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 ba9is 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 11 fIeld such as that described by the nonrelativistic Schrodinger equation (616) or by one of the relativistic equations (514) or (523) 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 (616) to (161) or (407) Because of this equivalence it might seem that the quantization of 11 fields merely pro-vides 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 meson-nucleon interaction) 490

THE QUANTIZATION OF WAVE FIELDS ell

This chapter is intended to serve al an introduction to luum

field theoryl We start in Sec 54 vith a discussion of the classicL1 lind quantum equations of motion for a wave field without specifying the detailed nature of the field The application to Eq (616) 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 con-verted 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 (rt) 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 Wly ns a system of particles is specified by the positional coordinates qi aud their dependence on the time The field evidently hfs an infinite n lm ber of degrees of freedom and istnalogous to a system that (1ousisiH of lUI infinite number of particles It is natuml then to lIKO tlw (rt) at all points r as coordinates in analogy with the nates qi(t) of Sec 24

It is not necessary however to pro(()od in thiK wny AK Itll alter-native W() can expand in some comnlcte orthonormal Ket of fUllctions

I For further discussion see P A M The Principles of Quantum liechanics 4t1l I chaps X XII (Oxford New York 1958) H Goldstein Classical Meehan-ieR hnp J1 (Addison-Wesley Reading Mass J950) i J D Bjorkcn and S D Droll HdiviHlic Quantum Fields (McGraw-Hili New York 1965) E Henley and W Thinilll( Elementary Quantum Ficld Theory (McGraw-Hili New York 1962) S H Hltliwc(cr An Introduction to Relativistic QUtmtum Field Theory (Harper amp Row Nw York 1961) J J Sakurai Advanced Quantum Mcehanics (Addison-W(IllY Mass 1967) bull W lIeimhcll( 111(1 W Pauli Z Physik 66 1 (1929) 69 168 (11)30)

1 I $

492 QUANTUM MECHANICS

Uk

tt(rt) = Skak(l)uk(r)

The expansion coefficients ak in (541) call be regarded as the field coordi-nates and the field equations can be expressed in termR of either tt or the ak We shall use the wave amplitudes at all points as the field coordi-nates 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 pl1rticilt theory both total and partial time derivatives were defined in cOlllleetioll wifb 1

function F(qiPt) of the coordinates momenta and time theBe deriva-tives are related through Eq (2422) Similarly both dcrivatjv(B were defined for a Heisenberg-picture operator and related to each otlOr as in Eq (2410) In classical field theory tt(r) is the analog of q and the only time derivative that can be defined is afi)t we refer to it aN f in analogy with qi in the particle case Thus in the claHsieal hamil-tonian equations of motion of the field (5419) beIOl] we illtcrploI f and also Ii as partial time derivatives However a functional P(tt1ft) can depend explicitly on the time as well as Oil the field so that it i important to distinguish between dFdt and aFat in (5420)

The same situation appears ill quantum field theory No dilill(-tion can be made between diftdt and aiftat and both are referred 10 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 (5423)

CLASSICAL LAGRANGIAN EQUATiON

The lagrangian L(qrjt) 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 (2417)

o(t L dl 0 =0it 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 bullall of a lagrangian L

f JL(oIVfft) dar (542) t Ill 1middot(IIIi1rlwd above is attat or dttdt The appearance of Vtt

HK HII ItrIlUIll()II( of L is a consequence of the continuoUH dependence of tt

[I 493

THE QUANTIZATION OF WAVE FIELDS

on r (continuously infinite number of of freedom) higher deriva-tives of y could also be present but (10 tlo scem to ariHe in problems of physical interest The variational Umt eorrcHponds to (2417)

is (543)f laquoiL) dllll 0 L dt = Ilf JL dt dr

to the restrictions where the infinitesimal variation oy or ift

(544) ily(rt1) = oy(ri2) = 0 If L has the form indicated ill its variation can be written

(545)aL aL ill((lift)oL ay oy + L iJiIy ilt) f iI1 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 dilftrellce between the original and varied and hence is 11)( tilll dmiddotJlvdive of the variation of yo This and the similar expressioll for hI (11 elm be written

iI a (ilf) iI (bift)Oif = -(oy) f fI1at Equation (543) then lecollwH

I

JraL at (oy) 1dt dar = 0 (540)11 (llift) clift ]Lay oy + L XIIZ

The summation terms ill (Id 10) tttl be integrated by parts with respect to the space COortiilllI(H 1111 lrrae( terms vanish either because y falls off rapidly enough ILl illiillik diNIIllee or becausey obeys periodic boundary conditions at Hw wnllH fIt Inrge but finite box The last term of (546) can be integrated hy Illd with respect to the time and the boundary terms vanish beeJIHI nr (pound11) Equation ()46) ean therefore

be written (547)

(12 JaL a r ilL a () oy di d3 = 0 ill ily)11 04 r )1 ax L5laquoift il

r

xyz

Since (543) is valid for an arhitrnry VIIIIuon oy at each point in space Eq (547) is equivalent to the dilTtmiddotrlllid equation

(548)al ii (ilL) 0ao f ax o(oylax) ill at

Equation (548) is the classical iied middotlIIILlIlI derived from the lagrangian

density L(4V4jt)

494 QUANTUM MECHANICS

FUNCTIONAL DERIVATIVE

In order to pursue further middotthe analogy with particle mechanics it is deOlirable to rewrite Eq (548) 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 iLjiN and iLjilJ 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(Yi (Vy) p tJ OTt

appronelwx Ii in the limit in which all the OT approach zero In ximilar ftulhioIl the t integrand in Eq (546) or (547) can be

rltlllltwed

f - a LOYi Or + f of OT

where the variation inL is now produeed by independent variations in the Yi and the pi Suppose now that all the OYi and ofi are zero except for a particular oYj 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 oYj we therefore define

ilL r oL aL a [ aL ] (549)ily oYj OTj = ay - L aX iJ(oyjox) xV

Similarly the functional derivative of L with respeet to f is defined by setting all the OYi and of equal to zero except for a particular ofi

iJLilL lim oL (54lO)iJf Irj-gtO of j Orj af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 049) and O4lO) into (548) gives

o iJI aL = 0 (5411)at iJp iJy

whieh dosely resembles the lagrangian equations (2418) for a system of partieltlH I

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

THE QUANTIZATION OF WAVE FIELDS ell

This chapter is intended to serve al an introduction to luum

field theoryl We start in Sec 54 vith a discussion of the classicL1 lind quantum equations of motion for a wave field without specifying the detailed nature of the field The application to Eq (616) 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 con-verted 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 (rt) 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 Wly ns a system of particles is specified by the positional coordinates qi aud their dependence on the time The field evidently hfs an infinite n lm ber of degrees of freedom and istnalogous to a system that (1ousisiH of lUI infinite number of particles It is natuml then to lIKO tlw (rt) at all points r as coordinates in analogy with the nates qi(t) of Sec 24

