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\rfc_aSf^ f Tfc[b `fcr\^a`h XSZX \ [b\ Yfb\ S^a\b\`a\V S^hX[` af s\ `a_VS\VU
X\ b\`Tf^`\` f aX\ YSZbf`ZfTSZ T[baSZc\`
3
Yfc\Z_c\`h[afY`h\c\Zabf^`4 f aX\ `fcr\^a
b\ _Sb\V af b\[ZX [ Z\ba[S^ \ _ScSsbS_Y r[c_\ f aX\ Tfc[bS[aSf^ X[r\ `T\ZS Z ZX[b[Za\bS`aSZ
aSY\`
3
W4UX\^ aX\ `fc_a\ ZX[bg\ VS`abSs_aSf^ r[bS\` [TTb\ZS[scp SaXS^ [ T\bSfV f aX\
`[Y\ fbV\b [` aX\`\ W`haX\ b\`Tf^`\ f aX\ YSZbf`ZfTSZ T[baSZc\` Scc ^fa s\ `_ZS\^acp
b[TSV af s_ScV _T [ ^\ \ _ScSsbS_Y Tfc[bS[aSf^h[^V aX\ [Za_[c r[c_\ f aX\ Tfc[bS[aSf^
Scc c[g s\XS^V aX\ ZX[^gS^g ZX[bg\ VS`abSs_aSf^U X\ [Za_[c Tfc[bS[aSf^
P (t)
S` V\a\bo YS^\V sp aX\ [\cc \cV S^ aX\ Y\VS_Y ^fa f^cp [a aSY\t
s_a [c`f [a [ Tb\rSf_` aSY\t
0
h S^ [ fbY \Tb\``\V sp aX\ S^a\gb[c~
P (t) =
Z
t
−∞
Vt
0
G(t − t
0
) ~
E(t
0
)
3 mU i4 X\b\ aX\ \b^\cG(t − t
0
)
S` aX\ `fcr\^a b\`Tf^`\ _^ZaSf^ [^V~
E(t
0
)
aX\ [\cc \cVU b[^` fbYS^g aX\ Zf^rfc_aSf^ S^a\gb[c \ U mU i44 af aX\ f_bS\b b\ _\^Zp `T[Z\ \fsa[S^
~
P (ω) =
ˆ(ω) − 1
4π
E(ω)
~
3 mU 4 X\b\ˆ(ω)
S` aX\ ZfYTc\ VS\c\ZabSZ T\bYSaaSrSapˆ(ω) =
0
(ω) − i
00
(ω)
3 mU j4 X\ b\[c T[ba0
(ω)
S` aX\ b\ _\^ZpoV\T\^V\^a VS\c\ZabSZ Zf^`a[^ah V\`ZbSsS^g aX\ ZfYo Tf^\^a f aX\ Tfc[bS[aSf^ V\^`Sap S^ TX[`\ SaX aX\ f`ZScc[aS^g \cV XSc\ aX\ ZfYTc\T[bah fb cf`` [Zafbh
00
(ω)
h V\a\bYS^\` aX\ ZfYTf^\^a f aX\ Tfc[bS[aSf^ SaX [ TX[`\ VS\b\^Z\ fπ/2
SaX b\`T\Za af aX\ [\cc \cVhgSrS^g bS`\ af aX\ cf`` f \^\bgp f aX\ \c\ZabSZ\cV S^ aX\ Y\VS_YU X\ b\ _\^ZpoV\T\^V\^a VS\c\ZabSZ Zf^`a[^a
0
(ω)
[^V aX\ cf`` [Zafb00
(ω)
Z[^ s\ V\a\bYS^\V sp \T\bSY\^a[c fb aX\fb\aSZ[c Y\aXfV`U fb Yf`a Tfc[b`fcr\^a`h[a cf b\ _\^Zph0
(ω)
S`\ _[caf aX\ `a[aSZ VS\c\ZabSZ Zf^`a[^a h[^V00
(ω)
S` \bfU
X\ \Tc[^[aSf^ f aX\ Zfbb\`Tf^V\^Z\ S^ aX\ s\X[rSfb f [cc Tfc[b ZfYTf_^V` S^ [^
