§1.1
!"
§1.1.1
L(~ri , ~r˙i , t)
#$
Lagrange
L(~ri , ~r˙i , t) = T (~r˙i , t) − V (~ri , t),
%'&
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9
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$G@UA1WVM,NO+>X>"YZNO+ 9[R]\^_`abcd>e>f
T =
n
P
i=1
i
1
m ~r˙ ·~r˙
2 i i i
(1.1)
V (~ri , t)
(i = 1, 2, · · · , n)
d
dt
∂L
∂ q̇a
−
r~i = ~ri (q1 , q2 , · · · , qg , t)
qa (a = 1, 2, · · · , g)
∂L
= 0,
∂qa
g
(a = 1, 2, · · · , g)
(1.2)
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Lagrange
(1.2)
qa (t0 )
q̇a (t0 )
L(~qb , ~q˙b , t),
qa = qa (~qb , t),
g
2g
g
2g
™š ƒjk1Wco
pa =
∂L
,
∂ q̇a
q̇a (q1 , q2 , · · · , qg , t)
9[UA›1Wcd H(qa , pa , t) =
œ ceo
Hamilton
 œ jk
q˙a =
∂H
,
∂pa
(a = 1, 2, . . . , g)
X
a
Hamilton
#$
pa q˙a − L(qa , q̇a , t),
(1.3)
(1.4)
(canonical equation),
ṗa = −
∂H
.
∂qa
(a = 1, 2, · · · , g).
(1.5)
r žd 4Ÿwœ jk¡œ¢g a£>, œ 4vwjk9
œ
 j0k0a œ c`> mn
0

0
ƒ
¥
9
0
¤
0
¦

mn1§ž¨
 4 mŒp
© 4q, œ mŒ,mn1 ™>ª>«
KL
2g
g
Hamilton
(qa , pa )
(canonial transformation)
2g
X
a
(Qa , Pa )
pa dqa − Hdt =
X
a1
2g
Pa dQa − Kdt + dG,
(1.6)
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#$
9WcdÃ@Ä
2
dG
G
Hamliton
K(Pa , Qa , t)
pa =
∂G
,
∂qa
Pa = −
∂G
,
∂Qa
(a = 1, 2, · · · , g)
(1.7a)
∂G
.
∂t
(1.7b)
ÅÆ  œ m nÂ1Wq,> œ j>k"
K(Qa , Pa , t) = H(pa , qa , t) +
Q̇a =
a œ g c ¼Ç<, Èo
G
∂K
,
∂Pa
Ṗa = −
∂K
.
∂Qa
(a = 1, 2, · · · , g)
9W"V1Wɐƒ
K≡0
Hamilton-Jacobi
∂G
∂G
H qa ,
,t +
=0
∂qa
∂t
,>Ê>4>Y>»>ƒ>1|‚[[j[k ,>Ÿ>4>Ë>Ì 4>¹>º>Í>Œ
‚c1 ,c}¹ºœ ͌>Ž[Ï>9|Ð Ñ_ÒÓ
9W¤dq, jk>ÔÕ>cƒ>1|‚
(1.8)
g
G
G



Qa = const = αa
∂G

Pa = const = −
= βa 
∂αa
qa = qa (αa , βa , t)
G
%@&
αa = qa (t0 )
1
Hamilton
βa = pa (t0 ).
c0d0œ Þ߄
T % 0€ 
 mn9œ
RA jk ç
(qa , pa )
(1.5),
>d Ù
Ý#$
(1.8)
αa
,>Î>ƒ œ
, m n1WÈ>o
G = G(qa , αa , t)
(qa , pa ) → (Qa , Pa )
K≡0
Ö Â>×>c š ƒØ„
ƒ9WÎÛÜ1Wc¼ "
(a = 1, 2, · · · , g),
βa = −
(αa , βa )
(1.9)
pa = ∂G/∂qa = pa (αa , βa , t).
W = W (qa , αa , t),
R
jk
∂W
,
∂αa
pa =
ˆ>ž> œ j>kØÚ
(1.10)
∂W
,
∂qa
,0à0+09âá0M@ã ,0ä0å01
F
t
q a , pa
-0ž0d0æ0Ê0€,
∀f (qa , pa , t),
df
∂f
=
+ {f, H},
dt
∂t
(1.11)
èéê@ëìAí
%@&
Ìî
§1.1
Poisson
F
c0d0Ãßħ1
3
"
{u, v}
X ∂u ∂v
∂v ∂u
{u, v} =
−
.
∂q
∂p
∂q
∂p
a
a
a
a
a
Poisson
ÔÕ1WFðñ,
Ì0î0ç0 œ m0n0žŽmŒ1ï‚<
(qa , pa )
T
(1.12)
(Qa , Pa )
"0 œ m0n0M01
X
X ∂u ∂v
∂v ∂u
∂u ∂v
∂v ∂u
−
=
−
.
∂qa ∂pa ∂qa ∂pa
∂Qa ∂Pa ∂Qa ∂Pa
a
a
Poisson
Ìîà+
ò à+ h žóô
œ mŒ,ó>ô>8÷ 9
ž
õ
"
Ÿ
ö

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ÿ6,NO+,
{qa , pb } = δab ,
{qa , qb } = {pa , pb } = 0.
(a, b = 1, 2, · · · , g)
(1.13)
(qa , pa )
Hamilton
X p2
i
2 2
H=
+ m i ωi q i
2mi
i
ß% & " 060(0Œ1 " %  œ .Œ>1 " %  œ 78>1 " 0 60,
Ã@ÄA1 %  œ j k@R c o"
mi
pi
qi
ωi
(1.5)
dqi
pi
=
,
dt
mi
ˆžco
(1.14)
>9
dpi
= −mi ωi2 qi .
dt
(1.15a)
d 2 qi
+ ωi2 qi = 0
2
dt
(1.15b)
¤d žÅ0Ÿø +d & ",.1W‚
1[R cdŸƒØ„ 9
& Ð_0p" yN0!O,,_>Ÿ`1|ÿÎ>Û,ž#" N$+@%,>ùAúŒ>,6ÿå>k& 6& 1Ð'>Œä>6àb(>,,)Ò*>Ó9 ~
+, -./ 012345#66879:;<2=>
?@AB,C4@ÚAF,h D h
D
E1
E
qi
qi ∼ e±iωi t
ωi
pi
§1.1.2
1.
• Rayleigh-Jeans
%@&
(1900
1905
uν dν =
kB
)
8πν 2
kB T dν,
c3
žÍ$1WVD h ‰FGHIHM>,JKLM>9
(1.16)
4
• Stefan-Boltzmann
%@&
R∞
u=
uν dν
D h QP
• Wein
T Å K D
%@&
0
(1879
E1
1884
E
u = aT 4 ,
"-/ŒOS1
E1
ESR
1893
1
σ = ac
4
)
"
Stefan
h
uν dν = ν φ
ν (1.18a)
uν dν ∝ ν 3 e−c2 /λT dν.
(1.18b)
c2
D
(1900
<WIHM1W‚
ν
Í$9
dν,
T
"ABIH1 !"3vAB͌1VU
,JKLh M9
(1.17)
1896
3
λ = c/ν
• Plank
N
¬@­A® ¯A°±³²´@µA¶·¸
E
(1.18b)
‰FGWIHM
)
XY1WF
uν dν =
8πν 2
hν
dν
3
hν/K
BT − 1
c e
Wien
D
(1.19)
h
8πh 3
hν
8πh 3
hc 1
∼
uν dν = 3 ν · exp −
dν = 3 · ν · exp −
·
dν.
c
kB T
c
kB λT
‚F
< Hh I H0M01
D
c2 = hc/kB ,
λ
%@& ž
X Y Z
h
ν→0
uν dν ≈
M01 F
Planck
Í$9
ehν/kB T → 1 + hν
kB T
ˆ0ž0F
Rayleigh-Jeans
8πν 2
hν
8πν 2
·
· kB T dν
dν
=
c3
c3
1 + khν
−1
BT
0]¤ d01
D h h ž0Ÿ040Ë>Ì¢
D h ~
D h [§, \}
Ç,?@ABhD 9
">¢>™ o>p D 1W\>"!>,>ž[1|•[–_^`_A_B[/>Œ[ž[Ÿ_a[Ÿa[,[1|‚[/[Œ
61 U ¢Ÿ4>T3>vAB>ÍWŒ>F>à,>Í>$1|‚ g % Í$ h
9
E b cl1
d>b>O e Þ D d>Œ>6
f ÄA1Wg
V cihž Œ6 ,gÓ c E b c 9
?@AB© jˆ AB>6,>N>O+ R
kl1W‚
þ,
jk1WF
Planck
Wien
Rayleigh-Jeans
Planck
Planck
1900
12
14
Planck
(1900
2.
Couloumb
~=0
∇·A
Berlin
12
(quanta)
14
)
Maxwell
~
1 ∂2A
~
∇ A − 2 2 = 0.
c ∂t
2
h = kB c2 /c
(1.20)
è éê@ëìAí
çˆ?@m1on`pqrst>K>LÐ
§1.1
(1.21)
λ
1
∇2 A~λ + 2 ωλ2 A~λ = 0,
c
žpq#$
ª «
%@&
U
u"
~
A
X
~ r , t) =
~ λ (~r),
A(~
qλ (t)A
%@&
A~λ
5
A~λ =
r
pqrstKL
∇ · A~λ = 0,
r
2
e~λ cos(k~λ · ~r),
V 0
(1.22a)
2
e~λ sin(k~λ · ~r).
V 0
2π
k~λ =
(nx , ny , nz )
L
(1.22b)
" Z$1 ™ F
’AV1WÞ cFv4 W1 ‚wv4xyj{z†1V|! ª>«
(1.22c)
nx , ny , nz
~k
ˆž1o}~
~e~k · ~kλ = 0,
(λ)
(1.22d)
ωλ2 = kλ2 c2 .
(1.22e)
e~λ
(1.20)
1
(1.21)
Ù
(1.22)
co
d 2 qλ
+ ωλ2 qλ = 0,
2
dt
(1.23)
V‚ÿ78 ,i.jk9WcdÃ@Ć1WV>M>Z>4+>,/>Œ"
qλ
Z
1
1X
[ε0 E 2 + µ0 H 2 ]d3~r =
2
2 λ
%@& " %ç/Œ,€>"
$%"
ˆ0ž01?
$"
V
1
2
Z
"
dqλ
dt
2
#
+ ωλ2 qλ2 ,
2
1 X dqλ
ε0 E d ~r =
,
2 λ
dt
(1.24a)
2 3
V
1
2
Z
µ0 H 2 d3~r =
1X 2 2
ω q
2 λ λ λ
@ A B0,‚ƒ„©j0ˆŸ+60,NO+,‚ ƒ„1
V
H=
X1
λ
2
p2λ + ωλ2 qλ2 ,
%
(1.24b)
(1.24c)
Hamilton
#
6
¬@­A® ¯A°±³²´@µA¶·¸
 œ j k"
RA{ˆ " $%Fv4BC>,xy1‡’†V p
p˙λ = −ωλ2 qλ .
q˙λ = pλ ,
ν
X
X
1λ = 2
nx ,ny ,nz
λ
ν + dν
,6-$"
∆nx · ∆ny · ∆nz .
þ†Ï6-$9o‡>78>þ>1W‚ˆ
nr =
co
q
L ~
L
L
n2x + n2y + n2z =
| kλ |= = ν
2π
λ
c
4nx 4ny 4nz =
‰
4πn2r dnr
X
’AV1oŠ@‹{[A,6>$O>S>"
R Åø
†db,
%@&
Væ1
D
Boltzmann
0
uνdν
·
L
L3 ν 2
dν = 4π · 3 dν,
c
c
8πν 2
1λ = L · 3 dν
c
3
R
He−H/kB T dpdq
U = R −H/k T
= kB T,
B dpdq
e
h9
ˆ
uν dν =
1884
8πν 2
· kB T dν
c3
E Ö A BŽ
p=
R∞
/ŒŒy b cf1o46,‚Œ>/Œ>"
Rayleigh-Jeans
u=
Lν
c
1 X
1
8πν 2
8πν 2
1λ = 3 · L3 · 3 dν = 3 dν.
V λ
L
c
c
¤d1Wco % /ŒO>S"
V‚
λ
= 4π ·
1
,
3u
"-/ŒOS1WِNO>3>v N
d(uV ) + pdV = T dS,
(1.25a)
(1.25b)
è éê@ëìAí
%@& ž‘
§1.1
Ö
Z
S
(entropy)
1 ž@‹1[R
V
∂
∂v
v ∂u
T ∂T
dS
7
,»xyop
∂
−
∂T
4u
3T
= 0,
u
∂u
=4
∂T
T
4
⇒ u = aT .
⇒
N
g ’ ò>N 1
ˆ
E
J
K
ˆ E Ö b g e@„ òh N 9
“@„A£,?@ABD P
çˆ/ŒT‚Œ, y” Œj ”SR 1‡R8†d>bc>f>1W"
V‚
Stefan-Boltzmann
(Stefan
1884
1879
)
3. Planck
(H − U )2 = kB T 2
ç
Wien
D
h
%@& žÍ$ˆžco
γ
c ν
2
U = γνexp −
,
cT
1
c
U
=−
ln ,
T
c2 ν γν
ˆž
• Ÿj1Wçˆ
¤d
Rayleigh-Jeans
(H − U )2 = U 2
D
h 1WF
1
kB
=
,
T
U
d
dU
1
c
=−
,
T
c2 νU
k B c2 ν
U.
c
9o–y”j~_"—>1|‚
(H − U )2 = U 2 +
Z
d
dU
(H − U )2 =
U = kB T,
’
dU
kB
=−
1
d
dT
dU T
d
dU
1
kB
=− 2
T
U
k B c2 ν
kB
,
U =−
d
1
c
dU T
1
kB
c
=−
=
k
c
ν
2
B
T
c2 ν
U (U + c )
1
U+
kB c 2 ν
c
1
−
U
!
,
Boltzmann
8
¬@­A® ¯A°±³²´@µA¶·¸
Ö Z co
%@&
%'&
}~
U + kBcc2 ν
1
c
=
ln
T
c2 ν
U
hν
kB c2 ν/c
= hν/k T
,
U = c2 ν/k T
B
B
e
−1
e
−1
⇒
ˆž
h = kB c2 /c.

uν dν = 
hD 1e@„Adþà>+
+$
Planck
co
hν
kB T
2
 · 8πν ,
c3
−1
9 G)<)ˆ
N = 6.175×1023 mol−1
9
Wien
%@& `p
(3)
exp
h = 6.55×10−34 J·S kB = 1.346×10−23 JK−1
(1)
(2)
hν

a=
Stefan
Z
Í$
∞
0
8π 5 k 4
,
15c3 h3
x3
π4
dx
=
.
ex − 1
15
1
2π 5 k 4
σ = ac =
.
4
15c2 h3
˜`Œ6^`1W‚^`AB6>,/>Œ
En = nE = nhν,
En exp − kEBnT
E
hν
=
=
U = nP
.
En
E
E
exp − kB T
exp kB T − 1
exp kB T − 1
P
n
PAR @™ ã & ,
§1.2
Maxwell
jk"
~
~ = ∂D ,
∇×H
∂t
~ = 0,
∇·B

~ 
∂B
~
∇×E =−
, 
∂t


~ = 0.
∇·D
(1.26a)
A¯ °±³²´@µA¶·¸
ñšà+
co
§1.2
爛I%£ f o1 œ
9
~ = µ0 H,
~
B
~−
∇2 E
~ = Ex e~x
E
F
Ex (z, t) =
%@&
~ = 0 E.
~
D
(1.26b)
~
1 ∂2E
= 0.
c2 ∂t2
X
(1.27)
Aj qj (t) sin(kj z),
(1.28a)
j
Aj =
™ ce@„
Hy =
†Ïco1_" $%,->/>Œ"
1
H=
2
%@&
s
X
2ωj2
,
V 0
Aj
j
Z
V
π
kj = j ,
L
(1.28b)
q̇j (t)0
cos(kj z).
kj
d3~r(0 Ex2 + µ0 Hy2 ) =
(1.28c)
1X 2
(p + ωj2 qj2 ),
2 j j
(1.28d)
ÖQP R P R c>d>Ñ؄†1 P†R"8$_%[,
Œ>Tx™ ž>6>,>GØU†1
™ F0G ˜0,0 œ 78~> œ .Œç
œ
9 Ÿ ¡0 Œ6åk&>1 Ð Œ0Ñ0_0Ï
F1W“”cdop dþ 1 Œ"ÏF 1
pj = q˙j .
1.29d
1.29e
(1.28e)
Hamilton
(qj , pj )
(
qj
pj
)
[qj , pj 0 ] = ih̄δjj 0 ,
ˆ
œo
Hamilton
Œ
qj , pj
[qj , qj 0 ] = [pj , pj 0 ] = 0.
aj e−iωj t =
s
a†j eiωj t
s
=
H = h̄
1
(ωj qj + ipj ) ,
2h̄ωj
1
(ωj qj − ipj ) .
2h̄ωj
X
j
1
ωj (a†j aj + )
2






(1.29)
(1.30)





(1.31a)
¬@­A® ¯A°±³²´@µA¶·¸
10
Ùçà+
G ˜Ü1WÐ
!_¢£1WÒÓÏF>9W œ 7>8~>.Œ>c>d@R†‘w¤>"
[aj , a†j 0 ] = δjj 0 ,
aj , a†j 0
[aj , aj 0 ] = [a†j , a†j 0 ] = 0.
s
h̄ −iωj t
qj =
aj e
+ a†j eiωj t ,
2ω
r j h̄ωj
pj = −i
aj e−iωj t − a†j eiωj t .
2
ˆž1WŒ6åÂ,¥%>,>yŒ>"
Ex =
P
j
Hy = −i0 c
Ej (aj e−iωj t +
P
Ej =
p
h̄ωj /0 V .
Ex ,
(1.33)
E~k =
%@&
~k
p
h̄ω~k /20 V
P
~
(λ)
ˆ~k E~k α~k e−iω~k t+ik·~r
+ c.c.,
nj (j = x, y, z)
Ej sin(kj z) cos(kj z)
j











h ~1 žŠy±Œ1
w¤³´9[R pqrstK>Lc>f
~k
~k = (kx , ky , kz )
c.c.
žZ$9[R
(1.32)
©ªPy01 y0Û0ç ˜« ¢ £
X ~k × ˆ~(λ)
1
~
k
~ r , t) =
E~k α~k e−iω~k t+ik·~r + c.c.,
H(~
µ0
ω~k
1





[Ex , Hy ] = 2i0
~k
P
(1.31c)






Ej (aj e−iωj t − a†j eiωj t) cos(kj z). 