It is not necessary however to pro(()od in thiK wny AK Itll alter-native W() can expand in some comnlcte orthonormal Ket of fUllctions

I For further discussion see P A M The Principles of Quantum liechanics 4t1l I chaps X XII (Oxford New York 1958) H Goldstein Classical Meehan-ieR hnp J1 (Addison-Wesley Reading Mass J950) i J D Bjorkcn and S D Droll HdiviHlic Quantum Fields (McGraw-Hili New York 1965) E Henley and W Thinilll( Elementary Quantum Ficld Theory (McGraw-Hili New York 1962) S H Hltliwc(cr An Introduction to Relativistic QUtmtum Field Theory (Harper amp Row Nw York 1961) J J Sakurai Advanced Quantum Mcehanics (Addison-W(IllY Mass 1967) bull W lIeimhcll( 111(1 W Pauli Z Physik 66 1 (1929) 69 168 (11)30)

1 I $

492 QUANTUM MECHANICS

Uk

tt(rt) = Skak(l)uk(r)

The expansion coefficients ak in (541) call be regarded as the field coordi-nates and the field equations can be expressed in termR of either tt or the ak We shall use the wave amplitudes at all points as the field coordi-nates 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 pl1rticilt theory both total and partial time derivatives were defined in cOlllleetioll wifb 1

function F(qiPt) of the coordinates momenta and time theBe deriva-tives are related through Eq (2422) Similarly both dcrivatjv(B were defined for a Heisenberg-picture operator and related to each otlOr as in Eq (2410) In classical field theory tt(r) is the analog of q and the only time derivative that can be defined is afi)t we refer to it aN f in analogy with qi in the particle case Thus in the claHsieal hamil-tonian equations of motion of the field (5419) beIOl] we illtcrploI f and also Ii as partial time derivatives However a functional P(tt1ft) can depend explicitly on the time as well as Oil the field so that it i important to distinguish between dFdt and aFat in (5420)

The same situation appears ill quantum field theory No dilill(-tion can be made between diftdt and aiftat and both are referred 10 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 (5423)

CLASSICAL LAGRANGIAN EQUATiON

The lagrangian L(qrjt) 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 (2417)

o(t L dl 0 =0it 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 bullall of a lagrangian L

f JL(oIVfft) dar (542) t Ill 1middot(IIIi1rlwd above is attat or dttdt The appearance of Vtt

HK HII ItrIlUIll()II( of L is a consequence of the continuoUH dependence of tt

[I 493

THE QUANTIZATION OF WAVE FIELDS

on r (continuously infinite number of of freedom) higher deriva-tives of y could also be present but (10 tlo scem to ariHe in problems of physical interest The variational Umt eorrcHponds to (2417)

is (543)f laquoiL) dllll 0 L dt = Ilf JL dt dr

to the restrictions where the infinitesimal variation oy or ift

(544) ily(rt1) = oy(ri2) = 0 If L has the form indicated ill its variation can be written

(545)aL aL ill((lift)oL ay oy + L iJiIy ilt) f iI1 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 dilftrellce between the original and varied and hence is 11)( tilll dmiddotJlvdive of the variation of yo This and the similar expressioll for hI (11 elm be written

iI a (ilf) iI (bift)Oif = -(oy) f fI1at Equation (543) then lecollwH

I

JraL at (oy) 1dt dar = 0 (540)11 (llift) clift ]Lay oy + L XIIZ

The summation terms ill (Id 10) tttl be integrated by parts with respect to the space COortiilllI(H 1111 lrrae( terms vanish either because y falls off rapidly enough ILl illiillik diNIIllee or becausey obeys periodic boundary conditions at Hw wnllH fIt Inrge but finite box The last term of (546) can be integrated hy Illd with respect to the time and the boundary terms vanish beeJIHI nr (pound11) Equation ()46) ean therefore

be written (547)

(12 JaL a r ilL a () oy di d3 = 0 ill ily)11 04 r )1 ax L5laquoift il

r

xyz

Since (543) is valid for an arhitrnry VIIIIuon oy at each point in space Eq (547) is equivalent to the dilTtmiddotrlllid equation

(548)al ii (ilL) 0ao f ax o(oylax) ill at

Equation (548) is the classical iied middotlIIILlIlI derived from the lagrangian

density L(4V4jt)

494 QUANTUM MECHANICS

FUNCTIONAL DERIVATIVE

In order to pursue further middotthe analogy with particle mechanics it is deOlirable to rewrite Eq (548) 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 iLjiN and iLjilJ 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(Yi (Vy) p tJ OTt

appronelwx Ii in the limit in which all the OT approach zero In ximilar ftulhioIl the t integrand in Eq (546) or (547) can be

rltlllltwed

f - a LOYi Or + f of OT

where the variation inL is now produeed by independent variations in the Yi and the pi Suppose now that all the OYi and ofi are zero except for a particular oYj 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 oYj we therefore define

ilL r oL aL a [ aL ] (549)ily oYj OTj = ay - L aX iJ(oyjox) xV

Similarly the functional derivative of L with respeet to f is defined by setting all the OYi and of equal to zero except for a particular ofi

iJLilL lim oL (54lO)iJf Irj-gtO of j Orj af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 049) and O4lO) into (548) gives

o iJI aL = 0 (5411)at iJp iJy

whieh dosely resembles the lagrangian equations (2418) for a system of partieltlH I

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

1 I $

492 QUANTUM MECHANICS

Uk

tt(rt) = Skak(l)uk(r)

The expansion coefficients ak in (541) call be regarded as the field coordi-nates and the field equations can be expressed in termR of either tt or the ak We shall use the wave amplitudes at all points as the field coordi-nates 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 pl1rticilt theory both total and partial time derivatives were defined in cOlllleetioll wifb 1

function F(qiPt) of the coordinates momenta and time theBe deriva-tives are related through Eq (2422) Similarly both dcrivatjv(B were defined for a Heisenberg-picture operator and related to each otlOr as in Eq (2410) In classical field theory tt(r) is the analog of q and the only time derivative that can be defined is afi)t we refer to it aN f in analogy with qi in the particle case Thus in the claHsieal hamil-tonian equations of motion of the field (5419) beIOl] we illtcrploI f and also Ii as partial time derivatives However a functional P(tt1ft) can depend explicitly on the time as well as Oil the field so that it i important to distinguish between dFdt and aFat in (5420)