\c\ZabfY[g^\aSZ \cV S` f_^V S^ aX\ [Za aX[a aX\ \c\ZabSZ Tfc[bS[aSf^ S` s_Sca _T f aXb\\
T[ba`haX\ fbS\^a[aSf^[chaX\ [afYSZh[^V aX\ \c\Zabf^SZ Tfc[bS[aSf^h\[ZX T[ba Zfbb\`Tf^Vo
S^g af YfaSf^` f [ VS\b\^a S^V f T[baSZc\
3
Yfc\Z_c\`h [afY`h \c\Zabf^`h b\`T\ZaSr\cp4
SaX VS\b\^a W`U ` aX\ b\ _\^Zp S^Zb\[`\`haX\ `foZ[cc\V^f^\ _ScSsbS_Y \\Za`[TT\[b
`_s`\ _\^acp S^ aX\ VS\b\^a Zf^abSs_aSf^` f aX\ Tfc[bS[aSf^U bS\^a[aSf^[cZf^abSs_aSf^
3
W s\cf
10
−12
`4 Scc b`a `a[ba af c[g s\XS^V aX\ r[bS[aSf^` f aX\ \c\ZabfY[g^\aSZ \cV ^\ah [afY` 3 W [bf_^V10
−14
`4 Scc ^fa s\ [sc\ af fccf aX\ \cVh [^V ^[ccph [a r\bp XSgX b\ _\^ZS\`h \c\Zabf^` 3 W [bf_^V10
−16
`4 [c`f Scc c[g s\XS^V aX\ \c\Zabfo Y[g^\aSZ \cVU bS\^a[aSf^[cTfc[bS[aSf^ [bS`\` bfY VS\b\^a b\c[[aSf^ TbfZ\``\` s_ahS\ [``_Y\ aX[a Sa Z[^ s\ V\`ZbSs\V S^ a\bY` f f^cp f^\ f aX\`\ TbfZ\``\`h ZX[b[Za\bS\V
sp [ b\c[[aSf^ aSY\
τ
haX\^ aX\ b\ _\^Zp V\T\^V\^Z\ f aX\ VS\c\ZabSZ Zf^`a[^a Y[p s\ [V\ _[a\cp V\`ZbSs\V sp aX\ R\sp\ \ _[aSf^0
(ω) =
∞
+
−
∞
1 + ω
2
τ
2
3 mU l400
(ω) =
( −
∞
)ωτ
1 + ω
2
τ
2
3 mU 4 X\b\∞
S` aX\ VS\c\ZabSZ Zf^`a[^a [a [ b\ _\^Zp [a XSZX aX\ fbS\^a[aSf^[c Tfc[bS[aSf^
Z[^ ^f cf^g\b fccf aX\ ZX[^g\` S^ aX\ \cV S^ faX\b fbV`h Sa S` aX\ VS\c\ZabSZ Zf^`a[^a
ZX[b[Za\bS`aSZ f aX\ S^V_Z\V Tfc[bS[aSf^U
cc f aX\ TX\^fY\^[ X\b\ aX\ `fc_a\ _^V\bgf\` [ `_VV\^ ZX[^g\ S^ Sa` ZX[bg\ VS`o
abSs_aSf^ b\ _Sb\ [ `T\ZS Z [^[cp`S` X\^ S^ aX\ Tb\`\^Z\ f [ Tfc[bS[sc\ `fcr\^aU ]^ aXS`
Z[`\h S^ [Zah Sa s\ZfY\` _^V[Y\^a[c af TbfT\bcp [ZZf_^a fb aX\ Vp^[YSZ b\`Tf^`\ f
\ `X[cc fZ_` f^ aX\ r\baSZ[c \c\Zabf^SZ ab[^`SaSf^`hs_a \c\Zabf^ [^V \^\bgp ab[^` \b`h fb
^_Zc\[b rSsb[aSf^`h[b\ [c`f \[YTc\` f aX\ `[Y\ Zc[``U
S`afbSZ[ccph S^ Zf^aS^__Y YfV\c` c[bg\ _`\ X[` s\\^ Y[V\ f aX\ [TTbfSY[aSf^ [Zo
