Quantize electromagnetic field

•Classical free field equations•Quantize•Coupling to charged particles•One-body operator acting on nucleons and photons

Quantization of the EM field

Maxwell:

! · E(x, t) = 4!"(x, t)! · B(x, t) = 0

!"E(x, t) = #1c

#

#tB(x, t)

!"B(x, t) =1c

#

#tE(x, t) +

4!

cj(x, t)

Scalar and Vector potential

Quantum applications require replacingE and B!

in terms of vector and scalar potentials.Homogeneous equations are automatically solved.

E = !"!! 1c

!A!t

B = "#A

Gauge freedom

Remaining equations

To decouple could choose (gauge freedom)

!2! +1c

!

!t(! · A) = "4"#

!2A" 1c2

!2A!t2

"!!! · A +

1c

!!!t

"= "4"

cj

! · A +1c

!!!t

= 0

! · A = 0

Decoupled equations

Resulting in

Radiation gauge useful for quantizing free field:

!2!" 1c2

!2!!t2

= "4"#

!2A" 1c2

!2A!t2

= "4"j

Instantaneous Coulomb

Yields

No sources ⇒ free field

and ⇒ solve

!(x, t) =!

Vd3x! !(x!, t)

|x! x!|

E = !1c

!A!t

B = "#A

!2A" 1c2

!2A!t2

= 0

Free field solutions

Periodic BC ⇒

Gauge ⇒

Future reference:

From Wave Eq.

A(x, t) =1!V

!

k

Ak(t) eik·x

k · Ak = 0

Ak =!

!=1,2

ek!Ak!

!2Ak(t)!t2

+ c2k2Ak(t) = 0

Harmonic solutions

Fourier coefficients ⇒

and

A real:

Fields:

!k = ck

Ak(t) = e!i!kt Ak

A(x, t) =1

2!

V

!

k

"Ak(t) + A!

"k(t)#eik·x

E(x, t) =i

2c!

V

!

k

!k

"Ak(t)"A!

"k(t)#eik·x

B(x, t) =i

2!

V

!

k

k#"Ak(t) + A!

"k(t)#eik·x

Energy in field

Generalfields are realIn terms of Fourier coefficients!

Vd3x E · E! =

14c2

"

k

!2k

##Ak(t)!A!"k(t)

##2

!

Vd3x B · B! =

14

"

k

k2##Ak(t) + A!

"k(t)##2

Hem =18!

!

Vd3x (E · E! + B · B!)

Expand Fourier coefficients along polarization vectors

Note: no time dependence!In order to quantize, introduce real canonical variables

Hem =18!

!

k!

k2 |Ak!|2

Qk(t) =i

2c!

4![Ak(t)"A!

k(t)]

Pk(t) =k

2!

4![Ak(t) + A!

k(t)]

Ak(t) = !ic"

!

!Qk(t) +

i

"kPk(t)

"Or

Hamiltonian

No time dependence ⇒ harmonic oscillators

Hem =12

!

k!

"P 2

k! + !2kQ2

k!

#

PhotonsStandard quantization procedure

Usual operators

with

[Pk!, Pk!!! ] = 0[Qk!, Qk!!! ] = 0[Qk!, Pk!!! ] = i!!k,k!!!,!!

ak! =1!

2!!k(!kQk! + iPk!)

a†k! =1!

2!!k(!kQk! " iPk!)

[ak!, ak!!! ] = 0!a†k!, a†k!!!

"= 0

!ak!, a†k!!!

"= !k,k!!!,!!

Each mode HONumber operatorHamiltonian

Momentum from Poynting vector

Single photon ⇒ m=0!

Nk! = a†k!ak!

Hem =!

k!

!!k

"Nk! + 1/2

#!

!

k!

!!kNk!

Pem =1

8!c

!

Vd3x (E!B"B!E)

="

k!

!k#Nk! + 1/2

$#

"

k!

!kNk!

Hema†k! |0! = !!k a†k! |0!

Pema†k! |0! = !k a†k! |0!