The same situation appears ill quantum field theory No dilill(-tion can be made between diftdt and aiftat and both are referred 10 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 (5423)

CLASSICAL LAGRANGIAN EQUATiON

The lagrangian L(qrjt) 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 (2417)

o(t L dl 0 =0it 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 bullall of a lagrangian L

f JL(oIVfft) dar (542) t Ill 1middot(IIIi1rlwd above is attat or dttdt The appearance of Vtt

HK HII ItrIlUIll()II( of L is a consequence of the continuoUH dependence of tt

[I 493

THE QUANTIZATION OF WAVE FIELDS

on r (continuously infinite number of of freedom) higher deriva-tives of y could also be present but (10 tlo scem to ariHe in problems of physical interest The variational Umt eorrcHponds to (2417)

is (543)f laquoiL) dllll 0 L dt = Ilf JL dt dr

to the restrictions where the infinitesimal variation oy or ift

(544) ily(rt1) = oy(ri2) = 0 If L has the form indicated ill its variation can be written

(545)aL aL ill((lift)oL ay oy + L iJiIy ilt) f iI1 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 dilftrellce between the original and varied and hence is 11)( tilll dmiddotJlvdive of the variation of yo This and the similar expressioll for hI (11 elm be written

iI a (ilf) iI (bift)Oif = -(oy) f fI1at Equation (543) then lecollwH

I

JraL at (oy) 1dt dar = 0 (540)11 (llift) clift ]Lay oy + L XIIZ

The summation terms ill (Id 10) tttl be integrated by parts with respect to the space COortiilllI(H 1111 lrrae( terms vanish either because y falls off rapidly enough ILl illiillik diNIIllee or becausey obeys periodic boundary conditions at Hw wnllH fIt Inrge but finite box The last term of (546) can be integrated hy Illd with respect to the time and the boundary terms vanish beeJIHI nr (pound11) Equation ()46) ean therefore

be written (547)

(12 JaL a r ilL a () oy di d3 = 0 ill ily)11 04 r )1 ax L5laquoift il

r

xyz

Since (543) is valid for an arhitrnry VIIIIuon oy at each point in space Eq (547) is equivalent to the dilTtmiddotrlllid equation

(548)al ii (ilL) 0ao f ax o(oylax) ill at

Equation (548) is the classical iied middotlIIILlIlI derived from the lagrangian

density L(4V4jt)

494 QUANTUM MECHANICS

FUNCTIONAL DERIVATIVE

In order to pursue further middotthe analogy with particle mechanics it is deOlirable to rewrite Eq (548) 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 iLjiN and iLjilJ 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(Yi (Vy) p tJ OTt

appronelwx Ii in the limit in which all the OT approach zero In ximilar ftulhioIl the t integrand in Eq (546) or (547) can be

rltlllltwed

f - a LOYi Or + f of OT

where the variation inL is now produeed by independent variations in the Yi and the pi Suppose now that all the OYi and ofi are zero except for a particular oYj 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 oYj we therefore define

ilL r oL aL a [ aL ] (549)ily oYj OTj = ay - L aX iJ(oyjox) xV

Similarly the functional derivative of L with respeet to f is defined by setting all the OYi and of equal to zero except for a particular ofi

iJLilL lim oL (54lO)iJf Irj-gtO of j Orj af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 049) and O4lO) into (548) gives

o iJI aL = 0 (5411)at iJp iJy

whieh dosely resembles the lagrangian equations (2418) for a system of partieltlH I

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

[I 493

THE QUANTIZATION OF WAVE FIELDS

on r (continuously infinite number of of freedom) higher deriva-tives of y could also be present but (10 tlo scem to ariHe in problems of physical interest The variational Umt eorrcHponds to (2417)

is (543)f laquoiL) dllll 0 L dt = Ilf JL dt dr

to the restrictions where the infinitesimal variation oy or ift

(544) ily(rt1) = oy(ri2) = 0 If L has the form indicated ill its variation can be written

(545)aL aL ill((lift)oL ay oy + L iJiIy ilt) f iI1 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 dilftrellce between the original and varied and hence is 11)( tilll dmiddotJlvdive of the variation of yo This and the similar expressioll for hI (11 elm be written

iI a (ilf) iI (bift)Oif = -(oy) f fI1at Equation (543) then lecollwH

I

JraL at (oy) 1dt dar = 0 (540)11 (llift) clift ]Lay oy + L XIIZ

The summation terms ill (Id 10) tttl be integrated by parts with respect to the space COortiilllI(H 1111 lrrae( terms vanish either because y falls off rapidly enough ILl illiillik diNIIllee or becausey obeys periodic boundary conditions at Hw wnllH fIt Inrge but finite box The last term of (546) can be integrated hy Illd with respect to the time and the boundary terms vanish beeJIHI nr (pound11) Equation ()46) ean therefore

be written (547)

(12 JaL a r ilL a () oy di d3 = 0 ill ily)11 04 r )1 ax L5laquoift il

r

xyz

Since (543) is valid for an arhitrnry VIIIIuon oy at each point in space Eq (547) is equivalent to the dilTtmiddotrlllid equation

(548)al ii (ilL) 0ao f ax o(oylax) ill at

Equation (548) is the classical iied middotlIIILlIlI derived from the lagrangian

density L(4V4jt)

494 QUANTUM MECHANICS

FUNCTIONAL DERIVATIVE

In order to pursue further middotthe analogy with particle mechanics it is deOlirable to rewrite Eq (548) 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 iLjiN and iLjilJ 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(Yi (Vy) p tJ OTt

appronelwx Ii in the limit in which all the OT approach zero In ximilar ftulhioIl the t integrand in Eq (546) or (547) can be

rltlllltwed

f - a LOYi Or + f of OT

where the variation inL is now produeed by independent variations in the Yi and the pi Suppose now that all the OYi and ofi are zero except for a particular oYj 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 oYj we therefore define

ilL r oL aL a [ aL ] (549)ily oYj OTj = ay - L aX iJ(oyjox) xV

Similarly the functional derivative of L with respeet to f is defined by setting all the OYi and of equal to zero except for a particular ofi

iJLilL lim oL (54lO)iJf Irj-gtO of j Orj af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 049) and O4lO) into (548) gives

o iJI aL = 0 (5411)at iJp iJy

whieh dosely resembles the lagrangian equations (2418) for a system of partieltlH I