Hy
~ r, t) =
E(~
%@& žç®¯4





a†j eiωj t ) sin(kj , z),
%'&
R h c)d¦¦’0œ 1§¥ %¨0F00~
~)Ò)Ó)ϦF)9 U M01 ~ GW<)ˆ)Ÿ)ç) m)Œ01
þ¬­I%,Œ6>å9
j
(1.31b)
°
ˆ~λk
(1.33)
α~k
ž®Œ²Œ1
~k = 2π (nx , ny , nz ),
L
Maxwell
’§V0T µ ¶0,0B0C0yŒ>Fv41 "
!¸d ¹1W‚
~k
jkcf
~k · ˆ(λ) = 0,
~k
(λ)
ˆ~k (λ = 1, 2).
2
X
~k
→2
L
2π
3 Z
œ 0p0r0Ÿ0*1
d3~k,
Ð
P
~k
·
å0" 0‹ y0É
(1.34)
9
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º78þ1
§1.2
’ Z 1o@‹"
Z
3~
dk=
‚
α~k → a~k
1
Z
2π
dφ
Z
,™@ã & 1 p
L3
0
π
sin θdθ
0
ω
dN = 2
g0h &½¼ ¾ ¢
11
(dω)2
L
2π
∞
0
4π
k dk = 3
c
2
[A,°$g»³"
ω + dω
Z
∞
ω 2 dω.
0
3 Z ω+dω
4π
L3 ω 2
∆
2
·
·
ω
dω
≈
dω = D(ω)dω,
2 c3
c3
π
ω
¿ Ù 0À w ¿
α~k∗ → a~+
k
Z
Âco
1 %@&
~ r , t) =
E(~
P
~k
D(ω) =
X0"Á°OS 9lÐ
L3 ω 2
π 2 c3
~
(λ)
ˆ~k E~k a~k,λ e−iω~k t+ik·~r
+ h.c.,
(λ)
1 X ~k × ˆ~k
~
~
H(~r, t) =
E~k a~k,λ e−iω~k t+ik·~r + h.c.
µ0
ω~k
%@& Fþ œ çà>+
~k
(1.34)






(1.36)
= δ~kk~0 δλλ0 ,
h
i h
i

a~k,λ , a~k0 ,λ0 = a~†k,λ , a~†k0 ,λ0 = 0. 
ˆž{" % ~$%Œ/Â>XÃo©ª>y1|‚
~ =E
~ (+) + E
~ (−) ,
E
(2) (2)
~ki ~kj + ~ki ~kj = δij −
h
%@&
i
0
~
Ej (~r, t), Hj (r , t) = 0
[Ej (~r, t), Hk (~r, t)] = −ih̄c2
j, k, l
zXŸ4ÇnÈ1oÉ
j = x, k = y
(1.37)
~ =H
~ (+) + H
~ (−) .
H
Z1WcdÃ@Ä P _>"Ä#ÅÆR
(1) (1)
Ï 0F å01








[a~k,λ , a†k~0 ,λ0 ]
(1.35)
œ
ki kj
,
k2
(1.38a)
(j = x, y, z),
(1.38b)
∂ (3)
δ (~r − r~0 ).
∂l
(1.38c)
l=x
Ê
j = y, k = z
œ
l=x
ÊW©9
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12
Fock
}~çà+
Ö
„
| ni
ž
Ë
1
H = h̄ω(a a + ) ,ñ̄1W‚
2
§1.3
Fock
†
1
H | ni = h̄ω(a† a + ) | ni = En | ni
2
[a, a† ] = 1
1WcdÃ@Ä
∆
1. Ha | ni = (En − h̄ω)a | ni = En−1 a | ni
2.
⇒
Z
a | ni
1
1
H | 0i = h̄ω(a† a + ) | 0i = h̄ω | 0i = E0 | 0i
2
2
3.
1
En = (n + )h̄ω,
2
a† a = n̂,
n̂ | ni = n | ni.
4.
√
a | ni = n | n − 1i,
√
a† | ni = n + 1 | n + 1i,
(a† )n
| ni = √ | 0i.
n!
ˆž
ž ,ñ̄
Ha | 0i = (E0 − h̄ω)a | 0i ⇒ a | 0i = 0
’AV
5. Fock
(1.39)
„zXŸ4ÍΟYÏ>+>1W‚
∀ | ψi
∞
X
n=0
1WF
爟4°xy,#"
| nihn |= 1,
$%ZÐ>1
| ψi =
hm | ni = δnm .
X
n
cn | ni.
~
cdÃ@Ä _"Ã@Ä
(
E(~r, t) = Eae−iωt+ik·~r + h.c.,
)
H
§1.3 Fock
Ñ
13
1.
hn | E | ni = 0
2.
2
hn | E | ni = 2E
ÎÛÜ1W<
| ni =| 0i
2
"ҙ„M1
h0 | E 2 | 0i = E 2 =
1
n+
2
h̄ω
.
20 V
rÙ f §Ä 1 ‚0È0žÒ ™„1Ó¥%SÁeÅFøÔ , Õ Ö P ×ØR 1 rsÕ֞@RAˆ¥%ñ
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Š á01EŒ06â’ß>1 ©©>9 þ“>”Ð>ŸŸãä `@ÚAF>,Ÿæ¥%>Œ6>å,>fçç
}dy è~ƒ Þ9
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Ï F01éê0ž % x ž0z G01ëÉ
1
ìíž
Ï F9
%
þ ¬î°
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0
Lamb
a† )
a,a†
1
(a − a† )
2i
Hermite
Fock
Hamiltonian
H=
%@&
X
H~k ,
~k
4°¤ç˜,ñ̄>"
ž
™F
1
(a +
2
Hermite
H~k = h̄ω~k
| n~k i
1WF
a~†k a~k
1
+
.
2
1
H~k | n~k i = h̄ω(n~k + ) | ni.
2
ˆž ,ñ̄˜ ò
H
| nk~1 i | nk~2 i · · · | nk~l i · · · =| nk~1 , nk~2 , · · · nk~l , · · · i =| {n~k }i,
p
ak~l | nk~1 , nk~2 , · · · n~kl , · · · i = n~kl | n~k1 , n~k2 , · · · n~kl − 1, · · · i,
q
a~†k | n~k1 , n~k2 , · · · n~kl , · · · i = n~kl + 1 | n~k1 , n~k2 , · · · n~kl + 1, · · · i.
VM@R ï 4°,ṉ̃>¤z>X,>q>,„± ª «  ÍÃ|YÏÎ>Ÿ,>à>+1|‚
l
X
~k1 ,~k2, ···~kl ···
| n~k1 , n~k2 , · · · n~kl , · · · ih· · · n~kl , · · · n~k2 , n~k1 |= 1,
14
UzX,q,
Hilbert
| Ψi =
&1
™@ã
X
1
=
rsw¤{ð
\} ] Ç Pòñ _Ä{ÅgR 9
cn~k
n~k ,n~k ,···n~k ,···
2
X
{n~k }
1oŒc@R
∀ | ψi
1
u"
,n~k ,···n~k ,···
2
l
¬@­A® ¯A°±³²´@µA¶·¸
| n~k1 , n~k2 , · · · n~kl , · · · i
l
c{n~k } |{n~k } i,
| Ψi =| Ψ~k1 i | Ψ~k i · · · | Ψ~k i · · ·
2
§1.4
Lamb
l
óõô
ö ÷øoùúûü 2S ý 2P þÿ
ü 2S ý 2P þÿ þ
1. Dirac ü ö ÷ ½ü 2S þ ÿ 2P þ ÿ þ 1.057GHz(∼ 6.6 ×
2. 10 J).
úû #$ û%&'( )*+, (a) ! þ"
û6$879: (QED) ;< ý=>
(b) -./102345
ýDE üGFHI JK E (renormalization) L
(c) @? 3ABC þ M 3NOP
MQ + Welton RSTU (V QED) WX Y Lamb Z[ ? U\
&]']^]($`_](ab c ù ú û#$½ûde ú û]fg Coulomb (# −e /4π r
h 7ij kl üm@n $8û h 7 Wo " ~r → ~r + δ~r ü δ~r p $@( abqr Ps
S abtuqr Coulomb v ab w sp Lamb Z[ ú mwx Myz ø
1/2
1/2
1/2
1/2
1/2
1/2
−25
2
∆V
m@F{ X ab| } @~ üm8€
‚ = V (~r + δ~r) − V (~r)
1
= δ~r · ∇V + (δ~r · ∇)2 V (~r) + · · ·
2
hδ~rivac = 0
ü @
0
(1.40)
1
h(δ~r · ∇)2 ivac = h(δ~r)2 ivac ∇2
3
(1.41a)
1
e2
2
2
h∆V i = h(δ~r) ivac ∇ −
.
6
4π0 r
at
(1.41b)
§1.4 Lamb
A
2S
ü
ƒ„
15
∇
2
e2
−
4π0 r
Z
e2
1
∗
= −
d3~rψ2s
(~r)∇2 ψ2s (~r)
4π0
r
2
e
=
|ψ2s (0)|2
0
e2
=
,
8π0 a30
H # † + ‡ˆ‰ y
at
1
∇
= −4πδ(~r),
r
s
1
ψ2s (0) =
,
8πa30
2
ü V AŠ‹Œ ψ (0) = 0 üm@€
MŽ h(δ~r) i P x Newton ‘’“”
A
(1.42)
2P
Lamb
2P
2
Z[P
vac
€ y• <–— ^( ˜™
€  ü
k > π/a P
m
ω~k
d2
~~ ,
(δ~r)~k = −eE
k
dt2
Born
ijš 6›8˜™ üGœ
(1.43)
ω~k > πc/a0
ü1œ
0
δ~r(t) ∼
= δ~r(0)e−iωt + h.c.
H #
(δr)~k ∼
=
e
E~
mc2 k 2 k
~
E~k = E~k (a~k e−iωt+ik·~r + c.c.).
‚ h(δ~r)ivac =
=
e 2
h0|E~k2 |0i
mc2 k 2
~k
X e 2 h̄ck X
~k
ž FŸü1š y ž F
2
h(δ~r) ivac
mc2 k 2
20 V
,
Z ∞
e 2 h̄ck V
2
= 2·
· 4π
dk · k
(2π)3
mc2 k 2
20 V
0
2 2 Z ∞
1
e
h̄
dk
=
·
,
2
20 π
h̄c
mc
k
0
(1.44)
¡ ¢@£ ¤@¥¦¨§© ª@«¬­
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< ü @ ü@¶ ! g^ Š· ‚ Compton
Š
·
Z
@
m
€
¸
¹
 ü1½ ‡
p/mc = h̄k/mc ≤ 1)
!º
dk/k »2¼
16
0
mc/h̄
π/a0
2
H#
ü
h(δ~r) ivac
e2
h̄c
α−1
∼
=
1
20 π 2
e2
h̄c
h̄
mc
2
ln
40 h̄c
e2
,
A ¾¿À Œ üÁgÂ@ÃÄ ZÅ M üÁ¾¿À Œ α = 2e h̄c = 7.2974 × 10
F Compton Š·P " n
h̄
= 137.04 λ =
= 3.8616 × 10 m
mc
2
−3
0
−13
c
2 2 e
e2
h̄
1
40h̄c
4
h∆V i ≈
ln
3
4π0
4π0h̄c
mc
8πa30
e2
ÆÇ Œ È  “ ½ø h∆V i ≈ 10 J ∼ 1GHz.
Æ ^:ÉÊ =>Ë ½ Lamb Z[ ? U †@³ @ÌÍÎÏЇ ü
?
Õ ^û 1s → 2s Ö×ØÙ ÏЇ 3 1s Lamb Z[ ? UP
−24
§1.5
Hänsch
(1.45)
ÑÒÓÔ
ÚÜÛÜÝ
Wß³à Þ û ø ³Þ
&'abãåä
+
Maxwell ‘’âá +
E
Rçèéëê =ìí á Spontaneous Emission Radiation ä ü Lamb Z[ î
þ
æ
. ^ï1ð ñ@ò ü ûó á Quantum Beat ä ôtõ x 4ö û » ÷ P
Äúû ? U üx8ü þÿ‰ý »þ P
sÿ ø!
ù
¿ ü þÿ úûø
S
úû ø
úû - ^( ø
| ψi = ca e−iωa t | ai + cb e−iωb t | bi + cc e−iωc t | ci.
| ψV (t)i =
| ψΛ (t)i =
(1.46)
X
ci | i, 0i + c1 | c, 1ω1 i + c2 | c, 1ω2 i,
(1.47a)
X
c0i | i, 0i + c01 | c, 1ω1 i + c02 | c, 1ω2 i.
(1.47b)
i=a,b,c
i=a,b,c
§1.5
«¬
17
a
a
b
Z1
Z2
Z2
Z1
c
b
c
V
/
³Þ A ^ó ˜ ÷ø
1.1: V-
úû
Λ-
úû
E (+) = E1 e−iω1 t + E2 e−iω2 t
H #
ü
⇒ |E (+) |2 = |E1 |2 + |E2 |2 + {E1∗ E2 ei(ω1 −ω2 )t + c.c.}
ãFó ˜ ã P
MŽ + û E ^( » Ð š ñ@ò Ð
3
4
(−)
E1
m@€
x y
(1.48)
= E1∗ a†1 eiω1 t ,
(−)
(1.50)
ý
(+)
E2
(−)
(+)
hψ(t) | E1 E2
| ψ(t)i
= E2 a2 e−iω2 t .
(+)
E1 (t)E2 (t) = E1∗ E2 a†1 a2 ei(ω1 −ω2 )t .
(1.48a)
(−)
“ ½
(+)
hψV (t) | E1 (t)E2
(−)
| ψV (t)i = κei(ω1 −ω2 )t hc | ci = κei(ω1 −ω2 )t ,
(+)
hψΛ (t) | E1 (t)E2
ü1H #
| ψΛ (t)i = κ0 ei(ω1 −ω2 )t hc | bi = 0,
(1.49)
(1.50a)
(1.50b)
18
H # ü
κ = E1∗ E2 c∗1 c2
g W ³Þ øA
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0
κ = E1∗ E2 c1∗ c02
V
(−)
(+)
hψ(t) | E1 (t)E2 (t) | ψ(t)i = κc ei(ω1 −ω2 )t ,
H # κ = E E c c ú mg
R¯ù Ä ¦ú ¦û]nø ¦ü ü ÿ
ü x –o (which-path)
• A V
ω ω
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S
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‡
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s
H
I
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G
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F
ì
í
Aš&'ab »IJ ü n BN * ö û OP ûó ˜ ü Mî Õ ^û !# ? UP ü
bQT½ 3T R + W ³Þß Aš&m@€'ü ag b ° ¸æ ¹ Z ü ö ^û û /S z 3 Ä R &'ü ag
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sø öa p
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V Í
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»þ ý D A 01 ý
IJP
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ø W ³Þ ø ßà û E + ³Þ^( (+ &'ab )
ÿ
1. ö ³Þû øßk à û E + ( û E
2. W
ò
^ $ c ø ñ@Wentzel
ü 1927 l
m . ìí ø
nop^ø
§1.6
§1.6
q¬ ªrstu
19
1.2:
êv ( ‘’
^û 0 1û ´ ø ´ ø
ü wx ìí ½ Planck y y ü 1900 l
(a) þ
0
1
Planck
^û 01 ´ ø Einstein ü1^ $ c ü 1905 l
(b)
hν = φ + T .
^
$
ø
ÿ
(c)
^i. c $ û ‘ ’ =ˆí ™ % Ç ¯6í ñ@^ò z •{ ^ Ç í‡ Ž % $ û| í E~}   }€ ii.
I Ž R¯ñò + ³Þ Ï 2B ü n + ^û‚ ƒ ‘ y »
A M P
§1.6.2 bcde
ñ@ò ø( W ³Þ ø š B &' aˆb ñ8ò E R‚ p ^ ê ^=û ìí P
êv ^(‘’ û († tu)¾01A J‡IJ
01ˆ
¶
P
@ E ü Bell Ž
§1.6.3 Šh‹Œ
W ³Þ ^+û &'%aH b → ° æ á ûü ó ˜ aåä
!#y
^ û !Dirac
ü# ?
“”  P! BcP “ê‘” S =$ 0!1 #^’ •s–S— þ+ !‚ A A ‚ Young
Bell °P
3.
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ˆ J‰ ü
c^:ü1n˜A÷Õü
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1. Š
20
Light
Matter
~ r, t)
E(~
ψ(~r, t)
2
Semiclassical
(∇2 −
Ouantum field
2
1 ∂ ~
∂ ~
)
E
=
−µ
P
0
c2 ∂t2
∂t2
i
ψ̇(~r, t) = − Hψ(~r, t)
h̄
Maxwell’s equation
Schrödinger’s equation
i
| ψ̇f i = − Hf | ψf i
h̄
P
~ r , t) = α~ (t)U~ (r)
E(~
k
k
i
| ψ̇m i = − Hm | ψm i
h̄
P
ψ̂(~r, t) =
ĉP~ (t)φP~ (r)
~k
~
P
û( * R¯ù Ä úû Dirac’s equation
V A ~ Š ( ‘ ’ ww Schrödinger ‘’
MŽ A
Schwinger’s equation
ih̄ψ̇(~x, t) = Hψ(~x, t)
¯ A ~ Š ( ‘ ’ ü H #
‘ ’P
ø «
(a) Klein-Gordon ‘’
3
H #
(b) Dirac
¯ ßà ( ü ý 1 ¯ ©ª ( á ^(ü1#}
(ü ê¬ F 0 ü à
(scalar)
2
‘’ ø ¬
( ä
m2 c 2
∂λ ∂ + 2 φ(~x, t) = 0,
h̄
λ
∂λ ∂ λ =
(spinor)
Bose
( ü ê­¬ F
h
1 ∂2
− ∇2 .
c2 ∂t2
1/2
ü à
mc i
iγ ∂µ −
ψ(~x, t) = 0,
h̄
µ
á $ ûü { $8ûü à û P ä
§1.6
q¬ ª rstu
H #
21




 I 0 
 0 ~σ 
γ0 = β = 
~γ = 
,
,
0 −I
−~σ 0






 0 1 
 0 −i 
 1 0 
σx = 
σy = 
σz = 
,
,
.
1 0
i 0
0 −1
( ‘’ ø ® (vector) (ü ê¬ F 1 ü
ßà ( ᶵ
ü
field) + »³´
* ©ª ( (gauge
ž ~ P g&' ø ý g Coulomb ©ª
U(1) ©ª°
(c) Maxwell
∂µ F µν = 0,
Bû à ·ä E~ü } ¯ ‚ #}Y°±(( ü¹K~
²
¸
º
)*+ M ü
(F µν = Aµ,ν − Aν,µ )
" n “ ½
∂ ∂ A = 0,
“,†Eü1š»½¼ à ( @? ¸g º ‘’ # P
S ‘¾ ø N/ þˆ‰ @ ø
V A ~þ¿À ‘’  E = T + V Schrödinger’s equation



E =p c +m c
Klein − Gordon



A] ~ à þ¿À ‘]’ ø E = ±pp c + m c Dirac equation
λ
2
λ
µ
2 2
2 4
2 2
•
2. Maxwell ‘’“MÁ
^( û E ’Æ ø
(a)
2 4





