-particle wave function that is antisymmetric with
to interchange of any pair of particle coordinates. The required
linear combination of products of one-particle wave functions can be
specified uniquely by stating the number of particles in each stat,e, provided that each of these numbers is either 0 or 1. The desired modificalimit the eigenvalues of each nnprfl.tor
bon of the
must,
Nk to 0 and 1.
Holill,iolll1 for
(iOtlHI.alll.
ANTICOMMUTATION RELATIONS
A review of the foregoing theory shows that the conclm;ion that the
values of each Nk arc the positive
arid zero stems from the com111lllation relations (.55.12) for the ak and akt. Equations (55.12) in turn
I H(',' W.
Heisenberg, "The Physical Principles of the Quantum Theory," App.,
see. 11 (University of Chicago Press, Chicago, 1930).
'\'
j
"
,
THE QUANTIZATION OF WAVE FIELDS
..
arise from the commutation relations (55.6) for"p and "pt. Thus we must
modify Eqs. (55.6) if we are to obtain a theory of particles that obey exclusion principle. It is reasonable to require that this modification be
made in such a way that the quantum equation of motion for"p is the wave
equation (55.2) when the hamiltonian has the form (55.5).
It was found by Jordan and Wigner 1 that the desired modification
consists in the replacement of the commutator brackets
[A,B]
==
AB - BA
in Eqs. (54.22) and (55.6) by anticommutator brackets
[A,BJ+
== AB,+ BA
This means that Eqs. (55.6) are replaced by
["p(r),Hr')]+
["pt(r),,,pt(r')l+
+ "p(r')"p(r)
"p(r)"p(r')
=
0
+ "pt(r')"pt(r) = 0
+ "pt(r')"p(r) = a (r -
= "pt(r)"pt(r')
["p(r),,,pt(r')l+ = "p(r)"pt(r')
3
(55.18)
r')
It then follows directly from Eqs. (55.11) and (55.18) that
[ak,ad+ = aka, + a,a", = 0
[akt,a,tl+ = a.l,ta,t + a,takt ... 0
[ak,a/]+ = akazt
+ a/ak
•
(55.19)
"" thl
We define Nk = a.ta", as before and notice first that each Nk commutes with all the others, so that they can be diagonalized simultaneously. The eigenvalues of Nk can be obtained from the matrix equation
Nk2
aktakaktak
= akt(1
- aktak)ak = aktak = Nk
(55.20)
where use has been made of Eqs. (55.19). ,If Nk is in diagonal form and
has the eigenvalues nr, ... ,it is apparent thM Nk2 is also in diagonal form and has the eigenvalUes • . •• Thus the matrix
equation (55.20) is equivalent to the algebraic equations
'2
nk
,.
n"2
k
n II.
k
for the eigenvalues. These are quadratic equations that have two roots:
1. Thus the eigenvalues of each Nk are 0 and 1, and the particles
obey the exclusion principle. The eigenvalues of N = T.Nk are the
positive integers and zero, as before. The earlier expressions (55.16) and
(55.17) for the hamiltonian are unchanged, and the energy eigenvalues
are T.nkE k •
o and
1
... .
I
nk
P. Jordan and E. Wigner"Z. Physik 47, 631 (1928) .
QUANTUM MECHANICS
508
effects of operating with ak and ak! on a ket
(,Jmt has the eigenvalue 11k (= 0 or 1) for the operator
Nt. 'I'lip d('.yir(·" f'('I:diolls would have the form (;j5.25) were it not that It
"pril';; of HII<'II ('I!,llIlioIlS (with subscripts added) would not agree with the
WO lilJ(1 LlI('
. . . ,II",
.
1.\\0 .. I' 1':q:l.
W('
.)
(f,!).1
proeeed in the following way. \Ve order the states k
but definite way: 1,2, . . . ,k, . . .. Then
has the form (55.2.5), except that l1 multiplyiiiI/, "hHI or IIlilltis sign is introduced, according as the kth state is preceded
ill II,,· 11',:1111111'.1 order by an even or an odd number of occupied states.
W.. 11111" !'I'plae!, the Einstein-Bose equations (55.15) bv the exclusioniI"
tlil'l'l,r.. l'!'
III" ...
,1
nk., . . .)
(1"1111,, ...
nk, . . .)
=
nk,.
.. ,1
Ok(l
.)
nk, . . .) (55.26)
k-l
Ih
(1)"
Vk
i
L
nj
1
As lm example, we calculate the effect of operating with akal and with
::lome ket, where we assume for definiteness that the order is such
Chat l > k. If each operation is not to give a zero result, both nk and nl
ill t.he original ket must equal unity. Operation with akal empties fir8t
the tth and t.hen the kth state and introduces a fltctor OIOk. Operation
wit,h alak empties the kth state first, so that. Ok is unchanged. But when
the lth stat.e is emptied iu this case, there is one less particle in the states
below the lth than there was in the previous case, since the kth state is
now empty, whereas it was occupied before. Thus the sign of .01 is
changed. We find in this way that
Ifl"f. 011
akad ...
nk • . • nl • • . )
=
-alakl . . .
nk . . • rll . • .)
in agreement with the first of Eqs. (5.5.19). In similar fashion, it can be
shown that Eqs. (55.26) agree with the result of operating with the other
two of Eqs. (.55.19) OD any ket. SiDce the aggregate of kets represents
all possible states of the many-particle system, they constitute a eomplete
sd, and Eqs. ([)f>.19) follow as operator equations from Eqs. (55.26).
561 IELECTROMAGNETIC FIELD IN VACUUM'
W.·
II \I'
the methods developed in Sec. 54 to the quantization of
field in vacuum. Since we are not coneerned with
II", 1....1.1,,·,· di"""N>lion of the material in this section aild the next, see the references
I',I,·d '" 1001"01... I. I'ag" 4!)], and also E. :Fermi, Rev. ilIod. Phys: 4, '137 (19:32); L.
1I0H,·"I',·ld .. 11111 / 118L 1l,.lI.l'i Poincare 1,25 (1981); W. Reitter, "The Quantum Theory
oj' i!/l.diu.lioll," :\d ..d. (Ox[Ol'(t, New York, 1954).