Vector potential operator

Acts on photon states: adds or removes one

Acts on charged particle at x and t(first quantization)

A(x, t) =!

hc2

!kV

"1/2 #

k!

$ak!ek!ei(k·x!"kt) + a†

k!ek!e!i(k·x!"kt)%

Coupling to charged particles

Lorentz

RewriteNoteandSo that with

F = e

!E +

1cv !B

"

F = e

!!"!! 1

c

!A!t

+1c

(v #"#A)"

v !"!A = " (v · A)# (v ·")A!A!t

+ (v ·!)A =dAdt

F = !"U +d

dt

!U

!v U = e!! e

cA · v

Check

Yields Lorentz from

Generalized momentumSolve for v and substitute

Hamiltonian for a charged particle

L = T ! U =12mv2 ! e! +

e

cA · v

d

dt

!L

!v! !L

!x= 0

p =!L

!v= mv +

e

cA

H = p · v ! L =!p! e

cA"2

2m+ e!

Nuclear Hamiltonian plus EM field plus coupling!

Coupling

Add by hand

H =!

i

"pi ! qi

c A(xi, t)#2

2m+

A!

i<j=1

V (i, j) + Hem

Hint =A!

i

"! qi

2mc(pi · A(xi, t) + A(i, t) · pi) +

q2i

2mc2A(xi, t) · A(xi, t)

#

Hspinint = !

A!

i

µisi · ["#A(x, t)]x=xi

Second quantizedUsing transversality

and

pi · A(xi, t) = A(xi, t) · pi

HIint = ! 1

m

!

!"

!

k#

"h

!kV

#1/2

ek# ·$""| e(1

2+ t3)ei(k·x!$kt) p |## a†

!a"ak#

+ ""| e(12

+ t3)e!i(k·x!$kt) p |## a†!a"a†

k#

%

Hspinint = ! e!

2m

!

!"

!

k#

"h

!kV

#1/2

(ik" ek#)

·$#"| gse

i(k·x!$kt) s |#$ a†!a"ak# + #"| gse

!i(k·x!$kt) s |#$ a†!a"a†

k#

%

gs =12(gp + gn) + t3(gp ! gn)

ek!eik·x · p =!

"m!

4!i"j"(kr)Y !"m!

(k)Y"m!(r)ek! · p

Y !!m!

(k) =!

2! + 14"

#m!,0

NucPhys

Some more details

Part of vector potential

Note operators

Take photon momentum along z-axis

Corresponding two polarization vectors can be chosen according to

can show px + py = !(e(1)1 p(1)

!1 + e(1)!1p

(1)1 )

ek! ! ek± = " 1#2(x± iy) $ e(1)

±1

NucPhys

formulas• It follows that

• Since projection is |1| there are no angular momentum 0 photons!

• Vector spherical harmonics are defined by

• and include

• Complete set of vector functions on a sphere

e(1)! Y"0(r) =

!

#

(! 0 1 " | # ")!#,"!(r)

e(1)0 = ez

!!,"µ(r) =!

!#

Y!k(r)e(1)# (! k 1 " | # µ)

NucPhys

Different components• Several possibibilities

• Parity good quantum number for nuclear states so photons have specific parity (instead of helicity)

• Photon is a vector particle so has intrinsic parity -1

• Two types of photons: electric

• magnetic

E!! ! =( "1)! " = !± 1

M!! ! =( "1)!+1 " = !

!!="!1,"µ = [!(2! + 1)]!1/2(r! + !r)Y"µ(r)

!!=","µ = !i[!(2! + 1)]!1/2(r "!)Y"µ(r)

!!="+1,"µ = [(! + 1)(2! + 1)]!1/2(r!! (! + 1)r)Y"µ(r)

NucPhys

Multipole moments• Conventional to write Hamiltonian using

• With the result

• Time dependence absorbed by applying Fermi Golden rule such that photon provides or removes the right amount of energy connecting two nuclear states --> transition rate

!(r) =!

i

e(12

+ t3(i))"(r ! ri)

j(r) =!

i

e(12

+ t3(i))"

12(vi"(r ! ri) + "(r ! ri)vi)

#

+e!2M

!

i

gs(i)"# si"(r ! ri)

H !int = !