 E = pc
Maxwell
equation
ü
¶
Ã
y
Schördinger ‘’Â
@
»ÄÅP
~ (+) =
E
X
~k
~ (+) =
E
X
ßà ( û E ’Æ ø á¶ÇÙ
~k
(b)
equation
(λ)
ˆ~k E~k α~k e−iω~k t U~k (t)
↓α→a
(λ)
ˆ~k E~k â~k e−iω~k t U~k (t),
û E ’Æ ä
ψ(~r, t) =
X
p
~
[â~k , â†k~0 ] = δ~kk~0 .
cp~ e−iωp~ t φp~ (~r)
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22
1
↓ φp~ (~r) = √ ei~p·~r ,
V
ψ̂p~ (~r, t) =
,¾ø
X
ĉp~ e−iωp~ t φp~ (~r),
p
~
cp~ → ĉp~
[ĉp~ , ĉ†p~0 ]± = δp~p~0 ,
RÙ û ûE ü ü p~ → −ih̄∂ ,
ÇÙ
E c → ĉ ,
[cp~ , c†p~0 ]± = δp~p~0 .
p
~
MŽ [ é ½ ‡ ^ û Š‹Œ¾ ¸º Schrödinger ‘’P
Ø Zü û ( ‘ ü ßà µ û Š‹Œ = È « /É %
Ψ(~r, t) = h~r | ψ(t)i,
H #
| ~ri = ψ̂ + (~(r)) | 0h,
X
+iωp~ t ∗
ψ̂ + (~r) =
ĉ+
φp~ (~r).
pe
m@۟
p
~
Ψ(~r, t) = h0 | ψ̂(~r) | ψ(t)i.
^ û Ê ¶ þ é €¿Ë H MŽ Ð Š ‹ŒP
P
| ψi =
c (t) | {n}i,
g | ψi ü1g ~r 2. = 1 ¯ ^û ½ ™ F
{n}
H #
Î
m@€ü á
{n}
P
{n}
| {n0 }ih{n0 } |= 1.
~ (−) (~r, t)E
~ (+) (~r, t) | ψi,
Pψ (~r, t) ∝ hψ | E
X (λ)
~ˆ (+) (~r, t) =
E
ˆ~k E~k a~k e−iω~k t U~k (~r),
ü1HÌ | 0ih0 | µ+
~k
| 1ih1 | + · · ·
− : Boson)
[xα , pβ ] = ih̄δαβ ,
~
x
p
~
(+ : Fermiion;
ô• *+Í 3 ä
~ˆ (−) (~r, t) | 0ih0 | E
~ˆ (+) (~r, t) | ψi.
Pψ (~r, t) ∝ hψ | E
~ E (~r, t) = h0 | E
~ˆ (+) (~r, t) | ψγ i
Ψ
® WV ŸüGœ
q ¬ ª rstu
I Ž Ï ‡ 1ü Ä^û
§1.6
H #
Θ(x)
F
| ψγ i
% Š‹Œ
~E
Ψ
23
F
r
r
Γ
(λ) E0
~
ΨE (~r, t) = ˆ~k
Θ t−
exp −i t −
ω−i
,
r
c
c
α
Heaviside
ÐыŒ ü
Γ
úû ‰ ŒP
~ E (~r, t) = h0 | E
~ˆ (+) (~r, t) | ψγ i
Ψ
s
X (λ) h̄ω~
~
k
= h0 |
ˆ~k
a~k,λ e−iω~k t+ik·~r | ψγ i.
20 V
~k,λ
m@€
H #
~ E (~r, t) =
Ψ
r
=
r
X (λ)
h̄ω
~
h0 |
ˆ~k a~k,λ e−iω~k t+ik·~r | ψγ i
20 V
~k,λ
h̄ω
ϕ
~ γ (~r, t),
20
~
ϕ
~ γ (~r, t) =
X
(λ)
ˆ~k h0
1ü ¶ “ MÒ _(zÓ A Š‹Œ
~k,λ
~ H (~r, t) = h0 | H
~ˆ (+) (~r, t) | ψγ i,
Ψ
s
~
X
~
h̄ω~k
e−iω~k t+ik·~r
k
(λ)
~ˆ (+) (~r, t) =
× ˆ~k
a~k,λ √
H
.
k
2µ0
V
m@€
‚ ü
e−iω~k t+ik·~r
| a~k,λ √
| ψγ i.
V
~k,λ
Maxwell
~ H (~r, t) =
Ψ
s
~
~k
h̄ω
e−iω~k t+ik·~r
(λ)
h0 | × ˆ~k a~k,λ √
| ψγ i
2µ0
k
V
=
s
h̄ω
χ
~ γ (~r, t).
2µ0
‘’“MÁ •
1 ∂ϕ
~γ
~ ×χ
∇
~γ =
,
c ∂t
~ ·χ
∇
~ γ = 0,
1 ∂~
χγ
~ ×ϕ
∇
~γ = −
,
c ∂t
~ ·ϕ
∇
~ γ = 0.
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24
H #



 ϕx 




ϕ
~ γ =  ϕy  ,




ϕz
‚ “ ½
ih̄
î


 χx 




χ
~ γ =  χy  ,




χz

Dirac
ih̄
Ô ¸º ¿ÀÕ ‘’
ÖØ×






~η   0
c~σ · p~   ϕ
~η 
∂  ϕ

=


∂t
χ
~η
c~σ · p~
0
χ
~η
Dirac
Š‹Œ“ x ¬
ϕ~η

sH ü1^û Š‹Œ F

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 ϕ
∇·
 = 0.
χ
~γ
‘’âá # ûåä
V Àü1H ## û

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−c~s · p~   ϕ
~γ 
∂  ϕ

=

,
∂t
χ
~γ
c~s · p~
0
χ
~γ

š y %

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Ψ


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Ψ


ϕ
~γ 
,
χ
~γ
ý
χ
~η

¿ • ü1œ
ϕ
~η 

χ
~η
(†)
(†)
~ (†)
Ψ
ϕγ , χ
~ γ ).
γ = (~
∂ ~†~
~ · ~j = 0,
Ψ Ψγ + ∇
∂t γ

~ † ~v Ψ
~ γ,
~j = Ψ
γ

 0 −~s 
~v = c 
.
~s 0
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e
ÖW×
kz = pz /h̄
å
1
ϕe (~r, t) = √ ei(kz z−ωt) ,
V
ωk = p2z /2mh̄
ñ/56å/7ù89:é
~k = kx~ex + kz ~ez .
æçå ùÛÜÝÞß?@å/ABCD6ÛçE
z
;<
x
x
;< =å/>è
FG å/>è
ei(kz z−ωt)
,
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V
56å/7ù89:é
x
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ei(kz z+kx x−ωt)
√
.
V
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√
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∂x
V
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å
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Baker-Hausdoff
1
eA+B = e− 2 [A,B] eA eB
1
z{| 7
2.
3.
7
[A, B] 6= 0
f (a, a† )
€ C
å
= e 2 [A,B] eB eA .
α
çéßå/>è
e−αA BeαA = B − α[A, B] +
a,a†
‚ßå z{
α2
[A, [A, B]] + · · ·
2!
}~
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26
(a)
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,
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∂f
,
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†
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z{|
4.
z{|
5.
7
= f (aeα , a† e−α ).
†
†
†
†
[a, e−αa a ] = (e−α − 1)e−αa a a,
[a† , e−αa a ] = (e−α − 1)e−αa a a† .
P
H = h̄ω(a† a + 1/2)
H=
En | nihn |
þÿ
X
n
a | αi = α | αi,
ÿŽ
n
eiHt/h̄ =
6.
†a
z{|
eiEn t/h̄ | nihn | .
1 2 ∞
|α| X αn
√ | ni.
| αi = e 2
n!
n=0
−
7. ~r, p~
8.
)à8‘å
z{|
~ = ~r × p~
L
~ = 0,
~r · L
)’8‘å z{|
~ = 0,
p~ · L
∂
a | αihα |= α +
| αihα |,
∂α
∂
| αihα | a = α +
| αihα | .
∂α∗
†
~ ×L
~ = ih̄L.
~
L
∗
“•”•–
žŸ¡¢¤£¤¥¤¦
§2.1
§¨©uܪ« ~ å è
56å/7è鬭Wö®
øµ
§
Schrödinger
(2.3)
~ þ·
þÿ z{
±W×
kçå/AB
v¸
±W×
—˜™š›œ™
~ = −i
A
X 1
~
ˆ~k E~k a~k e−iω~k t+ik·~r + h.c.,
ω~k
®¯°Wöuå/>è±ø²³´
J~
~k
Z
ih̄
Hamiltonian
(2.1)
)
~ r, t) · A(~
~ r , t)d3~r,
J(~
(2.2)
∂
| ψ(t)i = V (t) | ψ(t)i.
∂t
(2.3)
V =
;¶ )
™
i
| ψ(t)i = exp −
h̄
Z
t
0
0
dt V (t ) | ψ(0)i.
0
Y
Z
i t 0
0
exp −
dt V (t ) =
exp(α~k a~†k − α~k∗ a~k ),
h̄ 0
(2.4)
(2.5)
~k
1
α~k =
E~
h̄ω~k k
| ψ(0)i =| 0i
å/>è
Z
| ψ(t)i =
t
dt
0
Y
~k
0
Z
~ r , t)eiω~k t−i~k·~r .
d3~rˆ~k · J(~
exp(α~k a~†k − α~k∗ a~k ) | 0i~k ,
| {α~k }i =
Y
~k
| α~k i,
| αk i = exp(α~k a~†k − α~k∗ a~k ) | 0i~k .
(2.6)
(2.7)
(2.8a)
éP¹Pºå¼»P½¾%g¬­Wö®¨©Ûu1¿³øÀÁÛu励ÂX)ÃБ³´kÅ
ÁÆÇå/È
(2.8b)
| αi = exp(αa† − α∗ a) | 0i
∆
= D(α) | 0i.
27
(2.9)
ƒ É ÊËÌÍÎÏÌ
™×Ö¤Öؗ¤˜¤™
§2.2 ÐÑÒÓ£Ô¤Õ
þ ÿ z{ åM§ exp(−α a) | 0i =| 0i Ž Baker-Hausdorff }~ þ·
28
∗
D(α) = exp(αa† − α∗ a)
= e−
= e
Ž
| αi = e
§
−
|α|2
2
|α|2
2
|α|2
2
†
eαa e−α
∗
∗a
†
e−α a eαa ,
∞
X
αn
√ | ni.
n!
n=0
þÿ z { ÙxyÚ3
a | αi = α | αi.
ýÛÜJÃБéÝS | Þx1yÚ3
(2.10a)
(2.10b)
(2.11)
(4.10)
(2.12)
D † (α) = D(−α) = D −1 (α),
(2.13a)
D −1 (α)aD(α) = a + α,
(2.13b)
D −1 (α)a† D(α) = a† + α∗ .
(2.13c)
— ˜™ —˜¤ß¤à¤á
£
Á | ni ÷âãýÝÞß)
Fock
§2.3
U6åM')
p = −ih̄∂/∂q
å þä
φn (q) = hq | ni.
(2.14)
(2.15a)
∂
a= √
ωq + h̄
,
∂q
2h̄ω
1
∂
†
a =√
ωq − h̄
.
∂q
2h̄ω
1
å ')
þ·
a | 0i = 0,
hq | a | 0i = 0.
(2.15b)
Ê ËÌÊËæçè
éê)
§2.3
hq |
È
∂
ωq + h̄
∂q
° ;¶ fëéªì)
å §
29
| 0i =
∂
ωq + h̄
∂q
∂
ωq + h̄
∂q
hq | 0i =
∂
ωq + h̄
∂q
φ0 (q) = 0.
ω 41
ωq 2
φ0 (q) =
exp −
.
πh̄
2h̄
φ0 (q),
(2.16)
(2.17a)
(a† )n
| ni = √ | 0i.
n!
·
ù
| ni
Áv
φn (q) = hq | ni
(a† )n
= hq | √ | 0i
n!
n
1
1
∂
ωq − h̄
= √
φ0 (q)
n
∂q
n! (2h̄ω) 2
r 1
ω
=
q φ0 (q).
1 Hn
h̄
(2n · n!) 2
φn (q)
?@å þÿ z{
hqi = hn | q | ni = hn |
h̄
(a + a† ) | ni = 0,
2ω
r
h̄ω
hn | (a − a† ) | ni = 0,
2
h̄
h̄
1
† 2
2
hq i =
hn | (a + a ) | ni =
n+
,
2ω
ω
2
h̄ω
1
2
† 2
hp i = − hn | (a − a ) | ni = h̄ω n +
.
2
2
hpi = hn | p | ni = i
'5
r
1
(∆p) = hp i − hpi = h̄ω n +
,
2
h̄
1
2
2
2
(∆q) = hq i − hqi =
n+
.
ω
2
2
2
2
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30
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2
(∆q) (∆p) =
v¸þÿ³
íî
2
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φ0 (q)
ùµïðñòÁå/oè
1
n+
2
ψ(q, 0) =
>§
þÿ z{
h̄2 ,
1
n+
h̄
2
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2
h̄
.
2
±ö
(2.19)
ω 14
h ω
i
exp − (q − q0 )2 ,
πh̄
2h̄
2 2
∂
h̄ ∂
ω2q2
ih̄ ψ(q, t) = −
+
ψ(q, t),
∂t
2 ∂q 2
2
∞
X
ψ(q, t) =
an φn (q)e−iEn t/h̄ ,
n=0 1
h̄ω,
En = n +
2
(2.20)









(2.21)








ω 12
h ω
i
2
|ψ(q, t)| =
exp − (q − q0 cos ωt) ,
πh̄
h̄
(2.22)
ψ(q, 0) = hq | αi
∞
n
|α|2 X α
√ hq | ni,
= e− 2
n!
n=0
(2.23)
ω 12
α=
q0
2h̄
(2.24)
2
U6þÿ z{
(2.18)
NO çQå/ïðñòÝ÷ ψ(q, 0) O çøÀÁ÷âãýÝ Þßå N çÅÁ÷
 âãýÝÞßfé q øÄñ
0
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1.
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—˜™
§2.4
TÜß
31
£ûü
hα | a† a | αi = |α|2 = hni
(2.25a)
2
p(n) = hn | αihα | ni =
|α|2n e−|α|
hnin e−hni
=
,
n!
n!
p(n)
±ýö hni = |α| çøPÀPÁýPPõÜPøPþßÚÿ­ñ
ñ n < 1 6å ÷ n = 0 å
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2.
2
3. | αi
Z
∗ n
m −|α|2 2
(α ) α e
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6å÷ n = |α| ñ
n>1
2
∆p · ∆q =
éñ þÿ z1{
dα =
Z
∞
0
|α|
(2.25b)
h̄
.
2
(2.25c)
n+m+1 −|α|2
e
d|α|
Z
2π
ei(m−n)θdθ
0
= πn!δnm
(2.25d)
±ö´L α = |α|e å/#´L-føÛñ(§*%
å þ ÿ z {
Z
X
iθ
| αihα | d2 α = π
Èè
1
π
% O çøÀÁñ
ø ÀÁ#ñ þÿ z{
4.
| αihα | d2 α = 1.
(2.25e)
(2.25f)
1 2
1 02
0 ∗
hα | α i = exp − |α| + α α − |α | ,
2
2
(2.25g)
|hα | αi|2 = exp(−|α − α0 |2 ).
(2.25h)
0
V?è
Z
n
| nihn |= π,
.ÿå α = α 6å |hα | αi| = 1 å α 6= α 6å |hα | αi| 6= 0 åsè |α| |α |
vø!P6w3 è | < α | α > | → 0 ñ*§ %PPPSþÿ z{ ôå øÀÁþÿ´±"
øÀÁ  Cå/È
Z
0
2
0
2
0
2
1
d2 α0 | α0 ihα0 | αi
π
Z
1
1 2
1 02
0∗
2 0
0
=
d α | α i exp − |α| + α α − |α | .
π
2
2
| αi =
(2.25i)
ƒÉ ÊËÌÍÎÏÌ
32
%¿³øÀÁ#ñ
øÀÁð |
5.
øÀÁ笭Wö®¨©$Áå/vçøĐ%ùÅÁ³´$Áå$È
(a)
| αi = D(α) | 0i,
±ö
(b)
D(α) = exp αa† − α∗ a .
øÀÁçõu&'rÞvç()%w3 i*Áå/È
a | αi = α | αi
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p
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0
0
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1
∆A · ∆B ≥ |hci|.
2
Bè
1
(∆A)2 < |hci|,
2
> D.EÁ¿8FGÁ (queezed state) ñ/BU6Hè
EÁ¿³.IFJGÁñ
KL% X å X ì)
1
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2
M ½ìç6Nª
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1
∆A∆B = |hci|
2
1
X1 = (a + a† ),
2
O
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x
p
r
h̄
x=
(a + a† ),
2mω
X2 =
1
(a − a† ),
2i
p = −i
i
[X1 , X2 ] = ,
2
r
(2.26)
h̄ω
(a − a† ).
2
(2.27)
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33
1
∆X1 ∆X2 ≥ .
4
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E(~
= 2ˆ
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1
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4
U6å/B
π
2
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(i = 1, 2)
∆X1 ∆X2 =
> DEk.IFGÁñ¹ýÛ_÷øÀÁ
øÀÁ6å
| αi
1
4
O
Fock
Á
| ni
6
∆X1
(∆X1 )2 = hα | X12 | αi − (hα | X1 | αi)2
1 2
1
=
(α + α∗2 + 2|α|2 + 1) − (α2 + α∗2 + 2|α|2 )
4
4
1
,
=
4
È
(∆X2 )2 = hα | X22 | αi − (hα | X2 | αi)2
1
1
= − (α2 + α∗2 − 2|α|2 − 1) + (α2 + α∗2 − 2|α|2 )
4
4
1
,
=
4
∆X1 = ∆X2 = 1/2
å
∆X1 · ∆X2 = 1/4
ñ
Fock
Á6å è
(∆X1 )2 = hn | X12 | ni − (hn | X1 | ni)2
1
=
(2n + 1),
4
1
(∆X2 )2 = (2n + 1).
4
`
∆X1 = ∆X2 =
1√
2n + 1,
2
1
1
∆X1 · ∆X2 = (2n + 1) > .
4
4
O
∆X2
ñ
34
ƒÉ ÊËÌÍÎÏÌ
› œ
š›¤œ¤—¤˜¤™
Ò:a
´b#¯ ¶ ÆÇFGÁå óõcõ% Hamiltonian å è
§2.6
H = ih̄(ga†2 − g ∗ a2 )
kçå/õuÁ)
i
| ψ(t)i = e− h̄ Ht | 0i = e(ga
ÂXFG%
±ö
†2 −g ∗ a2 )
| 0i
(2.31)
1 ∗ 2 1 †2
ζ a − ζa
,
2
2
(2.32)
S † (ζ) = S −1 (ζ) = S(−ζ),
(2.33a)
S † (ζ)aS(ζ) = a cosh r − a† eiθ sinh r,
(2.33b)
S † (ζ)a† S(ζ) = a† cosh r − ae−iθ sinh r,
(2.33c)
S(ζ) = exp
ζ = reiθ
(2.30)
çéñ þÿ z1{
1.
2.
3.
±ö´L
ÿŽ
|
Baker-Hausdoff
eA Be−A = B + [A, B] +
1
1
[A, [A, B]] + [A, [A, [A, B]]] + · · · ,
2!
3!
∞
X r 2n
r2 r4
cosh r = 1 +
+
+··· =
,
2!
4!
(2n)!
n=0
∞
ÂX
X r 2n+1
r3 r5
+
+··· =
.
sinh r = r +
3!
5!
(2n + 1)!
n=0
>è
Y1 + iY2 = (X1 + iX2 )e−iθ/2 ,
(2.34)
±ö
S † (ξ)(Y1 + iY2 )S(ξ) = Y1 e−r + iY2 er .
(2.35)
1
i
i
†
Y1 =
a exp(− θ) + a exp( θ) ,
2 2
2 1
i
i
Y2 =
a exp(− θ) − a† exp( θ) .
2i
2
2







(2.36)
Î ÏdÍÎÏÊËÌ
FGøÀÁÂX)
§2.6
35
| α, ξi = S(ξ)D(α) | 0i,
± ö α = |α|e å ξ = |ξ|e ñ/§ ~ (4.36) ŽøPÀPÁPPÂXþeå øÀÁç1§
ý
[ f ³) e g !h«å/?FGÁi, a O a ø ;j ñ
ýÛ kúFGÁéÝS |
þÿ z{ å
iϕ
iθ
(2.37)
a
O
a†
†
hai = hα, ξ | a | α, ξi
= h0 | D † (α)S † (ξ)aS(ξ)D(α) | 0i
= hα | (a cosh r − a† eiθ sinh r) | αi
= α cosh r − α∗ eiθ sinh r.
(2.38a)
ha2 i = h(a† )2 i
= h0 | D † (α)S † (ξ)a2 S(ξ)D(α) | 0i
= h0 | D † (α)S † (ξ)aS(ξ)S † (ξ)aS(ξ)D(α) | 0i
= hα | S † (ξ)aS(ξ)S † (ξ)aS(ξ) | αi
2
= hα | [a2 cosh2 r − aa† eiθ cosh r sinh r − a† aeiθ cosh r sinh r + a† e2iθ sinh2 r] | α >
= α2 cosh2 r + (α∗ )2 e2iθ sinh2 r − 2|α|2 eiθ cosh r sinh r − eiθ cosh r sinh r
(2.38b)
ha† ai = |α|2 (cosh2 r + sinh2 r) − (α∗ )2 eiθ sinh r cosh r
kçå þÿ z{
− α2 e−iθ sinh r cosh r + sinh2 r
(2.38c)
(∆Y1 )2 = hY12 i − hY − 1i2
2
1
θ
θ
1
θ
θ 2
†
†
=
h a exp(−i ) + a exp(i ) i − h a exp(−i ) + a exp(i ) i
4
2
2
4
2
2
1 −2r
=
e ,
(2.39a)
4
1
(∆Y2 )2 = e2r ,
4
1
∆Y1 · ∆Y2 = .
4
(2.39b)
(2.39c)
36
§2.7
cZFG%
l:m
ƒÉ ÊËÌÍÎÏÌ
›œ™
S(ξ) = exp[ξ ∗ aω+ω0 aω−ω0 − ξa†ω+ω0 a†ω−ω0 ],
| α, ξi = S(ξ)D(α) | 0i.
f (),Æǐ‘ÂX)
>±
näk
1
b = √ (aω+ω0 + e−iδ aω−ω0 ),
2
1
b† = √ (a†ω+ω0 + eiδ aω−ω0 ).
2
X1
O
X2
þÿ z{
éùop‘ |
1
b1 = (b + b† ),
2
±ùµÂ)
(2.40)