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

494 QUANTUM MECHANICS

FUNCTIONAL DERIVATIVE

In order to pursue further middotthe analogy with particle mechanics it is deOlirable to rewrite Eq (548) 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 iLjiN and iLjilJ 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(Yi (Vy) p tJ OTt

appronelwx Ii in the limit in which all the OT approach zero In ximilar ftulhioIl the t integrand in Eq (546) or (547) can be

rltlllltwed

f - a LOYi Or + f of OT

where the variation inL is now produeed by independent variations in the Yi and the pi Suppose now that all the OYi and ofi are zero except for a particular oYj 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 oYj we therefore define

ilL r oL aL a [ aL ] (549)ily oYj OTj = ay - L aX iJ(oyjox) xV

Similarly the functional derivative of L with respeet to f is defined by setting all the OYi and of equal to zero except for a particular ofi

iJLilL lim oL (54lO)iJf Irj-gtO of j Orj af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 049) and O4lO) into (548) gives

o iJI aL = 0 (5411)at iJp iJy

whieh dosely resembles the lagrangian equations (2418) for a system of partieltlH I

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILTONIAN EQUATIONS

The momentum canonically conjugate to 1j can be defined as in particle mechanics to be the of oL to the infinitesimal change oh when all the other 0 and all the OYi are zero We thus obtain

p = Or 12) J Jo1j iJ j

It follows from (5411) and (15412) that

Pj Or (1413)

The analogy with Eq (2419) then gives for the hamiltonian

H = LPi L = L i Or - L (5414) i i iJ1

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

iJL ilL H = JHd3r H= L 71==- -

iJ il

The approximate hamiltonian (5414) with the relations (1412) and (1413) can be manipulated in precisely the same way as the hamiltonian for a system of particles Instead of showing this explicitly we now work with the true field hamiltonian H given in (5415) 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 (5411) and (1415)i oL = J 01 + 0) d3r J(ir01 + 710) dar

I = J[0(71) + iro1 - 07rJ dar = oH + aL + f(iro1 - 071) dar (5416)

r The variation of H produced the corresponding variations of 1 and 71 can be written

oH = J(iJH 01 + iJH 071) dar (5417)iJ1 iJ7r

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

bullbull

496 QUANTUM MECHANICS

It follow from (11111( discussion of functional derivatives that

W i) LilaH ilt ilt ur li(aNax)

rut (5418)illl 11 aH

111 illf - ax XUt

(Olllllllli11 or Eqs 16) and (5417) for arbitrary variations at and r1l 111111 the field equations in hamiltonian form

amp ilHif 11- (5419)i-l1r 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 IImiddot functional densijy F(t 11 t) which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t or 7f The foregoing analysis can be used to show that

dF = + J + aF + J(iJF aH _ ilF ilH) d3 at at iJ11 iJt r

= aF + FH (5420)

This equation also serves to define the Poisson bracket expression for two functionals of the field variables The right side of Eq (5420) is not changed if F also depends on Vt or Vrr Prob 2) It is apparent from (5420) 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 ti Pi suggests that we choose as quantum conditions for the lipid

[ttl = [PiPj ] = 0 = ihOij (5421)

1111111114 that we have converted the wave field from a real numerical IlIlIdioli 10 I lHrmitian operator in the Heisenberg picture

W( IIOW that the cell volumcs are very small Then Eqs I) 11111 Ion with the help of (5412) and in terms of

417 THE QUANTIZATIQN OF WAVE FIELDS

11 and 7r

[11 (rt)II(r [IT(rt) 7r(r =0 [II(rt)7r(r = tlio(rr)

where o(rr) = 10T ir r nnd r are in the same cell and zero otherwise The function o(rr) haM the property that ff(r)o(rr) d3r 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(rr) can bc replaced by the three-dimellsional Dirac 0 function 1l3 (r r) The Quantum conditions for tho canonical field variables then become

[II(rt)II(rtraquo) = [7r(rt)7r(rt)] 0 (5422)

[11 (rt) 7r(r = ihll 3(r

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 aF + 1 [F H) (5423)dt at

The commutator bracket can be evaluated with the help of (5422) when explicit expressions for F and H in terms of 11 and 7r are Thus Eqs

and (5423) 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

111 112 the lagrangian density has the form L(IIl 4111 Itt 113 4112 2 t) Then if each of the field components iH vlLried inlillpcnd ently the variational equation (543) leads to 1m eqlllltloll of UIIl form (548) or (5411) for ench of 111 112 A mOnl(llltlIlll conjugatc to each 11 can be defilled lUI ill Iq (M The hamiltonian -density lU11l the form

H = L7r - L (5424)

and the hamiltonian equations COlllliRt of It pail like (M19) for each 8 Equation (5423) is unchanged and tho commutation relations are replacedI

[II(rt)11(rt)] = [7r(rt)7rbullbull (rt)J = 0 (5425)

[II(rt)7r(rt)) = ihllo3(r - r)

I

498 QUANTUM MECHANICS

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 nonrelativutj(l Hehrodinger equation is a single ifi field that is complex or nonhermhil1ll ___c____c_____

middotc In the e1mlllienJ case we can express ifi in terms of real fields 1 and ifi2 as

e +iifi2) ifi 2-1(ifi1 - iifi2) (5426)

Wo HImI IirHI tim the lagrangian equations of the form (548) obtained hy ilIlaquoiopmllJolII variation of ifi and ifi are equivalent to those obtained vnrialioll 01 ifil ILnd ifi2 It follows from (5426) that

ll - i amp = 2-t + i ilifi ampifi1 ampifi2

)

ampifi1 ampifi2

1111111 LIIl ifi ifi equations are obtained by adding and subtracting the ifi1l ifi

III I-limilaf fashion the classical momenta canonically conjugate to ifi IIlld ifi arc seen to be

11 l- 1(111 - i1l2) if = 2-(111 + illZ) (5427)

1111 Il0(lOlld momentum is written as if rather than 11 in order to emphasize UIL fld 01111 it is defined as being canonically conjugate to ifi and is not 1I1IIIIllllllily Ihe complex conjugate of 11 Indeed as we shall see in the twx Hlllcioll if is identically zero for the nonrelativistic Schrodinger (l1uulioll Iinwever whenever the lagrangian is real 111 and 112 are inde-11111 h1ll I or (llIdl other and if = 11 In this case 1111 + 1122 = + 11 unci 1111 IlItlnil1onian is unchanged

1111 ((IfT()HpOnding quantum case is obtained from the commutation (fdlii) with 8 = 1 2 If 111 and 1l2 are independent then all

mlt of vHlillhles except the following commute

lifi(rI)IT(rl)] [ifit(rt)1ft(rt)] = ihQ3(r r) (5428)

51111QUANTIZATION OF THE NONRELATIVISTIC SCHRt)DINGER EQUATION

It tirl-Il mmrnple of the application of the field-quantization technique dllvplopOll ill Lhe preceding section we consider here the quantization of IIw lIolIlolntiviHLie Rchrodinger equation (616) The application implies Imt Wil 1110 LroaiitlfJ Eq (616) as though it were a classical equation that dOH(llihlll-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 Schlil-dinger equation (WI) 01 (middot107) For this reason field quantization is often called second (tluwlilllion this term implies that the transition from classical quantization