ZfbVS^g afXSZX SaS``_ZS\^aaf V\ZfYTf`\ aX\ b\`Tf^`\ _^ZaSf^ S^afaf a\bY`U SaXS^
aXS` [TTbfSY[aSf^ aX\ Tfc[bS[aSf^ r\Zafb
~
P
V\`ZbSs\V S^ aX\ Tb\rSf_` \ZaSf^ s\ZfY\`~
P ' ~
P
f ast
+ ~
P
slow
3 mU 04 X\b\f ast
S^VSZ[a\` aX\ T[ba f aX\ `fcr\^a b\`Tf^`\ aX[a [c[p` fccf` aX\ Vp^[YSZ` f aX\ TbfZ\`` [^Vslow
b\ \b` af aX\ b\Y[S^S^g S^\baS[c a\bYU _ZX `TcSaaS^g S^ aX\ Y\VS_Y b\`Tf^`\ gSr\` bS`\ af aX\ `f Z[cc\V ^f^\ _ScSsbS_Y b\gSY\U srSf_`cph X[a S` [`a[^V X[a S` `cf V\T\^V` f^ aX\ `T\Z S Z
Vp^[YSZ TX\^fY\^f^ _^V\b `a_VpU ]^ [ r\bp
[`a TbfZ\`` `_ZX [` aX\ r\baSZ[c ab[^`SaSf^ c\[VS^g af [ ZX[^g\ f aX\ `fc_a\ \c\Zabf^SZ
`a[a\ rS[ TXfaf^ [s`fbTaSf^ fb \YS``Sf^h
~
P
f ast
Z[^ s\ b\V_Z\V af aX\ a\bY b\c[a\V af aX\ b\`Tf^`\ f aX\ `fcr\^a \c\Zabf^`h X\b\[`~
P
slow
Zfcc\Za` [cc f aX\ faX\b a\bY` b\c[a\V af aX\ r[bSf_` ^_Zc\[b V\gb\\` f b\\VfY f aX\ `fcr\^aU X\ fT\b[aSr\ T[baSaSf^ f aX\afa[c Tfc[bS[aSf^ Z[^ s\ T\b fbY\V _`S^g af [ca\b^[aSr\h s_a \ _Sr[c\^ah `ZX\Y\`U ^\
`ZX\Y\h XSZX \ `X[cc Z[cc T[baSaSf^ ]h [` TbfTf`\V sp [bZ_` S^ m2jl S^ XS` c[^VY[b
T[T\b f^ aX\ ^f^\ _ScSsbS_Y Tfc[bS[aSf^ S^ [ Zc[``SZ[c Zf^aS^__Y `fcr[aSf^ b[Y\fbU
]^ aXS` `ZX\Y\h \ U 3 mU 04 S` b\bSaa\^ [`
~
P
A
= ~
P
or
E
+ ~
P
el
A
3 mU 24 X\b\ `_T\b`ZbSTa`E
[^VA
b\ \b af aX\ \[bcp 3fb S^SaS[c4 [^V aX\ [Za_[c aSY\ S^ b\c[aSf^
af aX\ ZXbf^fcfgSZ[c fbV\b f aX\ ab[^`SaSf^ TbfZ\``U ]^ aXS` T[baSaSf^h aX\ Tb\rSf_` `cf
[^V [`a S^VSZ\` [b\ b\Tc[Z\V sp aX\ `_s`ZbSTa`
or
[^Vel
b\ \bbS^g af fbS\^a[aSf^[c [^V \c\Zabf^SZ Tfc[bS[aSf^ b\`Tf^`\ f aX\ `fcr\^ahb\`T\ZaSr\cpU ]^ aX\ `\Zf^V `ZX\Y\hXSZX\ `X[cc S^VSZ[a\ [` T[baSaSf^ ]]h \ U 3 mU 04 X[` af s\ b\bSaa\^ [`
~
P
A
= ~
P
in
E
+ ~
P
dyn
A
3 mU jt4 X\b\ aX\ `_s`ZbSTa`in
[^Vdyn
b\ \b ^f af [^ S^\baS[c [^V [ Vp^[YSZ Tfc[bS[aSf^ b\`Tf^`\ f aX\ `fcr\^ahb\`T\ZaSr\cpU X\ _`\ f \ U3