(2.41)



1
b2 = (b − b† ).
2
1
∆b1 · ∆b2 ≥ .
4

1 −2r
δ θ
δ θ

2
2r
2
e cos
−
+ e sin
−
, 
(∆b1 ) =

4
2
2
2
2
1 2r
δ θ
δ θ

2
−2r
2
2

(∆b2 ) =
e cos
−
+ e sin
−
. 
4
2 2
2 2
2
U]å þÿÂXqZFGÁõu
S[ξ(ω)] =
±ö
ξ(ω) = r(ω)eiθ(ω)
Z
å è
dω 0
exp[ξ ∗ (ω 0 )aω+ω0 aω−ω0 − ξ(ω 0 )a†ω+ω0 a†ω−ω0 ] A#,
2π
| α(ω), ξ(ω)i ≡ S[ξ(ω)]D[α(ω)] | 0̃i.
r:a:s:t:;:u=<¤¡¤¢¤£=v=s
åJ|}~ )
§2.8.1 wxyz{ (P- z{
ù€‚„ƒ‘â
1.
§2.8
—¤˜
û
(2.42)
Œ†QR‡ˆ‰ÊË*ù
´ Fock Á  C
(a)
§2.8
ρ=
(b)
n,m
´øÀÁ  C
| nihn | ρ | mihm |=
X
n,m
ρnm | nihm | .
d2 α d2 β
| αihα | ρ | βihβ |
π π
Z Z 2 2
1
d αd β
∗
2
2
=
| αihβ | R(α , β) exp − (|α| + |β| ) ,
π π
2
±ö
(2.43)
Z Z
ρ =
1
2
2
R(α ) = hα | ρ | βi exp (|α| + |β| ) .
2
∗
øÀÁâÂX
Š‹Œ
2.
X
37
ON (a, a† ) =

X
cnm (a† )n am ,
(2.44a)
(2.44b)
(2.45)
n,m
hON (a, a† )i = Tr[ρON (a, a† )]
X
cnm Tr[ρ(a† )n am ].
=
(2.46)
n,m
ÂX%
Z
1
δ(α − a )δ(α − a) =
exp[−β(α∗ − a† )] exp[β ∗ (α − a)]d2 β
(2.47a)
π2
Z
1
exp[−iβ(α∗ − a† )] exp[−iβ ∗ (α − a)]d2 β,(2.47b)
=
π2
∗
†
þÿŽJ
hON (a, a )i =
Z
d2 α
=
Z
d2 αP (α, α∗)ON (α, α∗ ),
(2.48)
P (α, α∗ ) = Tr[ρδ(α∗ − a† )δ(α − a)]
(2.49)
†
±ö
X
n,m
cnm Tr[ρδ(α∗ − a† )δ(α − a)](α∗ )n αm
38
§
Trρ = 1
ƒÉ ÊËÌÍÎÏÌ
þÿ z{
Z
d2 αP (α, α∗ ) = 1,
(2.50)
Z
P (α, α∗ ) | αihα | d2 α.
(2.51)
Ž
ρ=
§! ~ þä
Z
P (α, α∗ )h−β | αihα | βid2 α
Z
2
∗
∗
−|β|2
= e
[P (α, α∗ )e−|α| ]eβα −β α d2 α.
h−β | ρ | βi =
 α=x
kçè
α + iyα
È
h−β | ρ | βie
å
|β|2
h−β | ρ | βie|β|
β = xβ + iyβ
=
2
ç
Z Z
å>
d2 α = dxα dyα
å
βα∗ − β ∗ α = 2i(yβ xα − xβ yα )
å
P (xα , yα∗ ) exp −(x2α + yα2 ) exp [2i(yβ xα − xβ yα )] dxα dyα
P (α, α∗ )e−|α|
2
‘
Fourier
’åJ“
Fourier
” ’·
2
2 Z Z
e(xα +yα )
P (α, α ) =
h−β | ρ | βi exp(x2β + yβ2 ) exp[2i(yα xβ − xα yβ )]dxβ dyβ
π2
2 Z
e|α|
2
h−β | ρ | βie|β| exp(−βα∗ + β ∗ α)d2 β
(2.52)
=
2
π
∗
3.
ýÛÜJéøÀÁâò%
uÁ
(a) •
± ö k ç
ý
_·
B
kç
— ?þä
– å
ρ=
exp[−H/kB T ]
,
Tr[exp(−H/kB T )]
ç
(2.53)
å o
1
H = h̄ω(a† a + )
2
X
h̄ω
nh̄ω
ρ=
1 − exp −
exp −
| nihn | .
kB T
kB T
n
Boltzmann
H
Hamiltonian
hni = Tr(a† aρ) =
ρ=
X
n
1
exp
h̄ω
kB T
.
−1
hnin
| nihn |,
(1 + hni)n+1
å þÿ
§2.8
Œ†QR‡ˆ‰ÊË*ù
39
ρnn = hn | ρ | ni =
hnin
.
(1 + hni)n+1
hnin
h−β | nihn | βi
n+1
(1
+
hni)
n
n
2
∞
e−|β| X (−|β|2 )n
hni
=
1 + hni n=0
n!
1 + hni


h−β | ρ | βi =
=
kç
2
2

|β|2 
e−|β|
,
exp 
−

1 
1 + hni
1+
hni



|β|2 
 · exp(−βα∗ + αβ ∗ )d2 β
−
exp 

1 
1+
hni
2
1
|α|
=
exp −
.
(2.54)
πhni
hni
P (α, α∗ ) =
(b)
e|α|
π 2 (1 + n)
X
øÀÁ
Z
ρ =| α0 ihα0 |,
(2.55)
h−β | ρ | βi = h−β | α0 ihα0 | βi
= exp −|α0 |2 − |β|2 − α0 β ∗ + βα0∗ ,
'5
Z
1 |α|2 −|α0 |2
∗
∗
∗
e
e−β(α −α0 )+β (α−α0 ) d2 β
P (α, α ) =
2
π
= δ (2) (α − α0 ),
∗
ÈøÀÁ
Á
(c) Fock
P
âç鐑
δ
ñ
ρ =| nihn |,
h−β | ρ | βi = h−β | nihn | βi
n
2n
2 (−1) |β|
= exp(−|β| )
,
n!
(2.56)
(2.57)
40
ƒÉ ÊËÌÍÎÏÌ
'5
2 Z
(−1)n e|α|
∗
∗
P (α, α ) =
|β|2n e−βα +β α d2 β
2
π n!
Z
|α|2
e
∂ 2n
∗
∗
=
e−βα +β α d2 β
2
n
∗n
π n! ∂α ∂α
2
∂ 2n
e|α|
δ (2) (α),
=
n! ∂αn ∂α∗n
∗
§2.8.2
' ) n > 0 å¼.ÿ˜™šK›Š%
+ž„Ÿ8ÂX¡ P â¢
|}~ª©
Q £¤z{¦¥¨§
OA =
X
P
âPœéPÂ›'?›
dnm an (a† )m ,
(2.58)
Fock
Á
(2.59)
n,m
Q(α, α∗ ) = Tr[ρδ(α − a)δ(α∗ − a† )].
Z
1
d2 α0 | α0 >< α0 |
π
Z
1
∗
Q(α, α ) =
Tr d2 α0 [ρδ(α − a) | α0 ihα0 | δ(α∗ − a† )]
π
1
hα | ρ | αi.
=
π
° ·
§8«¬›J­®LYï%
§
,
Tr(ρ) = 1
º8»
P-
° ·
ⱄ²8›
Z
Q-
Q(α, α∗ )d2 α = 1.
Z
†
hOA (a, a )i = Q(α, α∗ )OA (α, α∗ )d2 α.
✝³ž´¡¢JµW¶·¸¹
ρ=
¼½
º8»
Q-
X
ψ
â,
Q(α, α∗ ) =
P-
â¡·¸¹
ρ=
(2.61)
(2.62)
(2.63)
Pψ | ψihψ |
1X
1X
1
Pψ |hα | ψi|2 ≤
Pψ = .
π ψ
π ψ
π
Z
(2.60)
P (α, α∗ ) | αihα | d2 α
(2.64)
§2.8
¾¿†QÀR‡ˆ„Á‰Â„Ã*ù
41
1
hα | ρ | αi
πZ
1
0 2
=
P (α0 , α0∗ )e−|α−α | d2 α0 .
π
Q(α, α∗ ) =
Fock
Ä ¡
Q-
â »
1
hα | nihn | αi
π
2
1
e−|α| |α|2n
2
=
|hα | ni| =
.
π
πn!
(2.65)
Q(α, α∗ ) =
œÅÆ¢/ìÛH4_¯FG Ä ¡ Q- â¢
~Ë©
§2.8.3 Wingner-Weyl ÇÈz{¦¥¨ÉÊ
Ì͛JÎ P- O Q- Ï ¹
P (α, α∗ ) = Tr[ρδ(α∗ − a† )δ(α − a)],
(2.67a)
Q(α, α∗ ) = Tr[ρδ(α − a)δ(α∗ − a† )].
(2.67b)
MÐÑ ° ½ÒÓÔÕ Ö×¢ÙØ
1
δ(α − a )δ(α − a) = 2
π
∗
Ú
(2.67a)
†
¬ °Û
Z
1
P (α, α ) = 2
π
∗
܄Ý
exp[−iβ(α∗ − a† )] exp[−iβ ∗ (α − a)]d2 β,
Z
d2 βe−iβα
∗ −iβ ∗ α
†
²8ޛ
(2.67b)
ß »
1
Q(α, α ) = 2
π
Z
d2 βe−iβα
∗
∗ −iβ ∗ α
∗
¸àáâ„Ø
C (n) (β, β ∗ ),
C (n) (β, β ∗ ) = Tr(eiβa eiβ a ρ).
∗
܄Ý
(2.66)
C (a) (β, β ∗ ),
†
C (a) (β, β ∗ ) = Tr(eiβ a eiβa ρ).
Wingner-Weyl
ãäå„æ ¡ç苌é긡 ÔÕëì
C (s) (β, β ∗ ) = Tr(eiβa
† +iβ ∗ a
ρ),
(2.68)
(2.69)
(2.70)
(2.71)
(2.72)
í„î8ï „Ãðñ8òó„ð
42
Ú
W-
Ïô
W (α, α∗ )
ûÿ«
1
W (α, α ) = 2
π
Z
∗
õö
aa†
d2 βe−iβα
¡çè÷øùúûÅü¡¢þý·
1
haa† + a† ai =
2
W
Z
Ïôõ Ø8Åç ß
W (p, q) =
=
=
=
∗ −iβ ∗ α
1
2π
1
2π
1
2π
1
2π
2 Z
2 Z
2 Z
2 Z
dσ
dσ
dσ
dσ
Z
C (s) (β, β ∗ ).
(2.73)
W (α, α∗ )αα∗ d2 α.
x
p
(2.74)
Öס¹
dτ ei(τ p+σq) Tr[e−i(τ p̂+σq̂) ρ]
Z
(2.75)
1
dτ ei(τ p+σq) Tr[e−i(τ p̂ e− 2 h̄στ ρ]
Z
(2.76)
1
dτ ei(τ p+σq) Tr[e−i(τ p̂/2 e−i(τ p̂ ρe−i(τ p̂/2 ]e− 2 h̄στ
Z
dτ e
i(τ p+σq)
Z
i
dq 0 hq 0 | e−iτ p̂/2 e−iσq̂ ρe−iτ p̂/2 | q 0 ie− 2 h̄στ .
(2.78)
º8»
exp (−iτ p̂/2) | q 0 i =| q 0 −
°Û
W (p, q) =
º8»
­
y = −h̄τ /2
1
2π
2 Z
dσ
Z
1
2π
°Û
Z
1
W (p, q) =
πh̄
0
dτ dq 0 eiσ(q−q ) hq 0 +
#â Ïô
Z
ρ=π
h̄τ
i,
2
h̄τ
h̄τ iτ p
| ρ | q0 −
ie .
2
2
(2.79)
0
eiσ(q−q ) dσ = δ(q − q 0 ),
Z
∞
«¬¡ Ò J¸à Ò
§2.8.4
(2.77)
dye−i
−∞
WP-
2py
h̄
hq − y | ρ | q + yi.
Ïôö
Q-
(2.80)
Ïô Ú Q- Ïô ¡ùú¢
W- !"
P-
F Ω (α, α∗ )∆(Ω) (α − a, α∗ − a† )d2 α,
(2.81)
§2.8
܄Ý
¾¿$%&À'()„Á+*$„Ã+,
∆
(Ω)
Z
exp[Ω(β, β ∗ )] exp[−β(α∗ − α) + β ∗ (α − a)]d2 β,
F (Ω) (α, α∗ ) ≡ P (α, α∗ )
ÌÍ32
∆
∗
∗
†
(α − a, α − a ) =
=
=
=
=
=
=
Ü„Ý Ò67
1
π2
Ú
¬
(2.83)
Z
89
Z
eβ
∗ (α−α
(2.81)
∆
(Ω)
(Ω)
∗
∗
5 3
Z
|β|2
1
∗
†
∗
− 2
· e−β(α −a )+β (α−a) d2 β
e
2
π
Z
|β|2
1
∗
∗
†
∗
− 2
· e−βα +β α · eβa −β a d2 β
e
2
π
Z
2
2
1
− |β|
−βα∗ +β ∗ α
−β ∗ a βa† |β|
2
2
2 d β
e
·
e
·
e
e
e
π2
Z
1
∗
∗
†
eβ (α−a) e−β(α −a ) d2 β
2
π
Z Z
1
β ∗ (α−a)
−β(α∗ −a† ) 2
e
|
α
ihα
|
e
d βd2 α1
1
1
π3
Z Z
1
∗
∗
∗
eβ (α−α1 )−β(α −α1 ) | α1 ihα1 | d2 βd2 α1
3
π
1
| αihα |
π
1 )−β(α
∗ −α∗ )
1
(2.83)
d2 β = δ(α − α1 )δ(α∗ − α1∗ )
°Û
ρ=
Ω(β, β ∗ ) = |β|2 /2
2
2
d2 α1 δ(α − α1 )δ(α∗ − α1∗ ) | α1 ihα1 |=| αihα | .
¼½
2
∗
(Ω(0, 0) = 0)
Ω(β, β ∗ ) = −|β|2 /2
(Ω)
(2.82)
J¢ ý2·.32 Ω(β, β ) = −|β| /2 3
3 F (α, α ) ≡ Q(α, α ) ¢J¸à4µ
Ω(β, β ) = |β| /2
õ2Ï.- ÷2ø./10Ù32¡ ëÙì
Ω(β, β ∗ )
¶8¢
1
(α − a, α ) = 2
π
∗
43
Z
F (Ω) (α, α∗ ) | αihα | d2 α,
F (Ω) (α, α∗ ) = P (α, α∗ ).
5 3
Z
|β|2
1
−β(α∗ −a† )+β ∗ (α−a) 2
2
(α − a, α − a ) =
e
·
e
dβ
π2
Z
1
∗
∗
†
∗
=
e−β (α −a ) · eβ (α−a) d2 β
2
π
∗
(2.84)
†
(2.85)
í„î8ï „Ãðñ8òó„ð
44
: õ
Z
1
1 0
0
(Ω)
∗
0
−β(α∗ −a† )
β ∗ (α−a)
hα | ρ | α i =
F
(α,
α
)hα
|
e
·
e
| α0 id2 βd2 α
π
π2
Z
Z
1
β ∗ (α−α0 )−β(α∗ −α0 ∗ ) 2
(Ω)
∗
=
F (α, α ) 2 e
d β d2 α
π
Z
∗
=
F (Ω) (α, α∗ )δ (2) (α − α0 )d2 α = F (Ω) (α0 , α0 ),
;
1 0
∗
hα | ρ | α0 i = Q(α0 , α0 ),
π
<
F (Ω) (α, α∗ ) = Q(α, α∗ ).
2
Ω(β, β ∗ ) = 0
∆
∆
° ½ Û 6
W
Ïô ¢ º8»
1
(α − a, α − a ) = 2
π
Z
exp[−Ω(β, β ∗ )] exp[β(α∗ − a† ) − β ∗ (α − a)]d2 β,
1
(α − a, α − a ) = 2
π
Z
exp[Ω(β, β ∗ )] exp[−β(α∗ − a† ) + β ∗ (α − a)]d2 β,
−(Ω)
(Ω)
5 3= Ð > Û 6
(2.86)
∗
†
∗
†
Tr[∆(Ω) (α − a, α∗ − a† )∆−(Ω) (α0 − a, α0∗ − a† )] =
º@? ¼½
1 (2)
δ (α − α0 ),
π
F (Ω) (α, α∗ ) = Tr[ρ∆−(Ω) (α − a, α∗ − a† )]
Z
1
∗
W (α, α ) = 2 Tr[ρ exp(−βa† + β ∗ a)] exp(βα∗ − β ∗ α)d2 β.
π
¡ AB Ïô 2
¢ Ø«¬ °C W (α, α ) õ Tr[ρ exp(−βa
² 5FG 6 e õ (1/2π)e D Fourier ßE 3H
2
Fourier ßE
¸à åWæ
W (α, α∗ )
∗
−2|α|2
e
−2|α|2
º@? 2Ø@IJáÞ °C
܄Ý
1
=
2π
∗
W (α, α )e
C(β, β ∗ )
õ IJ3H
1
C(β, β ) = 3
2π
∗
Z
Z
−2|α|2
†
+ β ∗ a)]
|β|2
− 2
|β|2
exp −
2
=
Z
exp(βα∗ − β ∗ α)d2 β,
C(β, β ∗ ) exp(βα∗ − β ∗ α)d2 β,
†
∗
Tr{ρ exp[−(β − β1 )a , +(β −
β1∗ )a]} exp
|β1 |2
−
2
dβ1
D
¾ ¿$%&À'()„Á+*$„Ã+,
® 9KLMN ¯O 1/π R | αihα | d α = 1 Û
§2.8
45
2
1
C(β, β ) =
2π 5
∗
×
=
×
=
º@?
=
Z Z Z
Tr{ρ | β2 ihβ2 | exp[−(β − β1 )a† ] exp[(β ∗ − β1∗ )a] | β3 ihβ3 |}
1
1
2
2
exp − |β − β1 | − |β1 | d2 β1 d2 β2 d2 β3
2
2
Z Z Z
1
hβ3 | ρ | β2 ihβ2 | β3 i
2π 5
1
1
2
2
∗
∗
∗
exp −(β − β1 )β2 + (β − β1 )β3 − |β − β1 | − |β1 | d2 β1 d2 β2 d2 β3
2
2
Z Z Z
β
β
1
hβ
|
ρ
|
β
ih
|
β
ihβ
|
−
> d 2 β3
3
2
3
2
2π 4
2
2
1 β
β
h | ρ | − i.
2
2π 2
2
∗
W (α, α ) = e
=
§2.8.5
PQRST
Q
2|α|2
1
· 2
2π
Z
2 2|α|2
e
π2
h
β
β
| ρ | − i exp(βα∗ − β ∗ α)d2 β
2
2
h−β | ρ | βi exp[−2(βα∗ − β ∗ α)]d2 β.
ρ =| β, ξihβ, ξ |,
¼½
Q(α, α∗ ) =
܄Ý
¸àU¯
Z
1
1
hα | ρ | αi = |hα | β, ξi|2 ,
π
π
hα | β, ξi = hα | S(ξ)D(β) | 0i = hα | S(ξ) | βi
hα | S(ξ) | βi
¢
1
hα | a† S(ξ) | βi
α∗
1
=
hα | S(ξ)S † (ξ)a† S(ξ) | βi
α∗
1
=
hα | S(ξ)(a† cosh r − ae−iθ sinh r) | βi
α∗ 1
∂
1 ∗
−iθ
=
cosh r
+ β − e β sinh r hα | S(ξ) | βi,
α∗
∂β 2
hα | S(ξ) | βi =
(2.87)
í„î8ï „Ãðñ8òó„ð
46
: õ 3
1 ∗
∂
−iθ
∗
cosh r
− βe sechr +
β cosh r − α
hα | S(ξ) | βi = 0.
∂β
2
V ?WX °Û
܄Ý
°C
1
1
hα | S(ξ) | βi = K exp − |β|2 + α∗ βsechr + e−iθ β 2 tanh r
2
2
K
ö
α, α∗ , β ∗ , r, θ
YJ­„Ø
S(ξ)
DZ[\ á¢2Ø
hα | S(ξ) | βi∗ = hβ | S † (ξ) | αi = hβ | S(−ξ) | αi
(K ∗ = K ∗ (α, α∗ , β ∗ , r, θ))
1 2 1 iθ ∗ 2
∗
K exp − |β| + e (β ) tanh r
2
2
1 2 1 −iθ 2
∗
∗
= K(β, β , α , r, θ + π) exp − |α| − e α tanh r .
2
2
: õ 3
܄Ý
1 2 1 iθ(α∗ )2
tanh r ,
K(α, α , β , r, θ) = (sechr) exp − |α| − e
2
2
∗
1
(sechr) 2
cde¢
: õ 2Ø
∗
º O õ » 7]^_ Å`ab
1
π
K
Z
1
2
|hα | S(ξ) | βi|2 d2 α = 1
DÏf ¬ °Û
1
hα | S(ξ) | βi = (sechr) 2
1
1 iθ ∗ 2
× exp{− (|α|2 + |β|2 ) + α∗ βsechr −
e (α ) − e−iθ β 2 × tanh r}.
2
2
º@? sechr
exp{−(|α|2 + |β|2 ) + (α∗ β + βα∗ )sechr
π
1 iθ ∗2
−
e (α − β ∗2 )2 + e−iθ (α2 − β 2 ) tanh r}.
2
Q(α, α∗ ) =
ghi
p(n) = |hn | β, ξi|2,
§2.8
¾¿$%&À'()„Á+*$„Ã+,
hα | β, ξi =
Ø
∞
X
n=0
47
hα | nihn | β, ξi = e
exp(2zt − t2 ) =
°Û
− 12 |α|2
∞
X
(α∗ )n
√ hn | β, ξi.
n!
n=0
∞
X
Hn (z)tn
n!
n=0
,
n
βe−i 2
√
2 cosh r sinh r
(tanh r)n
1
exp{−|β|2 + e−iθ β 2 + eiθ (β ∗ )2 tanh r} Hn
p(n) = n
2 n! cosh r
2
βe−i 2
√
2 cosh r sinh r
º@?
Ø
θ
1
2
−iθ 2
hn | β, ξi = n
β tanh r) · Hn
1 exp − (|β| − e
2
2 2 (n! cosh r) 2
(eiθ tanh r) 2
hnr i =
Û
∞
X
nr p(n),
θ
!
!
(r = 1, 2)
n=0
(∆n)2 = hn2 i − hni2
= |β|2 [cosh 4r − cos(θ − 2φ) sinh 4r] + 2 sinh2 r cosh2 r.
² 5 J çjklm Ä nH
| 0, ξi
°Û
p(2n) =
1
tanh r
2
2n
,
p(2n + 1) = 0.
oqp@r
1.
(2n)!
(cosh r)−1
(n!)2
s µ
(a)
Z
1
δ(α − a )δ(α − a) =
exp[−β(α∗ − a† )] exp[β ∗ (α − a)]d2 β
π2
Z
1
=
exp[−iβ(α∗ − a† )] exp[−iβ ∗ (α − a)]d2 β.
2
π
∗
†
.
2
.
í„î8ï „Ãðñ8òó„ð
48
(b)
Z
1
δ(α − a)δ(α − a ) =
exp[β ∗ (α − a)] exp[−β(α∗ − a† )]d2 β
π2
Z
1
exp[iβ(α∗ − a† )] exp[iβ ∗ (α − a)]d2 β.
=
π2
∗
†
(c)
2.
δ(α)δ(α∗ ) = δ[Im(α)]δ[Re(α)]
µ„¶
܄Ý
µ ¶
„
3.
W (α, α∗ )
1
haa† + a† ai =
2
õ
Z
W (α, α∗ )|α|2 d2 α,
∂
α+
∂α
| αihα |,
∂
α+
∂α∗
| αihα | .
ãäÏô ¢
Wigner-Weyl
(a)
†
a | αihα |=
(b)
4.
| αihα | a =
µ„¶
(a)
Tr[D(α)] = πδ (2) (α),
(b)
Tr[D(α)D † (α0 )] = πδ (2) (α − α0 ),
(c)
∗
Tr[∆(Ω) (α − a, α∗ − a† )∆−(Ω) (α0 − a, α0 − a† )] =
܄Ý
∆
5.
6.
µ„¶
∆
(Ω)
1 (2)
δ (α − α0 ),
π
1
(α − a, α − a ) = 2
π
Z
exp[Ω(β, β ∗ )] exp[−β(α∗ − a† ) + β ∗ (α − a)]d2 β,
1
(α − a, α − a ) = 2
π
Z
exp[−Ω(β, β ∗ )] exp[β(α∗ − a† ) − β ∗ (α − a)]d2 β.
−(Ω)
∗
∗
U¯ gt Ä 3 u Ä D
†
†
2
W (α, α ) =
π
∗
Q
Ïôö
Z
W
P (β, β ∗) exp(−2|α − β|2 )d2 β.
Ïô ¢
vxwxy
z
-
§3.1
WXŒ
û Schrödinger @Ž 3
{}|}~}}€}}‚„ƒ„„|„†„‡
ƒ}}|}†}‡}‚„ˆ„‰„Š„‹
1. Schrödinger
ih̄
WXŒ
û Heisenberg @Ž 3
∂
| ψi = H | ψi.
∂t
2. Heisenberg
3.
dF̂
∂ F̂
1
=
+ [F̂ , Ĥ].
dt
∂t
ih̄
‘’q“ WX•”—– è˜
Ü„Ý ρ̂ = P P
Œ
4. @ŽßE
ψ
(a)
Ÿ
Liouville
WXš™›Œ û
Schrödinger
@Ž 3
1
∂ ρ̂
= [Ĥ, ρ̂] A#,
∂t
ih̄
ψ
| ψihψ |
Schrödinger
܄Ý
@Ž
˜ ‘’q“@œ
(Density Matrix Operater)
→ Heisenberg
ž
@Ž
| ψ(t)iH = Û −1 (t) | ψ(t)iS
| ψ(t)iS = Û (t) | ψ(t)iH
i
Û (t) = exp − Ĥt ,
h̄
”¢¡ Ó oqp ™
i.
ih̄
∂
| ψ(t)iH = 0.
∂t
ii.
dF̂H
=
dt
∂ F̂
∂t
!
+
1
[F̂H , Ĥ].
ih̄
ÜWÝ£+ F̂ = Û F̂ Û . ¤¥¦¨§ 3©û Schrödinger +Ž F̂ .
t 3H ∂F̂ /∂t = 0 3 dF̂ /dt = 0 ” Ò @ŽßE¦q§ ¡ Ó o¨p ª ™ ž «¬
H
S
−1
S
H
S
S
49
íq­8ï ® - ¯ °±²³´qµ@¶·°&¸
2 Û (t) ˜q¹@ (Unitary) ßE @ H Û (t) = Û (t) Û (t)Û (t) = 1 5 (H
iii.
3
Ĥ = Ĥ 5 )
50
†
−1
†
†
A.
hψ(t) | F̂S | ψ(t)iS = hψ(t) | F̂H | ψ(t)iH ,
B.
(b)
hψ(t) | ψ(t)iS = hψ(t) | ψ(t)iH .
@Ž →3g© º ÓÒ @Ž
˜¼»8Ø ö gº ÓÒ½¾ ã nH
Schrödinger @Ž
Hamiltonian Ïô
Ÿû
Schrödinger
Ĥ = Ĥ0 + ĤI ,
܄Ý
” oqp ™
ii.
iii.
‘œ
(a) ËÌ
(b)
2
i
Û0 = exp − Ĥ0 t ,
h̄
J܄Ý
Ĥ0
˜ÃÂÃÄ
3Á
dF̂I
=
dt
(Hermitian)
∂F
∂t
+
I
1
[F̂I , Ĥ0 ].
ih̄
œÃÃ5 ¿ÀÅÆÇÈ ëì DÉÊÆ ª ߞ
ρ̂ =| ψihψ |,
ÍÎÌ
ρ̂ =
X
Pψ | ψihψ | .
Ïq§ Œ 3ÐúÑ
û D O Ô ZÓ 4 D ÑU Z ;ÔÕÖ Ñ U Z ×
i. ËÌ5
D
Ò
ªÙ O Ô ZÃÓ 4 D ÑÃU Z ; Á ÕÖ u ¿ÃÀÓ 4 D ÃÑ U ÃZ ž
ii. ÍØÎØÌØ5
ψ
(c)
∂
| ψiI = V̂ | ψiI
V̂ = Û0−1 ĤI Û0 ,
∂t
F̂I = Û0−1 ĤS Û0
¿Àß ˜
i. ih̄
5.
| ψ(t)iI = Û0−1 (t) | ψ(t)iS
| ψ(t)iS = Û0 (t) | ψ(t)iI
§3.2
® -
¯ °±²³´