CLASSICAL LAGRANGIAN AND HAMILTONIAN EQUATIONS

The lagrangian dmlHiLy may he taken to be t2

L hljl Vj vljI V(rt)ljIljI (551)2m

As shown ai til( plld of the preceding section ljI and ljI can be varied separatdy 10 ublaill the lagrangian equations of motion The equationof tho f(llll ([11) Umt results from variation of ljI is

211ill 2m i2lj1 + V(rt)ljI

whiel iH Ul( wmplex conjugate of Eq (616) Variation of ljI givesEq (Iimiddot middot

2I i2lj1 + V(rt)ljI (552)lm

1111 IIlOlllontum canonically conjugate to ljI is ilL r thljl (553)ltI

How(w(1 dOIH not appear in the lagrangian density so that i ie identilally 110 It therefore impossible to satisfy the of tho conuHutllliOIl rdatjoflS (5428) (or the corresponding classical IO[-SOII-

bradwL nildioll) so that ljI i caIUlOt be regarded as a pail of conj lIllIto They can easily be eliminated from the hamilLonian sincer Illvtr and Eq (553) gives ljI in terms of r1

Tho Imllliitollinll density is

itt iH L -VrVljI - VlIlt2m II

I Tlw ()tWIIlHi IJlltl 11 identified with gt is related to the appearance of only th firH nlmiddot 1111 d vlltive in the wave equation (552) since in this case can be exprAd III 1IIIIiI of gtI nlld s space derivatives through the wave equation If the wave IlIltllo IH of HI order in the time derivative gt and are independent then If ill 11111lt1 I J IIdlllf thau to gt and both I 1lt and gt ii are pairs of canonical variablH 11 lIollrlllvHIe Hchrodinger equation and the Dirac equation are of the fOllIIer VI witor th relativistic Schrodinger equation is of the latter type

498 QUANTUM MECHANICS

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 nonrelativutj(l Hehrodinger equation is a single ifi field that is complex or nonhermhil1ll ___c____c_____

middotc In the e1mlllienJ case we can express ifi in terms of real fields 1 and ifi2 as

e +iifi2) ifi 2-1(ifi1 - iifi2) (5426)

Wo HImI IirHI tim the lagrangian equations of the form (548) obtained hy ilIlaquoiopmllJolII variation of ifi and ifi are equivalent to those obtained vnrialioll 01 ifil ILnd ifi2 It follows from (5426) that

ll - i amp = 2-t + i ilifi ampifi1 ampifi2

)

ampifi1 ampifi2

1111111 LIIl ifi ifi equations are obtained by adding and subtracting the ifi1l ifi

III I-limilaf fashion the classical momenta canonically conjugate to ifi IIlld ifi arc seen to be

11 l- 1(111 - i1l2) if = 2-(111 + illZ) (5427)

1111 Il0(lOlld momentum is written as if rather than 11 in order to emphasize UIL fld 01111 it is defined as being canonically conjugate to ifi and is not 1I1IIIIllllllily Ihe complex conjugate of 11 Indeed as we shall see in the twx Hlllcioll if is identically zero for the nonrelativistic Schrodinger (l1uulioll Iinwever whenever the lagrangian is real 111 and 112 are inde-11111 h1ll I or (llIdl other and if = 11 In this case 1111 + 1122 = + 11 unci 1111 IlItlnil1onian is unchanged

1111 ((IfT()HpOnding quantum case is obtained from the commutation (fdlii) with 8 = 1 2 If 111 and 1l2 are independent then all

mlt of vHlillhles except the following commute

lifi(rI)IT(rl)] [ifit(rt)1ft(rt)] = ihQ3(r r) (5428)

51111QUANTIZATION OF THE NONRELATIVISTIC SCHRt)DINGER EQUATION

It tirl-Il mmrnple of the application of the field-quantization technique dllvplopOll ill Lhe preceding section we consider here the quantization of IIw lIolIlolntiviHLie Rchrodinger equation (616) The application implies Imt Wil 1110 LroaiitlfJ Eq (616) as though it were a classical equation that dOH(llihlll-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 Schlil-dinger equation (WI) 01 (middot107) For this reason field quantization is often called second (tluwlilllion this term implies that the transition from classical quantization

CLASSICAL LAGRANGIAN AND HAMILTONIAN EQUATIONS

The lagrangian dmlHiLy may he taken to be t2

L hljl Vj vljI V(rt)ljIljI (551)2m

As shown ai til( plld of the preceding section ljI and ljI can be varied separatdy 10 ublaill the lagrangian equations of motion The equationof tho f(llll ([11) Umt results from variation of ljI is

211ill 2m i2lj1 + V(rt)ljI

whiel iH Ul( wmplex conjugate of Eq (616) Variation of ljI givesEq (Iimiddot middot

2I i2lj1 + V(rt)ljI (552)lm

1111 IIlOlllontum canonically conjugate to ljI is ilL r thljl (553)ltI

How(w(1 dOIH not appear in the lagrangian density so that i ie identilally 110 It therefore impossible to satisfy the of tho conuHutllliOIl rdatjoflS (5428) (or the corresponding classical IO[-SOII-

bradwL nildioll) so that ljI i caIUlOt be regarded as a pail of conj lIllIto They can easily be eliminated from the hamilLonian sincer Illvtr and Eq (553) gives ljI in terms of r1

Tho Imllliitollinll density is

itt iH L -VrVljI - VlIlt2m II

I Tlw ()tWIIlHi IJlltl 11 identified with gt is related to the appearance of only th firH nlmiddot 1111 d vlltive in the wave equation (552) since in this case can be exprAd III 1IIIIiI of gtI nlld s space derivatives through the wave equation If the wave IlIltllo IH of HI order in the time derivative gt and are independent then If ill 11111lt1 I J IIdlllf thau to gt and both I 1lt and gt ii are pairs of canonical variablH 11 lIollrlllvHIe Hchrodinger equation and the Dirac equation are of the fOllIIer VI witor th relativistic Schrodinger equation is of the latter type

THE QUANTIZATION OF WAVE FIELDS 4

resulting quantized field theory ie equivalent to a many-particle Schlil-dinger equation (WI) 01 (middot107) For this reason field quantization is often called second (tluwlilllion this term implies that the transition from classical quantization