mUjt4SaXS^ aX\ Zf^aS^__Y `fcr[aSf^
Y\aXfV` X[` s\\^ [TTcS\V sfaX af Zc[``SZ[c [^V af [TTbf[ZX\`U
X\ VS\b\^Z\` s\a\\^ aX\ af `ZX\Y\` [b\ b\c[a\V af aX\ [Za aX[ah S^ T[baSaSf^
]h aX\ VSrS`Sf^ S^af `cf [^V [`a Zf^abSs_aSf^` S` Vf^\ S^ a\bY` f TXp`SZ[c V\gb\\` f
b\\VfY
3
^[Y\cph aXf`\ f aX\ `fcr\^a ^_Zc\S [^V aXf`\ f aX\ `fcr\^a \c\Zabf^`4h X\b\[`
S^ T[baSaSf^ ]]h aX\ Zf^Z\Ta f Vp^[YSZ [^V S^\baS[c b\`Tf^`\ S` \TcfSa\VU XS` fbY[c
VS\b\^Z\ S` b\ \Za\V S^ aX\ fT\b[aSr\ \ _[aSf^` V\a\bYS^S^g aX\ af Zf^abSs_aSf^` af
~
P
[`hS^ ]]h aX\ `cf a\bY 3~
P
in
4 S^Zc_V\` ^fa f^cp aX\ Zf^abSs_aSf^` V_\ af aX\ `cf V\gb\\`k[baSaSf^ ] k[baSaSf^ ]]
~
P
A
= ~
P
E
or
+ ~
P
el
A
P
~
A
= ~
P
in
E
+ ~
P
dyn
A
~
P
E
or
=
χ
χ
or
P
~
E
()
P
~
E
in
= ~
P
E
(χ) − ~
P
din
E
(χ
el
)
~
P
A
el
= χ
el
E[ρ
~
A
, ~
P
or
E
, ~
P
el
A
]
P
~
dyn
X
(χ
el
) = χ
el
E[ρ
X
, ~
P
dyn
X
], X = A, E
−∇
2
V = 4πρ
A
−∇
2
V = 4πρ
A
−
∞
∇
2
V = 0
−
∞
∇
2
V = 0
V
in
− V
out
= 0
V
in
− V
out
= 0
∂V
∂n
in
−
∞
∂V
∂n
out
= 4πσ
E
or
∂V
∂n
in
−
∞
∂V
∂n
out
= 0
G
neq
= G
or−el
∞
+
R ρ
A
V
or
E
Vr
3
−
1
2
R ρ
E
V
E
or
Vr
3
−
1
2
R V
or
E
(σ
el
E
− σ
el
A
)
Vr
2
G
neq
= G
in−dyn
∞
+
R ρ
A
V
in
E
Vr
3
−
1
2
R ρ
E
V
in
E
Vr
3
G
or−el
∞
= E
0
A
+
1
2
R ρ
A
V
A
el
Vr
3
G
in−dyn
∞
= E
0
A
+
1
2
R ρ
A
V
A
dyn
Vr
3
[sc\ mU n1 [`SZ / _[aSf^` SaXS^ k[baSaSf^` ] [^V ]]U
χ = ( − 1)/4π
hχ
el
= (
∞
− 1)/4π
[^V
χ
or
= χ − χ
el
U /r\^ S ^fa S^VSZ[a\Vh aX\ Tfa\^aS[c
V
X[` aX\ `[Y\ `f_bZ\` f aX\ Zfbb\`Tf^VS^g \c\ZabSZ \cV~
E
Uf b\\VfY s_a [c`f aX\ T[ba f aX\ [`a ZfYTf^\^a aX[a S` S^ \ _ScSsbS_Y SaX aX\ `cf
Tfc[bS[aSf^h X\b\[`h S^ ]h aX\ c[aa\b ZfYTf^\^a S` Zf^a[S^\V S^ aX\ [`a a\bY
P
el
3 `\\[sc\ mU n X\b\ \ ZfYT[b\ aX\ \c\Zabf`a[aSZ \ _[aSf^` SV\^aS pS^g aX\ af `ZX\Y\`4U