Hamiltonian
3.1:
z
51
KÒÚÛ O öMÜÝÞ gº ÓÒ
{}|}~}}€} Hamiltonian
O à” ß  ™ ˜2ý ÝÞ E~ öKÒÚMÛ O gº Ó2Ò D
¥
K
Ò
Ú
Û
áâã3ä ¤¥ × å
§3.2
-
~
H = HA + HF − e~r · E,
܄Ý
HA =
X
Ei σii
HF =
æ 5
~k
e~r =
܄Ýç@áâèé ’q“@ê ˜
æ 5 3 ëìíqç Þ ˜ Œ
1
h̄ω~k (a~†k a~k + ).
2
X
X
i,j
e | iihi | ~r | jihj |=
X
î@?
X
~k
~ = h̄
−e~r · E
p~ij σij
i,j
~k
(3.3a)
ij
ˆ~k E~k (a~k + a~†k ).
XX
(3.2a)
(3.2b)
p~ij = ehi | ~r | ji,
~ r , t) =
E(~
û
(3.1)
σij =| iiij |,
i
Hamiltonian
g~kij σij (a~k + a~†k ),
(3.3b)
(3.3c)
íq­8ï ® - ¯ °±²³´qµ@¶·°&¸
52
܄Ý@ï Î ú ì ˜
: õ JúÑ Dð
Hamiltonian
H=
X
g~kij = −
˜
h̄ω~k a~†k a~k +
X
p~ij · ˆ~k E~k
h̄
Ei σii + h̄
i
~k
XX
i,j
g~kij σij (a~k + a~†k ),
~k
Ü ÃÝ £ ÃÕ ñÃò 7 óô Ò —” õ ì ª üö÷ ™ ùøq£@úûqç Þõ ëìí D
á p~ = p~ ” Hqç@á⠒q“ ˜ Hermite ýþ ™ 3
12
g~k = g~k12 = g~k21 .
p~ii = ehi | ~r | ii = 0
H=
à
Z¤C
X
~k
σij
”—ÿ èé ™ 3
h̄ω~k a~†k a~k + (E1 σ11 + E2 σ22 ) + h̄
DZ[ 4`
X
g~k (σ12 + σ21 )(a~k + a~†k ).
~k
Hamiltonian
2
X
i=1
ñò õ ì
ž ßÃü ú
21
: õ 2Ø :
Û
(λ = 1)
(3.4)
ž
E2 − E1 = h̄ω,
Á„Ø Û O Ì D Ú
| iihi |= σ11 + σ22 = 1,
1
1
E1 σ11 + E2 σ22 = h̄ω(σ22 − σ11 ) + (E1 + E2 )
2
2
(E1 + E2 )/2
3ø FG 6


 1 0 
σz = σ22 − σ11 =| 2ih2 | − | 1ih1 |= 
,
0 −1


 0 1 
σ+ = σ21 =| 2hi1 |= 
,
0 0


 0 0 
σ− = σ12 | 1ih2 |= 
.
1 0
„ñ¨µ@±²³´
§3.3
Ü Ý ÷ 6


H=
X
 0 
| 1i =  
1


 1 
| 2i =  
0
~k +
+
~k −
~k −
+
~k +
X
~k
§3.3

 0 0 
| 1ih1 |= 

0 1
)
53


 1 0 
| 2ih2 |= 

0 0
(3.5)
~k
[σij , σkl ] = σil δjk − σkj δli ,
(3.6a)
[σ− , σ+ ] = −σz ,
(3.6b)
[σ− , σz ] = 2σ− ,
(3.6c)
[σ+ , σz ] = −2σ+ .
(3.6d)
û gº ö ÷ Haliltonian Ý@ Œ
Oèé 6 ÒÚ Ý O
a σ :
O 6 ÒÚ "! Ý O.
a σ :
RWA )
O 6 ÒÚ Ý O
a σ :
Oèé 6 ÒÚ ! Ý O ž
a σ :
: õ Jû&) È ãä (RWA)  3
H=

X
1
h̄ω~k a~†k a~k + h̄ωσz + h̄
g~k (σ+ + σ− )(a~k + a~†k ),
2
æ 5 3 ¥  õ ÷çYú
~k
(Jaynes-Cummings
# ½ ÎÒ$% (H& È
# ½ ª ÎÒ$% ” H(
'˜
™
RWA
X
1
g~k (σ+ a~k + σ− a~†k ).
h̄ω~k a~†k a~k + h̄ωσz + h̄
2
(3.7)
~k
*,+,-,.,/,0,12*„{„|„‚„~„3 „€„
(Jaynes-Cummings
+
)
ç : MÜÝÞöKÒÚMÛ O gº ö÷ D éên Œ
45
H = H0 + HI
(3.8a)
1
H0 = h̄ω0 a† a + h̄ωσz ,
2
(3.8b)
HI = h̄g(σ+ a + a† σ− ).
(3.8c)
í ­8ï ® - ¯°±²³´qµ@¶·°&¸
q
#6 õ Jaynes-Cummings Ü7 (J-C Ü7 ) å„æ@D Hamiltonian ? 5 £ Õ8 ÷ 7 &)
È ã ä (RWA) áâãä#9 Ü7¤¥: ]; V ž û gº ö÷ +Ž n
54
h ÷
Baker-Hausdorff
V = eiH0 t/h̄ HI e−iH0 t/h̄ ,
<= 3H
α2
[A, [A, B]] + · · · ,
2!
eαA Be−αA = B + α[A, B] +
¤>
eiω0 a
: õ
4 5 ∆=ω−ω
JBA4 D; CBVE K £ Õ ;
WJL܌ É7ʞ
û gº ö÷ @Ž
0
† at
ae−iω0 a
† at
= ae−iω0 t ,
eiωσz t/2 σ+ e−iωσz t/2 = σ+ eiωt ,
V = h̄g(σ+ aei∆t + a+ σ− e−i∆t ),
(3.9)
è˜?@ (detuning) ž
>D gº ö÷ Hamiltonian ” gú öÃ÷ @Ž ™ ¤ ¥ ÷FGHI D W
íM WJ
Schrödinger WXN å
∂
ih̄ | ψi = V | ψi,
∂t
X
| ψ(t)i =
[c1,n (t) | 1, ni + c2,n (t) | 2, ni],
(3.10a)
(3.10b)
45 | i, ni OôÛ O û i ÒÚ ; ÝÞ n 9 Ý O (i = 1, 2) DPÌ "Q : õ û g º ö
÷  ŽÃ î@? c õBRSÃÉÃÊ íM ž Q RWA ¤ÃC gú ö÷ Hamiltonian ” 3.9 ™ =
Ð ÒBTBU | 2, ni | 1, n + 1i ÌWV D èÃé : õA = XÐY ; V c c Z 5WV
D[ ` X D V ž Q WX (3.9) (3.10a,b) ¤>
n
i,n
2,n
√
i∆t
1,n+1


, 
ċ2,n = −igc1,n+1 n + 1e
√

ċ1,n+1 = −igc2,n n + 1e−i∆t , 
V ?WX >
n
h
Ωn t
2
c1,n+1 (t) = c1,n+1 (0) cos
n
h
c2,n (t) = c2,n (0) cos Ω2n t −
+
i∆
Ωn
i∆
Ωn
sin
sin
Ωn t
2
Ωn t
2
i
i
−
−
(3.11)
√
2ig n+1
c2,n (0) sin
Ωn
√
2ig n+1
c1,n+1 (0) sin
Ωn
Ωn t
2
Ωn t
2
o
o
e
−i ∆t
2
∆t
ei 2 ,


, 


(3.12a)
„ñ¨µ@±²³´ (Jaynes-Cummings )
4W5 Ω = ∆ + 4g (n + 1) ž ßÃüB\B] õÃÛ O û | 2i Ì c = c (0) # ^ c (0) Ð õÝÞDÉÊ íMn
§3.3
2
n
2
2
2,n
55
n
c1,n+1 (0) = 0,



(3.12b)
n
h
c2,n (t) = cn (0) cos
î ˜
î@?
c1,n+1 (t) =
|ci,n (t)|2
(i = 1, 2)
i
∆t
Ωn t
− Ωi∆n sin Ω2n t ei 2 ,
2
√
∆t
−cn (0) 2ig Ωnn+1 sin Ω2n t e−i 2 .
Oô û t B5 _ ÝÃÞÝ O ì ˜
n


Û O û
| ii
ÌÃDÃÉÃÊ p(n) = |c1,n (t)|2 + |c2,n (t)|2
(3.13)
"
2
#
2 Ωn t
∆
Ωn t
4g n
Ωn−1 t
2
2
2
= ρnn (0) cos
+
sin
+ ρn−1,n−1 (0)
sin
2
Ωn
2
Ω2n−1
2
8BOO û B5 _ t úùAÃb B9\] n Ý O ÃD ÉÃ: Êg`t 4a5 3H ρ
Ý DÉʞ
5 ÝÞ
Ì
ρnn (0) =
nn (0)
hnin e−hni
n!
¤ ¥œc t 5 _d : ª æ De ìf ênH ª æ D
h ) O ì
X
W (t) =
(2.12b)
n
89¤>
∞
X
õÃÝÃÞ û \B] 5 A
= |cn (0)|2
∆
hni
g
HúÑ D VgV ž
∆2 4g 2 (n + 1)
+
cos(Ωn t) .
W (t) =
ρnn (0)
Ω2n
Ω2n
n=0
i Dõ 3H 8 \] 5_ÝÞ ûlm Ì (H
W (t) =
ρnn = δn0 )
9
(3.14)
[|c2,n (t)|2 − |c1,n (t)|2 ],
n
(3.15)
(3.16a)
j
n
o
1
2
2
2
2 12
∆
+
4g
cos[(∆
+
4g
)
t]
.
∆2 + 4g 2
(3.16b)
H j>k! Rabi íl ž # öm ÕÖn o c@Dp ª æ ž #6qrstuv TUw L
9x wy t ž{z Á|} c{~€‚ƒ„{€‡†Û„~Žˆ #‰ m ÕÖn o c{w x Ëw
Š ”Œ‹š™Ž ílq ª æ wž
bB t ˆ t ˆ t B‘ OB’ Š  íl ˆ“ƒ†Ûw”aVˆ“•– hni 1 ”Bˆ #B—
s ¤¥Ÿ (2.16a) ˜œc™ Œ
R
c
r
tR ∼
1
1
=
1 ,
Ωhni
(∆2 + 4g 2 hni) 2
(3.17a)
šq­› ® - ¯ °±²³´qµ@¶·°œ¸
56
î ˜
Poisson
 ”ˆ
∆n =
p
hni,
žŸ
(Ωhni+√hni − Ωhni−√hni )tc ∼ 1,
q
1
Ωhni+ hni − Ωhni−√hni
21
1
∆2
≈
1+ 2
,
2g
4g hni
tc ∼
æ ”ˆ î ˜
√
ž
(Ωhni − Ωhni−1 )tr = 2πm,
(3.17b)
(m = 1, 2, · · · )
p
12
2πm hni
2πm
∆2
.
tr =
≈
1+ 2
Ωhni − Ωhni−1
g
4g hni
¡ J¢Œ Heisenberg œ ¡ J
Q J-C Ü7 o c w Hamiltonian
Heisenberg
¡£ ¤¥¤>
1
ȧ = [a, H] = −iω0 a − igσ− ,
ih̄
σ̇− = iωσ− + igσz a,
b ½ 9¥¦ õ s œ t
(3.17c)
σ̇z = 2ig(a† σ− − σ+ a).