CLASSICAL LAGRANGIAN AND HAMILTONIAN EQUATIONS

The lagrangian dmlHiLy may he taken to be t2

L hljl Vj vljI V(rt)ljIljI (551)2m

As shown ai til( plld of the preceding section ljI and ljI can be varied separatdy 10 ublaill the lagrangian equations of motion The equationof tho f(llll ([11) Umt results from variation of ljI is

211ill 2m i2lj1 + V(rt)ljI

whiel iH Ul( wmplex conjugate of Eq (616) Variation of ljI givesEq (Iimiddot middot

2I i2lj1 + V(rt)ljI (552)lm

1111 IIlOlllontum canonically conjugate to ljI is ilL r thljl (553)ltI

How(w(1 dOIH not appear in the lagrangian density so that i ie identilally 110 It therefore impossible to satisfy the of tho conuHutllliOIl rdatjoflS (5428) (or the corresponding classical IO[-SOII-

bradwL nildioll) so that ljI i caIUlOt be regarded as a pail of conj lIllIto They can easily be eliminated from the hamilLonian sincer Illvtr and Eq (553) gives ljI in terms of r1

Tho Imllliitollinll density is

itt iH L -VrVljI - VlIlt2m II

I Tlw ()tWIIlHi IJlltl 11 identified with gt is related to the appearance of only th firH nlmiddot 1111 d vlltive in the wave equation (552) since in this case can be exprAd III 1IIIIiI of gtI nlld s space derivatives through the wave equation If the wave IlIltllo IH of HI order in the time derivative gt and are independent then If ill 11111lt1 I J IIdlllf thau to gt and both I 1lt and gt ii are pairs of canonical variablH 11 lIollrlllvHIe Hchrodinger equation and the Dirac equation are of the fOllIIer VI witor th relativistic Schrodinger equation is of the latter type

I SOD QUANTUM MECHANICS

The hamiltonian cqUlttjOllS of motion obtained from (5419) with the of (5418) arc

i poundhif - Vljt + 12ljtft 2m

i Viii 12 Il fI 11 2m 11

Tlw or llinHC (jquations is the same as (552) and the second equation tollolhmmiddot wilh (55a) is the complex conjugate of (552) We have thus AhoWIl 11O1l1 the point of view of classical field theory that the lagrangian

(11)1) and the canonical variables and hamiltonian derived from il Itl(l ill agreement with the wave equation (616) or (SS2)

QUANTUM EQUATIONS

as the hamiltonian (5423) as the equation of motion and linolt of (5428) 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 (5423) or (2410) 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 + Vljttp) dmiddotr (555)

and (2216) then shows that H is hermitian _ UltHlolU hamiltonian given in (555) is the operator that represents

the total energy of the field it is not to be confused with the operator (232) which is the energy operator for a single particle that is described by the wave equation (616) We have as given no explicit repre-sentation 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 lIi ve irtter in this section

ThH commutation relations are

=0 r)

1amp

I j

THE QUANTIZATIQN OF WAVE FIELDS lot

The omission of t from [IH tLll(llIIlOli t or the field varhthlJf implies hoth fields in a com III II [tttOI 1I1IWI(llI 11101 to Lhe same time In accord-ance with the earlinr (liJolflltllioll Ilio oqullioIl of motion for f is

rfH)

= [f J (1f Vf dil I If JVftf (557) where primes indiltmjo IIml 111 ill vnrinble r has been substituted for r The second tmlII 011 LlH evaluated with the help of (556) to gi vo

fV(fftf flVf) Itnl IT(IPf dar

JVf - r) (Pr (558)

f eommutes with V hill iii II IIIlllwImiddotjmtl function Evaluation of the first term Oil iho right JoIid bullbull of (MI7) iH Hilllplified by performing a partial integration on f(1fl bull Vf dil 10 ohLni II f f t V2f dJr the surface terms vanish bccaui-)() f Ilitlw vlIlIilllllH Lt infinity or obeys periodic houndary conditions W (l UtilI oill-Itill

[fJVft bull (1f (til11 - fNV2f = - tinl JcV2f) )3(r - r) d 3r -V2f (559)

Substitutioll or (oIioH) ILIlIi (MIIJ) illl) (557) yields Eq so that the eqmdjullK ublldunli 110111 (INNionl nnd quantum field theories agree A similur ealOlllntioll JoIhOWH 1llId UII oquatiOIl ihJt = [ftH) yields the hermitian adjoin of II)q it mm also be seen directly that this equation is tho hormililtll Itlijoillt of the equation [fH) so long as H is hermitian

If V is inUep()lIdollt of t I ImH 110 explicit dependence on the and Eq (542J) HhoWH 111101 1 ill 11 of the motion Thus the energy in the field iii UIIIlHIUIlL AlIoLller interesting operator is

N = Nt fd 8r

(55

The commutator of N with iIll V pnrl of II can be written as

JfV(ftfftf - ftfff) dlnPr

bull

502 QUANTUM MECHANICS

With the help of (55Jj) the parenthesis in the integrand is

1t - tt = t[t + (P(r - r)] _ tt tt + toJ(r - r) _ tt

+ to3(r r) _ tt

=0

since tho Il ililldion vanishes unless r r A similar but slightly more calculation shows that

11 vt V] [tv (vt)] VlJ(r - r)

TJUI dOllhle integral of this over rand r is zero Thus Eq (5510) shows Lhal N iH 1 constant of the motion

1( ew also be shown that the commutator brackets in (556) are COIIHItIltH of the motion so that these equations are always valid if they ILIO 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 me1nS of an expansion like (541) in terms of some complete orthonormal set of functions Uk(r) which we assume for definiteness to be discrete We put

(rt) = 2 ak(t)uk(r) t(rt) 2 akt(t)u(r) (5511) k k

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 I ak(t) Ju (r)(rt) d3r akt(t) JUk(r)t(rt) dar

Thus if we multiply the last of the commutation relations (556) 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) d3rd3r = Ilkl (5512)

of the orthonormality of the Uk In similar fashion it is apparent Ihnl ltk and al commute and that akt and alt commute for all k and 1 HuhHtiLution of (5511) into the expression for N shows that t

N=2 where Nk (5513) k

11 iM nu4ily thnt each Nk commutes vrith aJl the others so that they call ho diltlollnlizmi Himultaneollsly

101 THE QUANTIZATION OF WAVE FIELDS

CREATION DESTRlICTION AND NUMBER OPERATORS

The commutatioll relajiollH for the operators ak and akT woro solved in Sec 21) ill oOlllllidion the harmonic oscillator There it was found that tlw 8olution of (2510) in the representation in which ata is diagonal eOlHiHjs of the matrices (2512) It follows that the states of the qUltflLized field in the representation in which each Nk is diagonal are the kett

(5514)inln2 nk )

where each nk is an eigenvalue of Nk and must be a positive integer or zero We also have the relations