]a S` SYTfba[^a af `ab\`` aX[a aX\ af `ZX\Y\` ] [^V ]] VS\b S^ aX\ fbY f aX\ `cf
[^V [`aZfYTf^\^a`f aX\ Tfc[bS[aSf^hs_aaX\p [b\ Tb[ZaSZ[ccp[^V TXp`SZ[ccp \ _Sr[c\^a
3
S^ aX\ cSYSa f [ cS^\[b b\`Tf^`\ b\gSY\4 S^ aX\ `\^`\ aX[a aX\p gSr\ aX\ `[Y\ r[c_\ f
aX\ afa[c b\[ZaSf^ Tfa\^aS[c [^Vh aX_`h aX\ `[Y\ \\Za` f^ aX\ `fc_a\ [` \cc [` aX\ `[Y\
S^a\b[ZaSf^ \^\bgS\`U
SaXS^ aX\ Zf^aS^__Y `fcr[aSf^ b[Y\fbh [` S^ aX\ Z[`\ f S`fc[a\V Yfc\Z_c\`h Sa S`
Tb[ZaSZ\ af ZfYT_a\ aX\ \ZSa[aSf^ \^\bgS\`
3
fb s\aa\b \ZSa[aSf^ b\\ \^\bgS\`4 SaX af
VS\b\^a [TTbf[ZX\`1 aX\ `a[a\o`T\ZS Z
3
4 Y\aXfV [^V aX\ cS^\[bob\`Tf^`\
3 e4
Y\aXfVU X\ fbY\b X[` [ cf^g ab[VSaSf^ [^V S` b\c[a\V af aX\ Zc[``SZ[c aX\fbp f aX\
`fcr[afZXbfYSZ \\Za`
3
`\\ e\ U m24U X\ c[aa\b [` S^abfV_Z\V c[a\b S^ Zf^^\ZaSf^ SaX
aX\ V\r\cfTY\^a f aX\ cS^\[bb\`Tf^`\ aX\fbp fb Zf^aS^__Y `fcr[aSf^ YfV\c`U
X\
`a[a\o`T\ ZS
Z Y\aXfV `fcr\` aX\ ^f^cS^\[b ZXbVS^g\b \ _[aSf^ fb aX\ `a[a\ f S^a\b\`a
3
gbf_^V [^V \ZSa\V `a[a\4 _`_[ccp SaXS^ W] fb WW V\`ZbSTaSf^`h [^V Sa Tf`a_c[a\`
aX[a aX\ ab[^`SaSf^ \^\bgS\` [b\ VS\b\^Z\` s\a\\^ aX\ Zfbb\`Tf^VS^g r[c_\` f aX\ b\\
\^\bgp _^ZaSf^[ch aX\ s[`SZ \^\bg\aSZ _[^aSap f aX\ Zf^aS^__Y YfV\c`U X\ ^f^o
cS^\[b ZX[b[Za\b f aX\ b\[ZaSf^ Tfa\^aS[c b\ _Sb\` aX\ S^abfV_ZaSf^ S^ aX\ [TTbf[ZX\`
VS\b\^a [^[cp`S` [TTcS\` af aX\ e [TTbf[ZX
3
S^ \SaX\b [YYoR[^Zfh ekh fb
RR r\b`Sf^4 X\b\ aX\ \ZSa[aSf^ \^\bgS\`h V\ ^\V [` `S^g_c[bSaS\` f aX\ b\ _\^Zpo
V\T\^V\^a cS^\[b b\`Tf^`\ _^ZaSf^` f aX\ `fcr[a\V Yfc\Z_c\ S^ aX\ gbf_^V `a[a\h [b\
VSb\Zacp V\a\bYS^\V sp [rfSVS^g \TcSZSa Z[cZ_c[aSf^ f aX\ \ZSa\V `a[a\ [r\ _^ZaSf^U ]^
aXS` Z[`\h aX\ Sa\b[aSr\ `ZX\Y\ f aX\ [TTbf[ZX\` S` ^f cf^g\b ^\Z\``[bph [^V aX\ Xfc\
`T\Zab_Y f \ZSa[aSf^ \^\bgS\` Z[^ s\ fsa[S^\V S^ [ `S^gc\ b_^ [` fb S`fc[a\V `p`a\Y`U