(3.18)






N = a † a + σ + σ− ,
1
c = ∆σz + g(σ+ a + a† σ− ).
2
¤¥¦q§ Œ
[N, H] = [c, H] = 0,
4W5 N OB’ÃÛ t - Þ § Ñ wÃðB¨ k œ ˆ c OB’ B© õª ž Q
σ− (t) = [σ+ (t)]†
(3.19b)
(3.18)
«
(3.19a,b)
sin κt
sin κt
= e
e
cos κt + ic
σ− (0) − ig
a(0) ,
κ
κ
sin κt
sin κt
−iω0 t ict
a(t) = e
e
cos κt − ic
a(0) − ig
σ− (0) ,
κ
κ
−iω0 t ict
(3.19a)
¤B>
(3.20a)
(3.20b)
¬­¯°¨µ@±²³´
4 5 κ ® Lõª¯ t ˆ Á Ÿ
§3.3
(Jaynes-Cummings
∆2
κ=
+ g 2 (N + 1)
4
«
° >±
(3.21a,b)
12
)
57
,
(3.21a)
[c, κ] = 0.
”ˆ ÷ ±²
(3.21b)
∆2
c =
+ g 2 N,
4
2



(3.22)

gσz a = 2cσ− + ∆σ− − ga. 
Q = (3.20a,b) ˆ³>±² ° Heisenberg ´µ¶ ½ ÒÚÛ t ‰x Ü ÝÞ· º ö ÷¸ «wÛ
t «ÝÞ ¯ t w ¹º ˆ 4» w s¼ Ò½ #—ºa5 ; º {c ™¾ ß h ) t ª W (t) ˆ³Ÿ
W (t) = h2, α | σz (t) | 2, αi
= 2h2, α | σ+ (t)σ− (t) | 2, αi − 1,
45¿ Õ úû \] ”Û tÀ ¨ kÁ | 2i zÂà À ·Ä Á | αi ¾
Å ÷ Heisenberg ´µ¶w ¯"Æ ¡ J Ÿ L 9Ç"È w"ÉÊ"ˆÌËÍ"ÎÏ ; >Ð"”·"ÑÒ
ª ¾yӈ ÅÔ (2.20a,b) ÕÖ×ØcÙÚ - ÙÚ·ÑÒ ª
h2, α | σ+ (t)σ− (τ ) | 2, αi
∞
X
|α|2n
1
Ωn−1 τ
i∆
Ωn−1 τ
−iω0 t−|α|2
= e
×
cos
−
sin
2
n!
4Ω
2
2Ω
2
n−1
n
n
Ωn τ
τ
Ωn (τ + 2t)
2 −i Ωn
2
i
2
× (Ωn + ∆) e 2 + (Ωn − ∆) e 2 + 8g (n + 1) cos
.
2
¡ J FÛ"Ü Š
qÜ
B
Ó
B
Þ
B
§
ß
ì ² Ü Š ”íïîÝ
ðWñ ˆ
V (t)
¯ t ¡ J
”
V
[
Ý
Š wˆàËáŸâ㈠K ¡ J qäåx w¾àæ
¯ t
çBèBzBéBˆàêë o
V
U (t) = exp −i t ,
h̄
q ° ·BòBó Ô ´µ¶w
J-C
Hamiltonian
ˆôæ BõBö ôˆ Ë
V (t) = h̄g(σ+ a + a† σ− ).
(3.23)
∆=0
wB÷BøBzBéBˆôŸ
(3.24)
šù› ú - û üýþÿüœ
58
ÅÔ Ñ§
(σ+ a + a† σ− )2l = (aa† )l | 2ih2 | +(a† a)l | 1ih1 |
†
(σ+ a + a σ− )
Í
2l+1
† l
†
†
l



p
√
cos(gt a† a + 1) | 2ih2 | + cos(gt a† a) | 1ih1 |
√
√
†
sin(gt a† a + 1)
† sin(gt a a + 1)
√
√
a | 2ih1 | −ia
| 1ih2 | .
−i
a† a + 1
a† a + 1
u(t) =
(3.25)

= (aa ) a | 2ih1 | +a (a a) | 1ih2 |, 
(3.26)
| ψ(t) >= U (t) | ψ(0)i,
«
| ψ(0)i =
•Ÿ
| ψ(t)i =
q Ÿ
∞
X
n=0
∞
P
n=0
cn (0) | 2, ni
(
Á ®¨ Á
| 2i)
√
√
cn (0)[cos(gt n + 1) | 2, ni − i sin(gt n + 1) | 1, n + 1i],
√
c2,n (t) = h2, n | ψ(t)i = cn (0) cos(gt n + 1),
√
c1,n+1 (t) = −icn (0) sin(gt n + 1).
pÞ ‰ ¡ì w q w –
∆=0
” ¾
"!$#$%$& Weisskopf-Wigner '(
)+*+,+- ÂÃ÷.¶w/Õ01 t wîݾ234”1 tÀ 5 Õ0 | 2 > Á ˆ
Ÿ
§3.4
V = h̄
X
[g~k∗ (~r0 )σ+ a~k ei(ω−ω~k )t + h.c.]
(3.27a)
~k
«
| ψ(t)i = c2 (t) | 2, 0i +
467
c2 (0) = 1,
X
~k
c1,~k | 1, 1~k i,
c1,~k (0) = 0.
(3.27b)
(3.27c)
­ 89ûü:<;=> Weisskopf-Wigner œ
·òó Ô ´µ¶w Schrödinger ¡£
§3.4
i
| ψ̇(t)i = − V | ψ(t)i.
h̄
Í
ċ2 (t) = −i
ð ñ
Aº
Í
X
~k
Í
ðñ
|g~k (~r0 )|
Z
2
g~kij = −
IJ
ðñ ˆ
i(ω−ω~k )t
g~k (~r0 ) = g~k e−ik·~r0 ,
BC ±DE§ ª wFGH®
T
~k
~
(2.28)
θ
®
(2.29)
‰K ÃwLM¾ON”ˆQP
X
~k
R ®
V
→2
(2π)3
4~
p21
ċ2 (t) = −
2
(2π) 6h̄0 c3
R∞
ω~k ≈ ω,
dω~k
0
Z
∞
−∞
q 1 t w?@ ) ¾
t
0
dt0 ei(ω−ω~k )(t−t ) c2 (t0 ),
dφ
0
Z
Z
π
sin θdθ
0
∞
0
(3.29)
0
p~ij · ˆ~k E~k
h̄
2π
ÍV R ®
US 
(~r0
(3.28)



R ®S  ˆŸ
~k
Z




c2 (t),
ω~k 2
p~ cos2 θ
2h̄0 V 21
P
|g~k (~r0 )|2 =
p~ij
(3.27d)
g~k∗ (~r0 )ei(ω−ω~k )t c1,~k (t),
P
ċ1,~k (t) = −ig~k (~r0 )e
ċ2 (t) = −
ðñ
59
dω~k ω~k3
R∞
−∞
Z
Z
∞
k 2 dk,
0
t
0
dt0 ei(ω−ω~k )(t−t ) c2 (t0 ),
0
dω~k
ˆ
ω~k3
ÍV Ô
ω3
(3.30)
WX ˆ T ˆ
0
dω~k ei(ω−ω~k )(t−t ) = 2πδ(t − t0 ),
Γ
ċ2 (t) = − c2 (t),
2
(3.31a)
1 4ω 3p~221
.
4π0 3h̄c3
(3.31b)
Γ=
šù› ú - û üýþÿüœ
60
T
Ë À
ρ22 = |c2 (t)|2 = e−Γt ,
¨ Á w1 t PVY ª øHZ[ˆ ð\] ®
¶^ Aº
c1,~k(t)
1
.
Γ
τ=
¾QP W_ ¡£ Í
Z
t
0
0
c1,~k (t) = −ig~k (~r0 )
dt0 e−i(ω−ω~k )t −Γt /2
0 −i(ω−ω~ )t−Γt/2 k
1−e
= g~k (~r0 )
,
(ω~k − ω) + iΓ/2
T
| ψ(t)i = e
3  Âà Á
− Γt
2
| 2, 0i+ | 1i
X
g~k e
t Γ−1
~k
ˆŸ
X
g~k
1 − e−i(ω−ω~k )t−Γt/2
| 1~k i
(ω~k − ω) + iΓ/2
~k
Γt
e−ik·~r0
| 1~k i
(ω~k − ω) + i Γ2
| ψ(t)i = e− 2 | 2, 0i+ | 1i
¶^ Aº
−i~k·~r0
~
| γ0 i =
–
(3.31c)
→ | 1i | γ0 i.
t Γ−1
X
~k
~
g~k e−ik·~r0 ·
1
| 1~k i
(ω~k − ω) + i Γ2
”ÂÃw ` ·ÑÒ ª
G(1) (~r, ~r, t, t) = hψ | E (−) (~r, t)E (+) (~r, t) | ψi
= hγ0 | E (−) (~r, t)E (+) (~r, t) | γ0 i
ðñ Ô ±²
«
T
= hγ0 | E (−) (~r, t) | 0ih0 | E (+) (~r, t) | γ0 i,
∞
X
n=0
E (−) (~r, t)
∞
P
n=0
| nihn |= 1,
| n >= 0
ˆa I ®bŸ
∆
1
 t G(1) (~r, ~r, t, t) = |h0 | E (+) (~r, t) | γ0 i|2 = |Ψγ (~r, t)|2 .
(3.31d)
c úü9d
¶ ^ Aº Ψ (~r, t) Û
§3.5
61
γ
h0 | E
(+)
(~r, t) | γ0 i =
r
=
r
P A Ý®S  ˆQe BC ±
~0
1
h̄ X
e−ik ·~r0
~0
| 1~k i
h0 | ωk~20 ak~0 e−iωk~0 t+ik ·~r g~k
20 V
(ω~k − ω) + i Γ2
~k,k~0
1
h̄ X 21 −iω~k t i~k·(~r−~r0 )
ω~k g~k e
e
.
20 V
(ω~k − ω) + i Γ2
~k
~k = k(sin θ cos φx̂ + sin θ sin φŷ + cos θẑ)
T
ðWñ
h0 | E
(+)
icp21 sin η
(~r, t) | γ0 i =
8π 2 0 4r
Z
∞
0
dk · k 2 (eik4r − e−ik4r )
e−iω~k t
,
(ω~k − ω) + i Γ2
°
g FB’ _+hi ˆ I f+H ñ ñ s¿ e r
z
Í
V
j
k
l
¾
Wigner mno hpq
ˆ O3T ω w R ÝtAuº ˆÌË ω ≈ ω ˆ v VP
X ˆQePS  ¶vwx± −∞ ˆ ˆêëby
S 
−i(k∆r−ω~k t)
∆r = |~r − ~r0 |.
2
~k
Z
∞
−∞
dω~k
~k
Ô
Weisskopfω~k2
ω2
W
e−iω~k t+iωk 4r/c
.
(ω~k − ω) + i Γ2
† R Ò ª wz ª 3 p ™ Aº f{S  Í
ðñ
θ
T
h0 | E
(+)
Heaviside
E0
∆r
∆r
Γ
(~r, t) | γ0 i =
θ t−
exp −i t −
ω−i
,
∆r
c
c
2
`| Ò ª ¾ T
ω 2 p21 sin η
E0 = −
,
4π0 c2 ∆r
r0
|E0 |2
~r − ~r0 −Γ t− ~r−~
c
G (~r, ~r, t, t) =
θ t−
e
,
|~r − ~r0 |2
c
(1)
(3.32)
(3.33)
ñ ÍV}"~"ˆ `O| Ò ª ŠOT ÂO~OO€}O‚OƒO„ , O}† ÂOƒ O‡Oˆ €O‰ OŠ
½
f
H
‹ ¾
Œù ú - û üýþÿüŽ
62
´
3.2:
Â0‘
’ “”
ÓÔ´ (3.2) •Q0‘è–Õ01€O/æÕ0—˜"晚m›o h Z[œ
òó ´µ¶€ Hamiltonian ž
§3.5
V
= h̄
Xh
(1)
∗
g3,
r0 )σ+ a~k ei(ω32 −ω~k )t
~k (~
+ h.c.
~k
+h̄
ðñ
Xh
q~
(1)
σ+ =| 3ih2 |
•
(2)
σ+ =| 2ih1 |
¤¥
(3.34)
•
(3.35)
~k
c2,~k | 2, 1~k i +
¦·òó Ô§¨© €
ċ3 = −i
X
X
~k,~
q
Schrödinger
Γ2 .
·
(3.34)
c2,~k,~q | 2, 1~k , 1q~i.
(3.35)
+ h.c. .
¾QŸ T •Q¡€ Á Ò¢x£ž
X

i
(2)
∗
g2,~
r0 )σ+ aq~ ei(ω21 −ωq~)t
q (~
| ψ(t)i = c3 (t) | 3, 0i +
i
Γ1
¤¥ •Íª
∗
g3,
r0 )c2,~k ei(ω32 −ω~k )t ,
~k (~
(3.36a)
X
(3.36b)
~k
ċ2,~k = −ig3,~k (~r0 )c3 e−i(ω32 −ω~k )t − i
∗
g2,~
r0 )c1,~k,~qei(ω21 −ωq~)t ,
q (~
q~
ċ1,~k,~q = −ig2,~q (~r0 )c2,~k e−i(ω21 −ωq~)t
(3.36c)
c úü9d
«¬­ ޕ BC®¯
§3.5
Weisskopf-Wigner
63
°± ©
Γ1
∗
−i g3,
r0 )c2,~k ei(ω32 −ω~k )t = − c3 ,
~k (~
2
~k
P ∗
Γ2
−i g2,~
r0 )c1,~k,~q ei(ω21 −ωq~)t = − c2,~k .
q (~
2
q~
P
Ÿ T Í
Γ2
c ~,
2 2,k 




−i(ω21 −ωq~ )t

= −ig2,~q (~r0 )c2,~k e
.
ċ2,~k = −ig3,~k (~r0 )e−i(ω32 −ω~k )t−
fH•ÍV º
(3.37)











Γ1
ċ3 = − c3 ,
2
ċ1,~k,~q





Γ1
2
t
−
Γ1
(3.38)
Γ2
ei(ω~k −ω32 )t− 2 t − e− 2 t
,
c2,~k (t) = −ig3,~k (~r0 )
i(ω~k − ω32 ) − 21 (Γ1 − Γ2 )
¦
(3.39)
1
~
c1,~k,~q (t = ∞) = g3,~k g2,~qe−i(k+~q)·~r0 ·
i(ω~k − ω32 ) − 21 (Γ1 − Γ2 )
1
1
−
×
i(ω~k + ωq~ − ω31 ) − 21 Γ1 i(ωq~ − ω21 ) − 21 Γ2
~
²
=
t Γ1−1 , Γ−1
2
³ •
−g3,~k g2,~qe−i(k+~q)·~r0
[i(ω~k + ωq~ − ω31 ) − 21 Γ1 ][i(ωq~ − ω21 ) − 21 Γ2 ]
c3 → 0, c2,~k → 0,
.
(3.40)
Ÿ T Â ÁR ž
~
| γ, φi =
X
~k,~
q
−g3,~k g2,~qe−i(k+~q)·~r0
[i(ω~k + ωq~ − ω31 ) − 21 Γ1 ][i(ωq~ − ω21 ) − 21 Γ2 ]
| 1~k , 1q~i.
(3.41)
64
Œù ú - û üýþÿüŽ
´¶µ¸·
“ - ¹º»¼&$½$¾$¿$'$(
² /BÕ+0+1++€Õ0í2ÀÁÂÃÂÀĜG ® õöÅ Ï° õö • ÍV ÔÆ Õ0
³
Ô
Æ
¯
Î
p
1++ÁBÂBÃÇòó ¯ €K çèÈÊÉV© ËÌÍ
f• Ÿ Õ01Á"mnϞ 1/2 €+Ð
ß+ÔÑ+Ç+±+ҕN ³ • ÙÚ°± •³ËÓԀ iÕÔ Ö× Î Ÿ1¯ €Ø„•21ʐÁÓ Çò
óä €ÚÙ2Û+ñ ÁÜmnÏ+ž 1/2 €+Ý++Á+Þ ³+öß çÔÇèòó € p ¢àfáâãìÍ í³î •
Ó ß Ô €ÜmnϞ ö1/2
€QÝ++å++æ
•é1+Õ0€êëݐ¢ìå
±
ç
€ïðñžò Óà Rabi €÷øÍ Ô
Póôõ q ÓÁ1Çòó €ö r÷ Î p Í
- “ø&¹º»¼ Hamiltonian
ù K Ý  ¯ K Ô ñú à €Ðß Hamiltonian ža ä uDE1 p ß
³
§4.1
H=
1
~ r , t)]2 + eU (~r, t) + V (~r),
[~
p − eA(~
2m
ð ñ p~ ž äû Ãüýþ• A(~
~ r , t) •
I
€+Í ž p~ → −ih̄∇~ • e A
(4.1)
—˜F¤+ÿ ¥ - Ԁ  • V (~rR ) ž1
Schrödinger
€¥ψ(~r, t) R J ҕA ¤
• ³ •
ψ → ψ (~x, t) = ψ(~r, t) exp[iχ(~x, t)] ³ • Schrödinger
U (~r, t)
0
R •QUÑ
ðñ •
~ r, t)
A(~
•
¯ °± © •QÑ
i2
h̄2 h ~
e~
∂
∇ − i A(~r, t) + eU (~r, t) ψ(~r, t) = ih̄ ψ(~r, t)
−
2m
h̄
∂t
~ r , t) → A
~ 0 (~r, t) = A(~
~ r , t) + h̄ ∇χ(~
~ r, t),
A(~
e
h̄ ∂
U (~r, t) → U (~r, t) −
χ(~r, t),
e ∂t
U (~r, t)
—˜ž - Ԁ  •Ñ
~ = −∇U
~ − ∂A ,
E
∂t
~k · ~r 1(
1 ¯
~r0



(4.2)
(4.3)