) = nkln l nk - 1 )aklnl nk ) (n + 1)n nk + 1 ) (5515)

atlnt nk

Thus akt 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 (5510) that N = zlh is a constant The rate of change of Nk is given hy

ihNk [aktaII] where H is obtained from (555) and (5511)

H aal J Vui bull VUl + VUUI) dar

(5516) = aal JU ( - 72 + v) 111 dar

It is not difficult to show from (fiIU2) that a particular Nk is constant if and only if all the volume intogml in (5516) arc zero for which either j or l is equal to k These intogmlH are just the matrix elements of the one-particle hamiltonian (232) 140 Ihtt the necessary and sufficient condi-tion 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 (232) with eigen-values Ek is of particular illterliiL The integrals in (5516) are then E10 jh and the field hamiltonian IW(OU1CH (5517)

H LaktakEk LivkEk k k

This particular N representation ill t10 one in which H is also diagonal 1 ThiH for the quantized field is dORly related to the corresponding result containml in Eq (355) for the one-partido prol)llbility amplitude

504 QUANTUM MECHANICS

the kef In Ii) has jhe eigenvalue JnkEk for the tojal energy OPOIIOI fl I t it- nplllH1I t that all the are constant ill thi case

CONNECTION WITH THE SCHROOINGER EQUATION

Thn lt111111 Li 01 limiddotld UHorY is closely related to the many-particle Sehrcid-iltfJ4tmiddot cIlllitLioll in Sec 40 If the Uk are eigenfunctions of the Olt-IIIImiddotLled IlHluillolliall (232) the field theory shows that Holilliolll1 for which the number of particles n in the kth state is 11 (iOtlHIalll Imii Li VI i lIi-eger or zero and the energy is JnkEk bull Each solution (all 1- hH(Iilpd by ket nk these kets form a complete OllhollOIlIIII1 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 Iq (-iO1) can be written as a product of olle-particle wave fUlidiollH if there is no interaction hetween the 1114 linolLI combination of such products that is symmetric with

of any of particle 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 non-interacting 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 Fermi-Dirac statistics As discused 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 state pro-vided that each of these numbers is either 0 or 1 The desired modifica-bon of the must limit the eigenvalues of each nnprfltor

Nk to 0 and 1

ANTICOMMUTATION RELATIONS

A review of the foregoing theory shows that the conclmion that the values of each Nk arc the positive arid zero stems from the com-111lllation relations (5512) for the ak and akt Equations (5512) in turn I H( W Heisenberg The Physical Principles of the Quantum Theory App see 11 (University of Chicago Press Chicago 1930)

101 THE QUANTIZATION OF WAVE FIELDS

CREATION DESTRlICTION AND NUMBER OPERATORS

The commutatioll relajiollH for the operators ak and akT woro solved in Sec 21) ill oOlllllidion the harmonic oscillator There it was found that tlw 8olution of (2510) in the representation in which ata is diagonal eOlHiHjs of the matrices (2512) It follows that the states of the qUltflLized field in the representation in which each Nk is diagonal are the kett

(5514)inln2 nk )

where each nk is an eigenvalue of Nk and must be a positive integer or zero We also have the relations

) = nkln l nk - 1 )aklnl nk ) (n + 1)n nk + 1 ) (5515)

atlnt nk

Thus akt 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 (5510) that N = zlh is a constant The rate of change of Nk is given hy

ihNk [aktaII] where H is obtained from (555) and (5511)

H aal J Vui bull VUl + VUUI) dar

(5516) = aal JU ( - 72 + v) 111 dar

It is not difficult to show from (fiIU2) that a particular Nk is constant if and only if all the volume intogml in (5516) arc zero for which either j or l is equal to k These intogmlH are just the matrix elements of the one-particle hamiltonian (232) 140 Ihtt the necessary and sufficient condi-tion 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 (232) with eigen-values Ek is of particular illterliiL The integrals in (5516) are then E10 jh and the field hamiltonian IW(OU1CH (5517)

H LaktakEk LivkEk k k

This particular N representation ill t10 one in which H is also diagonal 1 ThiH for the quantized field is dORly related to the corresponding result containml in Eq (355) for the one-partido prol)llbility amplitude

504 QUANTUM MECHANICS

the kef In Ii) has jhe eigenvalue JnkEk for the tojal energy OPOIIOI fl I t it- nplllH1I t that all the are constant ill thi case

CONNECTION WITH THE SCHROOINGER EQUATION

Thn lt111111 Li 01 limiddotld UHorY is closely related to the many-particle Sehrcid-iltfJ4tmiddot cIlllitLioll in Sec 40 If the Uk are eigenfunctions of the Olt-IIIImiddotLled IlHluillolliall (232) the field theory shows that Holilliolll1 for which the number of particles n in the kth state is 11 (iOtlHIalll Imii Li VI i lIi-eger or zero and the energy is JnkEk bull Each solution (all 1- hH(Iilpd by ket nk these kets form a complete OllhollOIlIIII1 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 Iq (-iO1) can be written as a product of olle-particle wave fUlidiollH if there is no interaction hetween the 1114 linolLI combination of such products that is symmetric with

of any of particle 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 non-interacting 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 Fermi-Dirac statistics As discused 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 state pro-vided that each of these numbers is either 0 or 1 The desired modifica-bon of the must limit the eigenvalues of each nnprfltor

Nk to 0 and 1

ANTICOMMUTATION RELATIONS

A review of the foregoing theory shows that the conclmion that the values of each Nk arc the positive arid zero stems from the com-111lllation relations (5512) for the ak and akt Equations (5512) in turn I H( W Heisenberg The Physical Principles of the Quantum Theory App see 11 (University of Chicago Press Chicago 1930)

504 QUANTUM MECHANICS

the kef In Ii) has jhe eigenvalue JnkEk for the tojal energy OPOIIOI fl I t it- nplllH1I t that all the are constant ill thi case

CONNECTION WITH THE SCHROOINGER EQUATION

Thn lt111111 Li 01 limiddotld UHorY is closely related to the many-particle Sehrcid-iltfJ4tmiddot cIlllitLioll in Sec 40 If the Uk are eigenfunctions of the Olt-IIIImiddotLled IlHluillolliall (232) the field theory shows that Holilliolll1 for which the number of particles n in the kth state is 11 (iOtlHIalll Imii Li VI i lIi-eger or zero and the energy is JnkEk bull Each solution (all 1- hH(Iilpd by ket nk these kets form a complete OllhollOIlIIII1 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 Iq (-iO1) can be written as a product of olle-particle wave fUlidiollH if there is no interaction hetween the 1114 linolLI combination of such products that is symmetric with

of any of particle 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 non-interacting 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 Fermi-Dirac statistics As discused 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 state pro-vided that each of these numbers is either 0 or 1 The desired modifica-bon of the must limit the eigenvalues of each nnprfltor