caXf_gX Sa X[` s\\^ V\Yf^`ab[a\V aX[a fb [^ S`fc[a\V Yfc\Z_c\ aX\ [^V e
Y\aXfV` [b\ \ _Sr[c\^a
3
S^ aX\ cSYSaf aX\ \[Za `fc_aSf^ f aX\ Zfbb\`Tf^VS^g \ _[aSf^`4h
[ fbY[c ZfYT[bS`f^ fb Yfc\Z_c\` V\`ZbSs\V sp Zf^aS^__Y YfV\c` `Xf` aX[a aXS`
\ _Sr[c\^Z\ S` ^f cf^g\b r[cSVU p _`S^g [ ^f^\ _ScSsbS_Y [TTbf[ZX S^ sfaX b[Y\fb`h
Sa Z[^ s\ \`a[scS`X\V aX[a aX\`\ S^abS^`SZ VS\b\^Z\` [b\ b\c[a\V f^cp af aX\ S^a\b[ZaSf^ f
aX\ `fc_a\ SaX aX\ [`a b\`Tf^`\ f aX\ `fcr\^a Y\VS_Yh[^V aX\p [b\ Zf^^\Za\V SaX aX\
s[`SZ [``_YTaSf^ f [ [bab\\ T[baSaSf^ s\a\\^ `fc_a\ [^V `fcr\^aU \b\ \ [^aSZST[a\
aX[a S^ \ZaSf^ iUih \ Tb\`\^a [^ [TTbf[ZX S^ XSZX [ Zfbb\ZaSf^ af aX\ e Y\aXfV
S` S^abfV_Z\V af SYTbfr\ aX\ b\`_ca` Sa Z[^ TbfrSV\ af[bV aX\ f^\ TbfrSV\V sp aX\
Y\aXfVU
76F FEH7 6EEFH 69 H7
\b\ \ fZ_` f^ aX\ e [TTbf[ZXh s\Z[_`\ Sa`X[cc s\ VSb\Zacp _`\V S^ aX\ ^\a WX[Ta\b`U
]^ g\^\b[chaX\ aSY\oV\T\^V\^a \\ZaSr\ [YScaf^S[^ Z[^ s\ bSaa\^ [`
H
ef f
(t) = H
M
0
+ V
int
(t) + W (t)
3 mU jm4X\b\
H
0
M
S` aX\ [YScaf^S[^ f aX\ _^T\ba_bs\V `fc_a\h
V
int
aX\ `fc_a\o`fcr\^a S^a\b[Zo aSf^ a\bY [^VW (t)
[ g\^\b[c aSY\oV\T\^V\^a T\ba_bs[aSf^ a\bY aX[a VbSr\` aX\ `p`a\YU X\ aSY\ V\T\^V\^Z\ f aX\ `fc_a\o`fcr\^a S^a\b[ZaSf^V
int
(t)
fbSgS^[a\` bfY Vp^[YSZ[c TbfZ\``\` S^rfcrS^g S^\baS[c V\gb\\` f b\\VfY f aX\ `fcr\^aU X\ aSY\ `Z[c\ f aX\`\TbfZ\``\` S` fbV\b` f Y[g^Sa_V\ XSgX\b aX[^ aX\ aSY\ `Z[c\ f aX\ \c\Zabf^ Vp^[YSZ` f
aX\ `fc_a\h[^V [^ [VS[s[aSZ [TTbfSY[aSf^ Z[^ s\ _`\V af fccf aX\ \c\Zabf^SZ `a[a\ f aX\
`fc_a\h XSZX Z[^ s\ fsa[S^\V [` \Sg\^`a[a\ f aX\ aSY\oV\T\^V\^a \\ZaSr\ [YScaf^S[^
3 \ U
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i
∂Ψ(t)
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(t)Ψ(t)
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