~ =∇
~ × A.
~
B
(4.4)
)
i~k·(~r0 +~r)
~ r0 + ~r, t) = A(t)e
~
A(~
i~k·~r
~
= A(t)e
(1 + ~k · ~r0 + · · · )
i~k·~r0
~
~
≈ A(t)e
= A(t)(~
r0 , t)
65
(4.5)
Œ ú - û üýþÿŽ
66
VÑ
(
)
2
ie ~
∂
h̄2 ~
−
∇ − A(~r0 , t) + V (~r) ψ(~r, t) = ih̄ ψ(~r, t)
2m
h̄
∂t
J ³! É _ " 1 #$€
V (~r),
U%& " o h
W_
(3.6)
ñ)
(4.8)
~ r0 , t)]φ(~r, t),
ih̄φ̇(~r, t) = [H0 − e~r · E(~
(4.9)
H0 =
~ = −A
~˙
E
ÍQŸ T • J ³ Ð*€
ðñ
B}C}® }
p~2
+ V (~r),
2m
Hamiltonian
0
ðñ
) V+,
H = H0 + H1 ,
(4.10)
~ r0 , t),
H1 = −e~r · E(~
(4.11)
H 0 = H0 + H2 ,
(4.12)
T ¯ o h - © •Ìæ
T e~
− A(~
r , t) · ~r ÍON ³ •
1
Hamiltonian
5h̄,3 ) Í © ^õ q 6 7 ® Hamiltonian €
Hamiltonian
R 0 • ( χ(~r, t) =
i/. ¢ ó " / /
Á¤ 2 - ,-3 €}ü ) ( I ž E~ Á22O€24
Í (4.1)
2
e
~ r0 , t) + e A
~ 2 (~r0 , t),
p~ · A(~
m
2m
~
[~
p, A] = 0,
Coulomb
H2 = −
Á
® (4.13) H ³8! r & ® "
€ Í :; • A~ g t u• ) Vïjk•QŸ T
2
ih̄
(4.7)
ie ~
ψ(~r, t) = exp
A(~r0 , t) · ~r φ(~r, t)
h̄
ðñ
e& ® "
€67'
~ · A(~
~ r , t) = 0.
∇
U (~r, t) = 0,
(
(Coulomb)
(4.6)
o h 9 ¤ €¥R A
Schrödinger
ž
∂
e
~ r0 , t)]ψ(~r, t),
ψ(~r, t) = [H0 − p~ · A(~
∂t
m
e
~ r0 , t)
H = H0 − p~ · A(~
m
e
~ r0 , t)
H2 = − p~ · A(~
m
0





(4.13)
~ ·A
~=0
∇
T
(4.14)
(4.15)
<=>89ûü?<@úA ýþÿ
67
aì " €Çòó& Hamiltonian BCÈ T áâ Ns€•QU ¯ 1O
(3.11)  (3.15) H
€ÜT m Hamiltonian æ+™+€D | ii © ® €EFG T N€ÍIHJKL• ( ~r = 0 •
§4.2
Ÿ
0
Ÿ T
H1

H2
—˜ž
H1 = −e~r · E~ cos ω0 t,
H2 =
m
p~ = m~r˙ = [~r, H0 ]
ih̄
•QV¦
I JMN áâ õö ÷ø (
˜Í
§4.2
1.
]œ^ ¤_
`a
~
~ t) = − E sin ω0 t.
A(0,
ω0
~ t) = E~ cos ω0 t,
E(0,
e
p~ · E~ sin ω0 t,
mω0
H0 | ii = h̄ωi | ii
ω
hf | H2 | ii
= ,
hf | H1 | ii
ω0
ω 6= ω0 )
•Q/ 7 N€
Hamiltonian



| ψ(t)i = c2 (t) | 2i + c1 (t) | 1i, 



∂
ih̄ | ψ(t)i = H | ψ(t)i,

∂t





H = H0 + H1 .
2
P
i=1
)c
(ω = ωf − ωi )
(4.16)
T N€•QÑOu€P
QSRSTSUSVSWYXYZYQY[$“$øY\$¹$º$»$¼
H0 =
b(
)



h̄ωi σii , 
~
H1 = −e~r · E.
(4.17a)
(4.17b)



~ r , t) = êx E cos ω0 t,
E(~
ΩR
ċ2 = i e−iφ c1 ei(ω−ω0 )t ,
2
ΩR iφ −i(ω−ω0 )t
ċ1 = i e c2 e
.
2





(4.18)
68
Œ d - e fghijkŽl
`a
ΩR =
|~
p21 |E
,
h̄
(t)
p~21 = |~
p21 |eiφ ,
(0)
(t)
c2 = c2 eiω2 t ,
c1 = c01 eiω1 t ,
p~21 = e~r21 = p~∗12 ,
ω = ω2 − ω1 .
¯ c ® (4.18) m ³ • !onpq " exp[±i(ω + ω )t] r+•s pq " N Ïët°±r ()
u ‹ L•vÁâüwxyz|{Ïët°±}~{ ) ÍQŸ}€ (3.18) )c
0
c2 (t) = (a1 e
`a
c1 (t) = (b1 e
∆ = ω − ω0
•
Ω=
iΩt/2
iΩt/2
+ a2 e
+ b2 e
p
Ω2R + (ω − ω0 )2
−iΩt/2
−iΩt/2
•
ΩR
)e


, 
−i∆t/2
ðž

. 
(4.19)
 ç Ĝ•Q¦
Rabi


1

a1 = 2Ω
[(Ω − ∆)c2 (0) + ΩR e−iφ c1 (0)], 






1
−iφ
a2 = 2Ω [(Ω + ∆)c2 (0) − ΩR e c1 (0)], 
1
[(Ω
2Ω
Ÿ} ) u
)e
i∆t/2
(4.20)
−iφ


b1 =
+ ∆)c1 (0) + ΩR e c2 (0)], 





1
−iφ

b1 = 2Ω [(Ω − ∆)c1 (0) + ΩR e c2 (0)]. 
c
Ωt
i∆
Ωt
ΩR −iφ
Ωt
−
sin
+ i e c1 (0) sin
,
c2 (t) = e
c2 (0) cos
2
Ω
2
Ω
2
Ωt
i∆
Ωt
ΩR iφ
Ωt
−i∆t/2
c1 (t) = e
c1 (0) cos
+
sin
+ i e c2 (0) sin
.
2
Ω
2
Ω
2
i∆t/2
‚ƒ„
|c1 (t)|2 + |c2 (t)|2 = 1,
2
2
W (t) = |c2 (t)| − |c1 (t)| =
u ¦ †‡ˆ
∆2 − Ω2R
Ω2
sin
2
Ωt
2
+ cos
2
Ωt
2




(4.21)

. 

P~ (t) = ehψ(t) | ~r | ψ(t)i
iΩR
Ωt
i∆
Ωt
Ωt iφ iω0 t
= 2Re
p~21 cos
+
sin
sin
e e
.
Ω
2
Ω
2
2
< =>‰Šef?<@dA‹kghij
²Œ Ž ∆ = 0 ŽQÑ
§4.2
m ) Bz
2. ǔ•&–—
˜

(3.21)
69
W (t) = W (0) cos(ΩR t),
Rabi
Đ{‘w’“Í
∂
ih̄ | ψ(t)iI = V (t) | ψ(t)iI ,
∂t
V (t) = U0† H1 U0 (t),
~
H1 = −e~r · E,
i
U0 = exp − H0 t .
h̄
)c
i
UI (t) = T̂ exp −
h̄
}™šý›Ž b Ñ
i
T̂ exp −
h̄
œ|
pq
Z
t
V (τ )dτ
0
i
= 1−
h̄
H0 = h̄ω1 | 1ih1 | +h̄ω1
2ih2 |)
}
(4.22)










| ψ(t)iI = UI | ψ(0)iI
`a
T̂











ŽŸÑ
Z
t
0
Z
∞
V (τ )dτ
0
i
dt1 V (t1 )+ −
h̄
) už c
| 2ih2 | (
H0
U0 (t) = exp −i t
h̄
(4.23a)
2 Z
t
dt1
0
(4.23b)
Z
t1
0
dt2 V (t1 )V (t2 )+· · ·
H0n = (h̄ω1 )n | 1ih1 | +(h̄ω2 )n |
= e−iω1 t | 1ih1 | +e−iω2 t | 2ih2 |,
V (t) = −h̄ΩR U0† (e−iφ | 2ih1 | +eiφ | 1ih2 |)U0 (t) cos ω0 t
h̄ΩR −iφ
= −
[e
| 2ih1 | ei∆t + eiφ | 1ih2 | e−i∆t
2
+ e−iφ | 2ih1 | ei(ω−ω0 )t + eiφ | 1ih2 | e−i(ω−ω0 )t ],
exp[±i(ω + ω0 )t]
r (N Ï ët¡¢r ) )c
V (t) = −
(4.23c)
h̄ΩR −iφ
(e
| 2ih1 | +eiφ | 1ih2 |),
2
(4.24)
(4.25)
70
£ ¤ d - efghijk¥l
` a
Œ
!n¦§ } Ž ∆ = 0 {¨©ª«’¬ (­® )
2n
V
)c
V
(t) =
2n+1
2n
h̄ΩR
2
n
(| 1ih1 | + | 2ih1 |) =
h̄ΩR
(t) = −
2
2n+1
h̄ΩR
2
2n
,
(e−iφ | 2ih1 | +eiφ | 1ih2 |),
ΩR t
UI (t) = cos
(| 1ih1 | + | 2ih2 |)
2
ΩR t
+i sin
(e−iφ | 2ih1 | +eiφ | 1ih2 |).
2
Ž )c
¯° ›±²³ ´µ¶ Ž
œ·
(4.26)
| ψ(0)i =| 2i
ΩR t
ΩR t iφ
| ψ(t)i = UI (t) | 2i = cos
| 2i + i sin
e | 1i.
2
2
3.
¸¹º ° ›{» ˆ EF
ΩR t iφ
c1 (t) = h1 | ψ(t)i = i sin
e ,
2
ΩR t
c2 (t) = h2 | ψ(t)i = cos
.
2
¼½¾¿ rŽ&~À ¾¿ÁÂ
Ë 
`a
{Γ, ρ} = Γρ + ρΓ
ρ̇ij =
ª }
∂ρ
1
= [H, ρ]
∂t
ih̄
Γ
ÃÄŎÇÆ ½
hn | Γ | mi = γn δnm
È » ˆ ÁÂÉÊ
∂ρ
1
1
= [H, ρ] − {Γ, ρ}
∂t
ih̄
2
(3.27)
Ì u Ë ÍÎ ÉÊ ŽÏ
1 X
1X
(Hik ρkj − ρik Hkj ) −
(Γik ρkj + ρik Γkj ).
ih̄ k
2 k
(4.27)
(4.28)
Ð ¸¹º ° ›ÑҎ ½
ρ = | ψihψ |
= ρ22 | 2ih2 | +ρ11 | 1ih1 | +ρ21 | 2ih1 | +ρ12 | 1ih2 | .
(4.29a)
Ó ÔՉŠefÖÓ×d؋kghij
`a
§4.2
ρ22
ρ21
71



= h2 | ρ | 2i = |c2 (t)|2 






∗
= h2 | ρ | 1i = c2 (t)c1 (t) 
ρ12 =
ρ∗21
ρ11 = h1 | ρ | 1i = |c1 (t)|2
Æ ½Ù †‡ˆ
(4.29b)










P~ (z, t) = c2 (t)c∗1 (t)P~12 + h.c. = ρ21 (z, t)P~12 + h.c.
Ú
ρ̇22
ρ̇11
ρ̇21
`a
i
~ 12 − h.c.],
= −γ2 ρ22 + [~
p21 · Eρ
h̄
i
~ 12 − h.c.],
p21 · Eρ
= −γ1 ρ11 − [~
h̄
i
~ 22 − ρ11 ).
= −(iω + γ21 )ρ21 − p~21 · E(ρ
h̄
1
γ21 = (γ2 + γ1 )
2
γ2 = ha | Γ | 2i,
γ1 = h1 | Γ | 1i.
ÛÜÝÞßà° ›™{áâãäåŽ
ρ̇21







(4.29c)
(4.30)






ræ ½Ë †
ρ̇21 = − [iω + iδω(t) + γ21 ] ρ21
}
ρ21 (t) = exp −(iω + γ21 )t − i
Z
t
0
0
dt δω(t ) ρ21 (0).
çèé Á hexp[−i Z dt δω(t )]i ªëêì δω íîíï{ðñ8òŽ œ Ñ
òŽ δω(t) { Ë †ó γ ô Ž œ· Ì u ¦ ¯
t
0
0
0
0
(4.31)
hδω(t)i = 0
Ž
−1
ab
hδω(t)δω(t0 )i = 2γphδ(t − t0 )
`8a γ }~Àìõª ¯ δω(t) Ì uö Gauss ÷øù Êúû Ž } ˜
¦
Í{ § w (Moment theorem)
Ìüþýÿ­®
ph
Z t
0
0
exp −i
dt δω(t )
= exp(−γph t),
0
(4.32)
Gauss
ù Ê
(4.33)
£¤ d - e fghijk¥l
72
}
vŽ
(3.30)
ρ21 (t) = exp[−(iω + γ21 + γph )t]ρ21 (0)
Ë 
i
~ 22 − ρ11 )
ρ̇21 = −(iω + γ)ρ21 − p~21 · E(ρ
h̄
`a
(4.34)
}~À{¹º÷øù Ê { õª
γ = γ21 + γph
Maxwell-Schrödinger
§4.3
 ÄÅ
ÜÝ ¹º ° ›!{"#{ñ”• ö Ž `a$% { ° › ö » ˆ Á
ρ(z, t, t0 ) =
X
ραβ (z, t, t0 ) | αihβ |
$ À ° › & t (Ž ')
Ž
ρ (z, t, t ) }~À
µ * ñ”• ö {» ˆ ͋Î+ ª-, ρ (z, t , t ) = ρ È
`a
α,β
α, β = 1, 2
αβ
0
αβ
0
ρ(z, t0 , t0 ) =
X
. ½ ° › {/0Ì u (˜ $ À ° ›€ c ¬ŽÏ
α,β
ρ(z, t) =
`a
Z
(0)
αβ
0
z
Ž b
t0
& (0)
ραβ | αihβ | .
t
dt0 ra (z, t0 )ρ(z, t, t0 )
XZ t
=
dt0 ra (z, t0 )ρα,β (z, t, t0 ) | αihβ | .
(4.36)
−∞
(4.37)
−∞
} ° ›12ª Ù †‡ˆ 
α,β
ra (z, t0 )
(4.35)
Ð ¸¹º ° ›Ž-,
Z
t
dt0 ra (z, t0 )Tr[p~ˆρ(z, t, t0 )]
−∞
XZ t
=
dt0 ra (z, t0 )ραβ (z, t, t0 )~
pβα .
P~ (z, t) =
Ž È ½
α,β
p~21 = p~12 = p~
(4.38)
(4.39)
−∞
P~ (z, t) = p~[ρ21 (z, t) + ρba (z, t)].
(4.40)
§4.3 Maxwell-Schrödinger
}Ì c
ρ̇11
ρ̇21
çè ÜÝ5 † Ž ˜
73
i
~ 12 − h.c.),
= λ2 − γ2 ρ22 + (~
p · Eρ
h̄
i
~ 12 − h.c.),
= λ1 − γ1 ρ11 − (~
p · Eρ
h̄
i
~ 22 − ρ11 ),
= −(iω + γ)ρ21 − p~ · E(ρ
h̄
ρ̇22
`a
34
(0)
λ2 = r2 ρ22 ,
Maxwell
ÉÊ
~ ·B
~ = 0,
∇
Ú ‘6 ÉÊ
~ = 0 E
~ + P~ ,
D
Ì c
¯ }78Æ ¦
x, y




(4.42)
~
J~ = σ E,
(4.43)



~
~
∂E
∂2E
∂ 2 P~
+ µ0 0 2 = −µ0 2 ,
∂t
∂t
∂t
~ r , t) = E(z, t)êx ,
E(~
È ½
−
,
9:ŽÏ
(0)
~ = µ0 H,
~
B
~ + µ0 σ
∇ × (∇ × E)
(4.41)






λ1 = r1 ρ11 .
~
~ ×E
~ = − ∂B ,
∇
∂t
~
~ ×H
~ = J~ − ∂ D
∇
∂t
~ ·D
~ = 0,
∇







(4.44)
(4.45)
∂2E
∂E
1 ∂2E
∂2P
+
µ
σ
+
=
−µ
0
0
∂z 2
∂t
c2 ∂t2
∂t2



E(z, t) =
+ c.c. 



1
−i[ω0 t−k0 z+φ(z,t)]
P (z, t) = 2 P(z, t)e
+ c.c. 




i[ω0 t−k0 z+φ(z,t)]

P(z, t) = 2pρ21 e
.
1
E(z, t)e−i[ω0 t−k0 z+φ(z,t)]
2
;Ë ¡¢ ç ŽÏ
E˙ ω0 E,
∂z φ k 0 ,
∂z E k0 E,
Ṗ ω0 P,
φ̇ ω0 .
∂z P k0 P.
(4.46)
£¤ d - e fghijk¥l
74
Ì už z
Ú
}Ì c
`a
∂
1∂
+
∂z c ∂t
∂
1∂
−
+
∂z
c ∂t
∂
1∂
− +
∂z c ∂t
E = −µ0 σ
∂E
∂2P
− µ0 2 .
∂t
∂t
E ≈ −2ik0 E
∂E 1 ∂E
1
+
= −κE −
k0 ImP,
∂z
c ∂t
20
ω0 −
∂φ 1 ∂φ
+
= k0 −
k0 E −1 ReP.
∂z
c ∂t
c 20
c }7<=>õª
É κ Ê = σ/2
Ú (3.45) 6!~?A@ ÉÊ ŽBC 
(3.39)
EF ´ wx G{ HIª
0