!d3r j(r) · A(r)

wi![f ] =2!

! | !f | H "int |i" |2"

NucPhys

Multipole moments• For electric photons

• noting that

• Magnetic photons

M(M!, µ) = ! (2! + 1)!!ck!(! + 1)

!d3r j(r) · (r "!)j!(kr)Y!µ(r)

!! (r !!)j!(kr)Y!µ(r) = kj!+1(kr)!!

! + 12! + 1

"1/2

!"=!+1,!µ

" kj!!1(kr)(! + 1)!

!

2! + 1

"1/2

!"=!!1,!µ

M(E!, µ) =!i(2! + 1)!!ck!+1(! + 1)

!d3r j(r) ·!" (r "!)j!(kr)Y!µ(r)

NucPhys

Long wavelengths• For nuclear processes with photons

• simplifies spherical Bessel functions

• Also from continuity equation

• allows to replace current density by charge density where useful

kR = 6.1! 10!3A1/3E! [MeV ]

! kr " 1

j!(kr) =(kr)!

(2! + 1)!!

!1! 1

2(kr)2

2! + 3...

"

! · j +!

!t" = ! · j ! ikc" = 0

NucPhys

Final operators• Electric transitions

• Magnetic transitions

• From golden rule and photon density of states (see book)

• Total decay rate in terms of reduced transition probabilities

M(E!, µ) =!

d3r "(r)r!Y!µ(r)

M(M!, µ) = ! 1c(! + 1)

!d3r j(r) · (r "!)r!Y!µ(r)

T (E!; I1 ! I2) =8"(! + 1)

![(2! + 1)!!]21!k2!+1B(E!; I1 ! I2)

T (M!; I1 ! I2) =8"(! + 1)

![(2! + 1)!!]21!k2!+1B(M!; I1 ! I2)

NucPhys

Reduced transition probabilities• Electric

• Magnetic

• Total decay rates

• Units

B(E!; I1 ! I2) =!

µM2

|"I2M2|M(E!;µ) |I1M1#|2

B(M!; I1 ! I2) =!

µM2

|"I2M2|M(M!;µ) |I1M1#|2

T (E1) = 1.59! 1015E3B(E1)T (E2) = 1.22! 109E5B(E2)T (E3) = 5.67! 102E7B(E3)T (M1) = 1.76! 1013E3B(M1)T (M2) = 1.35! 107E5B(M2)T (M3) = 6.28! 100E7B(M3)

B(E!) ! e2(fm)2!

B(M!) !!

e!2Mc

"2

(fm)2!!2

NucPhys

Single-particle operators• Electric• Magnetic (angular momentum operators dimensionless)

• Single-particle matrix elements

• last transition has selection rule

• Weisskopf units: assume

• constant normalized wave functions

M(E!, µ) =!

i

e ( 12 + t3(i)) r!

i Y!µ("i, #i)

M(M!, µ) =!

i

e!2Mc

"gs(i)si +

2g!(i)! + 1

!i

#·!i

$r"Y"µ("i, #i)

%

B(E!; j1 ! j2) =e2

4"(2! + 1) (j1 1

2 ! 0 |j2 12 )

2 "j2| r! |j1#2

B(M!; j1 ! j2 = ! + j1) =!

e!2Mc

"2 !gs "

2! + 1

g!

"2

!2 2! + 14"

(j1 12 ! 0 |j2 1

2 )2 #j2| r"!1 |j1$2

!2 = !1 + "! 1

j1 = ! + 12

j2 = 12

!j2| r! |j1" # 3! + 3

R!

NucPhys

W.U.• Electric

• Magnetic

• Actual transitions: much larger --> collective

BW (E!) =(1.2)2!

4"

!3

! + 3

"3

A2!3 e2(fm)2!

BW (M!) =10"

(1.2)2!!2

!3

! + 3

"2

A2!!2

3

!e!

2Mc

"2

(fm)2!!2