Nk to 0 and 1

ANTICOMMUTATION RELATIONS

A review of the foregoing theory shows that the conclmion that the values of each Nk arc the positive arid zero stems from the com-111lllation relations (5512) for the ak and akt Equations (5512) 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 (556) forp and pt Thus we must modify Eqs (556) 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 forp is the wave equation (552) when the hamiltonian has the form (555)

It was found by Jordan and Wigner1 that the desired modification consists in the replacement of the commutator brackets

[AB] == AB - BA

in Eqs (5422) and (556) by anticommutator brackets

[ABJ+ == AB+ BA

This means that Eqs (556) are replaced by

[p(r)Hr)]+ p(r)p(r) + p(r)p(r) = 0 [pt(r)pt(r)l+ = pt(r)pt(r) + pt(r)pt(r) = 0 (5518) [p(r)pt(r)l+ = p(r)pt(r) + pt(r)p(r) = a3(r - r)

It then follows directly from Eqs (5511) and (5518) that

[akad+ = aka + aa = 0 [aktatl+ = altat + atakt 0 (5519) [aka]+ = akazt + aak thl

bull We define Nk = ata as before and notice first that each Nk com-

mutes with all the others so that they can be diagonalized simultane-ously The eigenvalues of Nk can be obtained from the matrix equation

Nk2 aktakaktak = akt(1 - aktak)ak = aktak = Nk (5520)

where use has been made of Eqs (5519) If Nk is in diagonal form and has the eigenvalues nr it is apparent thM Nk2 is also in diag-onal form and has the eigenvalUes bull bullbull Thus the matrix equation (5520) is equivalent to the algebraic equations

2 I 2 IInk nk nk nk

for the eigenvalues These are quadratic equations that have two roots oand 1 Thus the eigenvalues of each Nk are 0 and 1 and the particles obey the exclusion principle The eigenvalues of N = TNk are the positive integers and zero as before The earlier expressions (5516) and (5517) for the hamiltonian are unchanged and the energy eigenvalues are TnkEk bull

1 P Jordan and E WignerZ Physik 47 631 (1928)

508

Wmiddot II I

QUANTUM MECHANICS

WO lilJ(1 LlI( effects of operating with ak and ak on a ket II ) (Jmt has the eigenvalue 11k (= 0 or 1) for the operator

Nt Ilip d(yir(middot f(Idiolls would have the form (j525) were it not that It pril of HIIltII (IllIlioIlS (with subscripts added) would not agree with the

10 I 1ql (f)1 W( tlilllr l proeeed in the following way Ve order the states k

but definite way 12 k Then has the form (5525) except that l1 multiply-

iiiI hHI or IIlilltis sign is introduced according as the kth state is preceded ill IImiddot 1111111111 order by an even or an odd number of occupied states

11111 Iplae the Einstein-Bose equations (5515) bv the exclusion-W

iI III nk ) 1 nk )

(11111 nk ) = Ok(l 1 nk ) (5526) k-l

Ih (1) Vk L nj i 1

As lm example we calculate the effect of operating with akal and with Iflf 011 lome ket where we assume for definiteness that the order is such Chat l gt k If each operation is not to give a zero result both nk and nl ill the original ket must equal unity Operation with akal empties fir8t the tth and then the kth state and introduces a fltctor OIOk Operation with alak empties the kth state first so that Ok is unchanged But when the lth state 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

akad nk bullbull nl bullbull) = -alakl nk bull rll bull)

in agreement with the first of Eqs (5519) In similar fashion it can be shown that Eqs (5526) agree with the result of operating with the other two of Eqs (5519) OD any ket SiDce the aggregate of kets represents all possible states of the many-particle system they constitute a eomplete sd and Eqs ([)fgt19) follow as operator equations from Eqs (5526)

561 IELECTROMAGNETIC FIELD IN VACUUM the methods developed in Sec 54 to the quantization of

field in vacuum Since we are not coneerned with II 1 11middotmiddot diNgtlion of the material in this section aild the next see the references

IImiddotd 100101 I Iag 4)] and also E Fermi Rev ilIod Phys 4 137 (1932) L 1I0HmiddotImiddotld 11111 118L 1llIli Poincare 125 (1981) W Reitter The Quantum Theory oj ildiulioll d d (Ox[Ol(t New York 1954)

508

Wmiddot II I

QUANTUM MECHANICS

WO lilJ(1 LlI( effects of operating with ak and ak on a ket II ) (Jmt has the eigenvalue 11k (= 0 or 1) for the operator

Nt Ilip d(yir(middot f(Idiolls would have the form (j525) were it not that It pril of HIIltII (IllIlioIlS (with subscripts added) would not agree with the

10 I 1ql (f)1 W( tlilllr l proeeed in the following way Ve order the states k

but definite way 12 k Then has the form (5525) except that l1 multiply-

iiiI hHI or IIlilltis sign is introduced according as the kth state is preceded ill IImiddot 1111111111 order by an even or an odd number of occupied states

11111 Iplae the Einstein-Bose equations (5515) bv the exclusion-W

iI III nk ) 1 nk )

(11111 nk ) = Ok(l 1 nk ) (5526) k-l

Ih (1) Vk L nj i 1

As lm example we calculate the effect of operating with akal and with Iflf 011 lome ket where we assume for definiteness that the order is such Chat l gt k If each operation is not to give a zero result both nk and nl ill the original ket must equal unity Operation with akal empties fir8t the tth and then the kth state and introduces a fltctor OIOk Operation with alak empties the kth state first so that Ok is unchanged But when the lth state 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

akad nk bullbull nl bullbull) = -alakl nk bull rll bull)

in agreement with the first of Eqs (5519) In similar fashion it can be shown that Eqs (5526) agree with the result of operating with the other two of Eqs (5519) OD any ket SiDce the aggregate of kets represents all possible states of the many-particle system they constitute a eomplete sd and Eqs ([)fgt19) follow as operator equations from Eqs (5526)

561 IELECTROMAGNETIC FIELD IN VACUUM the methods developed in Sec 54 to the quantization of

field in vacuum Since we are not coneerned with II 1 11middotmiddot diNgtlion of the material in this section aild the next see the references

IImiddotd 100101 I Iag 4)] and also E Fermi Rev ilIod Phys 4 137 (1932) L 1I0HmiddotImiddotld 11111 118L 1llIli Poincare 125 (1981) W Reitter The Quantum Theory oj ildiulioll d d (Ox[Ol(t New York 1954)