Maxwell-Schrödinger
(4.47)
ÉÊ ªD}
JLKLM
€ Q¼
1.
RQ ,
[A, B] 6= 0
NOPN
Ž ½
e−αA BeαA = B − α[A, B] +
f (α) = e−αA BeαA
ŽŸ ½
f (α) =
`a
α2
[A, [A, B]] + · · ·
2!
∞
X
f (n) (0)
n=0
n!
αn ,
∂ n f (α)
|
∂αn α=0
f (0) (α) = e−αA BeαA ,
f (0) (0) = B
f (n) (0) =
f (1) (α) = e−αA (−AB)eαA + e−αA (BA)eαA
= −e−αA [A, B]eαA ,
f (1) (0) = −[A, B].
f (2) (α) = e−αA (A[A, B])eαA + e−αA (−[A, B]A)eαA
= e−αA [A, [A, B]]eαA ,
Ÿ
f (2) (0) = [A, [A, B]].
¦§
f (k) (α) = (−1)k e−αA [A, · · · [A, B] · · · ]eαA
Ž `a k
À Ð ƒ › Ž È ½
f (k+1) (α) = e−αA {(−1)k+1 A[A, · · · [A, B] · · · ]}eαA + e−αA {(−1)k [A, · · · [A, B] · · · ]A}eαA
Ï ½
` a
= e−αA (−1)k+1 [A, · · · [A, B] · · · ]eαA ,
(5.1)
m
f (k+1) = (−1)k+1 (0)(−1)k+1 [A, · · · [A, B] · · · ],
(5.2)
m
k+1
e−αA BeαA = B − α[A, B] +
À Ѓ ›Ž œ Ñ
α2
α3
[A, [A, B]] − [A, [A, [A, B]]] + · · ·
2!
3!
75
(5.1)
(5.2)
(5.3)
76
€ Q¼
2.
[A, B] 6= 0
Ž-W
[A, [A, B]] = [B, [A, B]] = 0
1
RQ ,
œ
Ž È ½
1
eA+B = e− 2 [A,B] eA eB = e 2 [A,B] eB eA .
È ½
f (α) = eαA eαB
(5.4)
df (α)
= eαA AeαB + eαA eαB B.
dα
[A, [A, B]] = [B, [A, B]] = 0
ŽŸ ½
e−αA BeαA = B − α[A, B],
 Ñ
BeαA = eαA {B − α[A, B]}
œ· Ž
df (α)
= eαA eαB {A − α[B, A]} + eαA eαB B
dα
= eαA eαB {A + B + α[A, B]}
Ï
. u
œ
£S¤ TUVT
ŽŸ
(5.6)
= f (α){A + B + α[A, B]},
(5.7)
df (α)
= {A + B + α[A, B]}dα
f (α)
(5.8)
f (α) = f (0)e(A+B)α+
f (0) = 1
(5.5)
f (α) = eα(A+B)+
α2
2
[A,B]
Ž . u
α2
2
[A,B]
.
(5.9)
1
f (1) = eA eB = eA+B+ 2 [A,B] ,
œ·
3.
€RQQ-X[a,Y eé Á
,
−αa† a
1
1
eA+B = e− 2 [A,B] eA eB = e 2 [A,B] eB eA .
†
] = (e−α − 1)e−αa a a
ae
−αa† a
ª
†
ª
eαa a ae−αa
†a
∆
= e−αA BeαA
(5.10)
77
`a
A = −a† a,
È ½
B=a
[A, B] = [−a† a, a] = a
[A, [A, B]] = [−a† a, a] = a
············
[A · · · [A, B] · · · ] = a
Ÿ
†
eαa a ae−αa
†a
œ· Ž
∞
X
(−α)n
1 2
α a +··· = a
= ae−α ,
2!
n!
n=0
= a − αa +
ae−αa
. u Ž
†a
(5.11)
†
= e−αa a ae−α .
†
†
†
[a, e−αa a ] = ae−αa a − e−αa a a
†
= e−αa a a(e−α − 1)
†
= (e−α − 1)e−αa a a.
òwÌ c
4.
†
†
[a† , e−αa a ] = (eα − 1)e−αa a a† .
€
(1)
(2)
R(3)Q ,
(1)
∂f
;
∂a†
∂f
[a† , f (a, a† )] = − ;
∂a
†
†
e−αa a f (a, a† )eαa a = f (aeα , a† e−α ).
P
f (a, a† ) =
cn,m (a† )n am
[a, f (a, a† )] =
n,m
Ž È ½
[a, f (a, a† )] =
`a
X
cnm [a, (a† )n ]am ,
n,m
[a, (a† )n ] = n(a† )n−1 .
(5.12)
(5.13)
78
£S¤ TUVT
Ÿ ½
[a, f (a, a† )] =
X
ncnm (a† )n−1 am =
X
cnm (a† )n [a† , am ]
n,m
(2)
[a† , f (a, a† )] =
∂f
.
∂a†
(5.14)
n,m
(3)
çè ÜZ
= −
†
†
†
†a
e−αa a f (a, a† )eαa a ,
e−αa a f (a, a† )eαa
˜
X
n,m
f (a, a† )
=
˜] m
Ìü
cnm (a† )n mam−1 = −
X
∂f
.
∂a
(5.15)
{[\mÌü
†
†
cnm e−αa a (a† )n am eαa a .
n,m
e−αA BeαA = B − α[A, B] +
†
e−αa a (a† )n eαa
†a
α2
[A, [A, B]] + · · ·
2!
= (a† )n − nα(a† )n +
∞
X
(−nα)m
= (a )
m!
m=0
α2 2 † 2
n (a ) + · · ·
2!
† n
= (a† )n e−nα .
(5.16)
òw
e
−αa† a m αa† a
a e
œ Ñ
m2 α 2
= a
1 + mα +
+···
2!
= am emα .
m
†
(5.17)
†
e−αa a (a† )n = (a† )n e−αa a e−nα ,
am eαa
. u
†
†a
e−αa a (a† )n am eαa
†
= eαa a am emα .
†a
= (a† )n am e−(n−m)α
= (a† e−α )n (aeα )m ,
(5.18)
79
Ÿ ½
†
e−αa a f (a, a† )eαa
†a
=
X
cnm (a† e−α )n (aeα )m
n,m
5.
? ˜^_ {
à Ì u ` !
Hamiltonian
(a) H = H
∞
P
n=0
(b) eiHt/h̄ =
En | nihn |,
∞
P
n=0
eiEn t/h̄ | nihn |
H | ni = En | ni,
Ú
∞
P
n=0
(a)
H=H
| nihn |
∞
X
n=0
(b)
Ÿ ½
(5.19)
1
H = h̄ω(a† a + )
2
€ Q
RQ2˜
= f (aeα , a† e−α ).
Hn =
∞
X
m=0
e
iHt/h̄
ŽÌü
| nihn |=
Em | mihm |
=
∞
X
n=0
!n
∞ n
X
it
=
En | nihn | .
∞
X
m=0
n
Em
| mihm |,
Hn
h̄
X it n
n
=
Em
| mihm |
h̄
n,m
"∞ #
∞
X
X iEm t n
=
| mihm |
h̄
m=0
n=0
n=0
=
=
∞
X
m=0
∞
X
n
| mihm | e
e
iEn t
h̄
iEm t
h̄
| nihn |
(5.20)
80
6.
£S¤ TUVT
˜ÉÊ
R ? ˜ a ™ {
~˜
1 ∂E
~˜
= ∇ × H,
c ∂t
~˜
1 ∂H
~˜
−
= ∇ × E,
c ∂t
Maxwell

ih̄
u Ú
~˜ = √0 E
~
E
Ž
(5.21)
~˜ = 0.
∇·H
(5.22)
ÉÊ Ì u` !b ç <m




~γ   0
−c~s · p~   ϕ
~γ 
∂  ϕ

=

,
∂t
χ
~γ
c~s · p~
0
χ
~γ

· ³
~˜ = 0.
∇·E
~˜ = √µ0 H
~
H

∇·
ϕ
~γ
vecχγ
ª bR
(5.23)


 = 0.
(5.24)
~ =∇×V
~,
~s · ∇V
`a


 0 0 0 




sx =  0 0 −1  ,




0 1 0
RQ2˜


 0 0 1 




sy =  0 0 0  ,




−1 0 0
(5.25)


 0 −1 0 




sz =  1 0 0  . (5.26)




0 0 0
~ = s x ∂x + s y ∂y + s z ∂z
~s · ∇






 0 0 0 
 0 0 1 
 0 −1 0 












=  0 0 −1  ∂x +  0 0 0  ∂y +  1 0 0  ∂z












0 1 0
−1 0 0
0 0 0


 0 −∂z ∂y 




=  ∂z
0 −∂x 




−∂y ∂x
0
81
Ìü
~ V~
~s · ∇

 0 −∂z ∂y


=  ∂z
0 −∂x


−∂y ∂x
0


 ∂ y Vx − ∂ z Vy 




=  ∂ z Vy − ∂ x Vz 




∂ x Vz − ∂ y Vx


  Vx 




  Vy 




Vz
= ~ex (∂y Vz − ∂z Vy ) + ~ey (∂z Vx − ∂x Vz ) + ~ez (∂c Vy − ∂y Vx ).
òŽ ˜
(5.27)
~ex ~ey ~ez
~ ×V
~
∇
=
∂x ∂y ∂z
Vx Vy Vz
= ~ex (∂y Vz − ∂z Vy ) + ~ey (∂z Vx − ∂x Vz ) + ~ez (∂c Vy − ∂y Vx )
Ì c
~V
~ =∇
~ ×V
~
~s · ∇
(5.29)
~ =∇
~ ×.
~s · ∇
(5.30)
1 ∂ϕ
~γ
~ ×χ
~ χγ ,
=∇
~ γ = ~s · ∇~
c ∂t
(5.31)
c ώ
}Ž ½
d Ñ
e ¢fŽ ˜
Ì c
ih̄
(5.28)
∂
ϕ
~ γ = −c~s · p~χ
~γ,
∂t
−
~
(~
p = −ih̄∇)
1 ∂~
χγ
~ ×ϕ
~ ϕγ ,
=∇
~ γ = ~s · ∇~
c ∂t
ih̄
∂
χ
~ γ = c~s · p~ϕ
~γ.
∂t
82
£S¤ TUVT
œ Ñ
ih̄
òŽ ˜
Ì c` ÍÎ < m 





~γ   0
−c~s · p~   ϕ
~γ 
∂  ϕ

=

.
∂t
χ
~γ
c~s · p~
0
χ
~γ
~ ·χ
∇
~ γ = 0,


(5.32)
~ ·ϕ
∇
~ γ = 0.

  
~
ϕ
~γ   ∇ · ϕ
~γ   0 
~ ·
∇

=
 =   = 0.
~
χ
~γ
∇·χ
~γ
0
(5.33)
gh Ž-bi c ¬ ç è { É Ê jlk
1 ∂ϕ
~γ
~ ×χ
∇
~γ =
,
c ∂t
~ ·χ
∇
~ γ = 0,
1 ∂~
χγ
~ ×ϕ
∇
~γ = −
.
c ∂t
~ ·ϕ
∇
~ γ = 0.
çè  Maxwell ÉÊ c ¬m è {nÀ É Ê ª «’¬
p Ž E~˜ = √ E~ H~˜ = √µ H~ å Ë 
0
(5.35)
ÉÊ • Þ Ë ào
0
œ· Žrqs Ë t ¬
uv
Maxwell
(5.34)
~˜ (±)
~ ×H
~˜ (±) = 1 ∂ E ,
∇
c ∂t
˜
~ ·H
~ (±) = 0,
∇
~˜ H)
~˜
(E,
(~
ϕγ , χ
~γ)
~˜ (±)
~ ×E
~˜ (±) = − 1 ∂ H .
∇
c ∂t
˜
~ ·E
~ (±) = 0.
∇
{:ª ˜
~ E (~r, t) = h0 | E
~ (+) (~r, t) | ψγ i
Ψ
s
X (λ) h̄ω~
~
k
= h0 |
ˆ~k
a~ e−iω~k t+ik·~r | ψγ i
20 V k,λ
~k,λ
s
X (λ)
h̄ω~k
~
∼
h0 |
ˆ~k a~k,λ e−iω~k t+ik·~r | ψγ i
=
20 V
~k,λ
s
h̄ω~k
∆
=
ϕ
~ γ (~r, t),
20 V
~
ϕ
~ γ (~r, t) =
X
~k,λ
(λ)
ˆ~k h0
e−iω~k t+ik·~r
| a~k,λ √
| ψγ i
V
(5.36)
(5.37)
83
. u
20
~ (+) (~r, t) | ψγ i
h0 | E
h̄ω
r
2
~˜ (+) (~r, t) | ψγ i.
=
h0 | E
h̄ω
ϕ
~ γ (~r, t) =
uv ö ¬ Þ
r
~ (+)
~˜ (+) = √0 E
E
ª e ¢fŽÌw
r
2µ0
~ (+) (~r, t) | ψγ i
h0 | H
h̄ω
r
2
~˜ (+) (~r, t) | ψγ i.
=
h0 | H
h̄ω
χ
~ γ (~r, t) =
}Ž ½
r
2
~ ×E
~˜ (+) (~r, t) | ψγ i
h0 | ∇
h̄ω
r
2
1 ∂
~˜ (+) (~r, t) | ψγ i
=
· −
h0 | H
h̄ω
c ∂t
1∂
= −
χ
~ γ (~r, t),
c ∂t
~ ×ϕ
∇
~γ =
2
~ ×H
~˜ (+) (~r, t) | ψγ i
h0 | ∇
h̄ω
r
2
1∂
~˜ (+) (~r, t) | ψγ i
=
·
h0 | E
h̄ω
c ∂t
1∂
=
ϕ
~ γ (~r, t),
c ∂t
r
2
~ ·E
~˜ (+) (~r, t) | ψγ i = 0,
~ ·ϕ
∇
~γ =
h0 | ∇
h̄ω
r
2
~
~ ·H
~˜ (+) (~r, t) | ψγ i = 0.
∇·χ
~γ =
h0 | ∇
h̄ω
~ ×χ
∇
~γ =
}Ìw
r
~γ
1 ∂ϕ
~ ×χ
∇
~γ =
,
c ∂t
~ ·χ
∇
~ γ = 0,
1 ∂~
χγ
~ ×ϕ
∇
~γ = −
,
c ∂t
~ ·ϕ
∇
~ γ = 0.
(5.38)
(5.39)
84
£S¤ TUVT
c Ï
ih̄




~γ   0
−c~s · p~   ϕ
~γ 
∂  ϕ

=

,
∂t
χ
~γ
c~s · p~
0
χ
~γ

7.
xüŽ
ϕ
~γ
χ
~γ
ϕ
~˙ γ = c~s · ∇~
χγ ,

ϕ
~γ 
~ ·
∇

 = 0.
χ
~γ
½ b ç : Q
χ
~˙ γ = −c~s · ∇~
ϕγ ,
y ž z » ˆ{ É Ê -Ž | }R½
RQ-XY Ž ˜

χ
~˙ †γ = −c∇ϕ~γ † · ~s† .
~s† = −~s.



 0 0 −1 




†
sy =  0 0 0  = −sy ,




1 0 0


 0 1 0 




†
sz =  −1 0 1  = −sz ,




0 0 0
Ìü
Ψ†γ Ψγ
Ú
Ψ†γ ~v Ψγ
Ž uv
~s† = −~s.

(a)
Ψ†γ Ψγ =
(5.40)
(5.41)
ϕ
~˙ †γ = c∇χ~γ † · ~s† ,
 0 0 0 




†
sx =  0 0 1  = −sx ,




0 −1 0
çè~ €z

ϕ
~ †γ χ
~ †γ

(5.43)

 0 −~s 
~v = c 
.
~s 0

(5.42)
~γ 
∆
 ϕ
~ †γ · ϕ
~γ + χ
~ †γ · χ
~ γ = ρ.

=ϕ
χ
~γ
85
(b)
Ψ†γ ~v Ψγ =
=
(c)
Y €z
ϕ
~ †γ χ
~ †γ





 0 −c~s  




c~s 0
c~
χ†γ ~s c~
ϕ†γ ~s

ϕ
~γ
vecchiγ

~γ 
 ϕ


χ
~γ







∆
= c(~
χ†γ ϕ
~γ − ϕ
~ †γ χ
~ γ )~s = ~j.
∂ρ/∂t
∂ρ
= ϕ̇†γ ϕγ + ϕ†γ ϕ̇γ + χ̇†γ χγ + χ†γ χ̇γ
∂t
˜
~ γ,
ϕ̇γ = c~s · ∇χ
.€
~ γ.
χ̇γ = −c~s · ∇ϕ
~ †γ · ~s† = −c∇χ
~ †γ · ~s,
ϕ̇†γ = c∇χ
~ † · ~s† = −c∇ϕ
~ † · ~s.
χ̇†γ = c∇ϕ
γ
γ
œ Ñ
(d)
(e)
€z
‚ƒ
†
∂ρ
~ † · ~sϕγ + cϕ† ~s · ∇χ
~ γ + c∇ϕ
~ † · ~sχγ − cχ† ~s · ∇ϕ
~ γ
= −c∇χ
γ
γ
γ
γ
∂t
~ †γ · ~sϕγ + ϕ†γ ~s · ∇χ
~ γ + ∇ϕ
~ †γ · ~sχγ − χ†γ ~s · ∇ϕ
~ γ }.
= c{−∇χ
~ · ~j
∇
~ · ~j = c{∇χ
~ † · ~sϕγ + χ†γ ~s∇ϕ
~ γ − ϕ†γ ~s · ∇χ
~ γ − ∇ϕ
~ †γ · ~sχγ }.
∇
(c),(d)
æ µ„
~ · ~j + ∂ρ = 0,
∇
∂t
(5.44)
∂ †
~ · (Ψ† ~v Ψγ ) = 0,
(Ψγ Ψγ ) + ∇
γ
∂t
(5.45)
86
£S¤ TUVT
uv

8.

 0 −~s 
~v = c 
.
~s 0
y êù P ê ê ‡ ’ˆ‰ à ~v Š ‹Œ R
y žz çè Š ÉÊ
ê = ~k/k
i i i
1
(1) (1)
P
(2) (2)
~k ~k + ~k ~k +
b ö‘’“ õ
ÃÄŔ•
k, θ, φ
~k
ŽÏ
i êi êi
=1
ŽŽb ½
~k~k
= 1.
k2
(1)
(2)
ê1 = ˆ~k ,ê2 = ˆ~k ,
(5.46)
~k = k(sin θ cos φ, sin θ sin φ, cos θ),
¸À–— ٘ ‰ à 
(1)
ˆ~k ≡ (sin φ, − cos φ, 0),
(2)
ˆ~k ≡ (cos θ cos φ, cos θ sin φ, − sin θ),
y™š
(1) (1)
uv
žÚ ’ ª P
~v · ê = v
~v =
v ê Ìü
š RQ-X› Y œŽ  ˜
i, j
P
êi êi = 1
i
,
(a)
i
(5.47)
i
i i
i
X
êi êi =
Ÿ $ ' Á ›ª çè
(1)
ê1 = ˆ~k
ki kj
.
k2
(Cartesian)
~v ·
Ÿ
(2) (2)
~ki ~kj + ~ki ~kj = δij −
Ž
i
(2)
ê2 = ˆ~k
Ž
X
i
ê3 =
1 =
(~v · êi )~ei =
X
vi êi = ~v
i
Ž È ½
~k
k
X
êi êi
i
(1) (1)
(2) (2)
= ˆ~k ˆ~k + ˆ~k ˆ~k +
Ï
(1) (1)
(2) (2)
ˆ~k ˆ~k + ˆ~k ˆ~k +
~k~k
k2
~k~k
=1
k2
(5.48)
87
‘’ ç Ž ½
(b)
~k ≡ k(sin θ cos φ, sin θ sin φ + cos θ)
(1)
ˆ~k = (sin φ, − cos φ, 0)
(2)
ˆ~k = (cos θ cos φ, cos θ sin φ, − sin θ)
ŸÌw
(1) (1)
ˆky ˆky = cos2 φ;
(1) (1)
(1) (1)
ˆkx ˆky = − sin φ cos φ;
ˆkx ˆkx = sin2 φ,
(1) (1)
ˆkz ˆkz = 0,
(1) (1)
(1) (1)
ˆkx ˆkz = 0,
ˆky ˆkz = 0;
(2) (2)
ˆky ˆky = cos2 θ sin2 φ;
(2) (2)
(2) (2)
ˆkx ˆky = cos2 θ cos φ sin φ;
(2) (2)
ˆky ˆkz = − cos θ sin θ sin φ;
ky ky
= sin2 θ sin2 φ
k2
kx ky
= sin2 θ cos φ cos φ;
2
k
ky kz
= sin θ cos φ sin φ.
k2
ˆkx ˆkx = cos2 θ cos2 φ,
(2) (2)
ˆkz ˆkz = sin2 θ,
Ÿ ½
ˆkx ˆkz = − cos θ sin θ cos φ,
kx kx
= sin2 θ cos2 φ,
k2
kz kz
= cos2 θ,
2
k
kx kz
= sin θ cos φ cos φ,
k2
(1) (1)
(2) (2)
(2) (2)
ˆkx ˆkx + ˆkx ˆkx = sin2 φ + cos2 θ cos2 φ
= sin2 φ + (1 − sin2 θ) cos2 φ
= 1 − sin2 θ cos2 φ,
Ÿ
δxx −
kx kx
= 1 − sin2 θ cos2 φ,
k2
(1) (1)
(2) (2)
(1) (1)
(2) (2)
ˆkx ˆkx + ˆkx ˆkx = 1 −
òÌ š u¡ ¨©Ž-¢£Ž ½
kx kx
.
k2
~ki ~kj + ~ki ~kj = δij −
¤¥ mŽ-m û¦§¨©½ª —Ä û š R«-˜
(1) (1)
(2) (2)
~k ~k + ~k ~k +
ki kj
.
k2
~k~k
=1
k2
(5.49)
88
gh«¬ G­®¯¸°±‹²
êi êj
ý
êi , êj
£ S¤ TUVT
~  Ÿ³´ ‘’ ç Š¸ÀH‰µ « È ½
~k~k
(1) (1)
(2) (2)
l.h.s. = ~k ~k + ~k ~k + 2
k
→
−
r.h.s. = 1 : êi êj
!
: êi êj ,
¶ |‰‹²· È Ìü « Ÿ ½
(1) (1)
(2) (2)
~ki ~kj + ~ki ~kj = δij −
ki kj
.
k2
(5.50)