Ожигов Ю.С. Квантовые вычисления
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|e
j
i |λ
j
|
2
¯a ∈ H
|¯ai
{e
j
} h¯a|
|¯ai
¯a
¯
b
h¯a| · |
¯
bi =
P
j
¯
λ
a
j
λ
a
j
λ
j
¯a
¯
b
h¯a|
¯
bi ¯a h¯a|¯ai
¯a |¯aih¯a| = ρ
a
× N
H α
ρ
2
= |¯aih¯a|¯aih¯a| = ρ ¯a h¯a|¯ah= 1
{|e
j
i}
{|e
0
j
i}
Ψ
{|e
0
j
i}
|e
0
j
i |he
0
j
|Ψi|
2
| Ψi =
X
j
λ
j
|e
j
i, | Ψi =
X
j
λ
0
j
|e
0
j
i,
λ
j
= he
j
| Ψi
he
j
|
Ψ
0
U
t
1
Ψ
1
t
1
|Ψ
pol
i
|Ψ
pol
i = λ
ver
|e
ver
i+λ
hor
|e
hor
i |e
ver
i |e
hor
i
|hΨ
pol
| e
gor
i|
2
λ
gor
λ
ver
U
t
|Ψi
t U
t
|Ψi U
t
|kP sii
|Ψi
t he
j
| U
t
Ψi
Ψ U
t
U
t
H
0
U
t
= e
−iH
0
(t)
δt
U
t
≈ (U
δt
)
t/δt
= (e
−iH
0
(δt)
)
t/δt
= e
−iH
0
(δt)t/δt
= e
−iHt
H h
H
t t
0
|Ψ
t
i = e
−iHt
|Ψ
t
0
i
i
∂|Ψi
∂t
= H|Ψi
H λ I I
δx x
j
Ψ
i
∂Ψ(x
j
)
∂t
= AΨ(x
j
) + BΨ(x
j+1
) + CΨ(x
j−1
)
A, B, C
B = C λI
i
∂Ψ(x
j
)
∂t
= a(2Ψ(x
j
) − Ψ(x
j+1
) − Ψ(x
j−1
))
δx −→ 0
i
∂Ψ(x, t)
∂t
= a
∂
2
Ψ(x, t)
∂x
2
.
a −h
2
/2m h m
i
V (x)
p =
h
i
∂
∂x
p p
V (x)
ih
∂|Ψi
∂t
= (−
P
2
2m
+ V )|Ψi
P
2
= −h
2
(
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
) P =
h
i
(
∂
∂x
+
∂
∂y
+
∂
∂z
)
n
ih
∂|Ψi
∂t
= (
n
X
j,k=1
P
2
j
2m
j
+ V
j
+ V
j,k
)|Ψi
V
j
j V
j,k
j k
A = (A
1
, A
2
, A
3
, iA
0
)
¯p −→ ¯p −
e
c
A, E −→ E −eA
0
¯p = (p
1
, p
2
, p
3
)
ih
∂Ψ
∂t
= (
1
2m
(¯p −
e
c
¯
A) + V )Ψ.
E =
p
M
2
c
4
+ c
2
p
2
|h
∂Ψ
∂t
|, |eA
0
Ψ| |Mc
2
Ψ|
i
∂Ψ
∂t
= HΨ
H
H = (h
α,β
), h
α,β
= k(δ
αβ
−
k
α
k
β
k
2
),
¯
k = (k
1
, k
2
, k
3
) k = |
¯
k| f
¯
k
¯
kf
¯
k
= 0
rot
¯
E = −
∂H
∂t
, rot
¯
H =
∂E
∂t
, div
¯
H = div
¯
E = 0,
E = −
∂
¯
A
∂t
− grad A
0
, H = rot
¯
A
E =
R
E
k
a
i
¯
k¯r
d
3
k, H =
R
H
k
a
i
¯
k¯r
d
3
k
E
¯
k
d
dt
E
k
f
¯
k
+f
∗
¯
k
f
¯
k
−f
∗
−
¯
k
H = H
0
Ψ(t) = exp(−iHt)Ψ(t
0
)
H H
Ψ(t) = exp(−i
t
Z
t
0
H(t)dt)Ψ(t
0
).
H Ψ
Ψ
0
= −i exp(−i
t
Z
t
0
H(t)dt)H(t)Ψ(t
0
).
H(t)Ψ(t
0
) H(t)
t
˜
H = H
0
+ H
1
(t)
H
0
t H
1
Ψ(t + t
0
) =
exp(−iH
0
t)Φ(t) Φ(t) Φ(0) = Ψ
0
=
Ψ(t
0
)
H = e
iH
0
t
H
1
(t)e
−iH
0
t
Φ
i
∂
∂t
Φ = HΦ, Φ(0) = Ψ
0
,
S
Φ(t) = S(t, t
0
)Φ(t
0
).
t = +∞, t
0
= −∞
S t
0
= 0
i
∂
∂t
S(t) = H(t)S(t), §(0) = I,
S(t)
I n − 1 S
n−1
n
S
n
= −i
t
Z
0
H(t)S
n−1
(t)dt
S =
∞
X
n=0
(−i)
n
t
Z
0
H(t
1
)dt
1
t
1
Z
0
H(t
2
)dt
2
. . .
t
n
Z
0
H(t
n
)dt
n
(−i)
n
Z
t≥t
1
≥t
2
≥...≥t
n
≥0
H(t
1
)H(t
2
) . . . H(t
n
)dt
1
dt
2
. . . dt
n
,
(−i)
n
n!
T {
t
Z
0
t
Z
0
. . .
t
Z
0
H(t
1
)H(t
2
) . . . H(t
n
)dt
1
dt
2
. . . dt
n
},
T { H(t
1
)H(t
1
) . . . H(t
n
)} = H(t
i
1
)H(t
i
2
) . . . H(t
i
n
),
t ≥ t
1
≥ t
2
≥ . . . ≥ t
n
≥ 0.
n H(t
i
), i = 1, 2, . . . , n
n!
S
S =
∞
X
n=0
(−i)
n
n!
T {
t
Z
0
t
Z
0
. . .
t
Z
0
H(t
1
)H(t
2
) . . . H(t
n
)dt
1
dt
2
. . . dt
n
},
(−1)
P
P
ω
0
= {a
0
, a
1
, . . . , a
m
}, ω
1
= {q
0
, q
1
, . . . , q
k
}
ω
0
a
0
ω
1
ω
0
q
0
q
k
q
k
a
i
, q
j
−→ a
k(i,j)
, q
l(i,j)
, (R, L)
i = 0, 1, . . . , m; j = 0, 1, . . . , k
a
i
q
j
a
k(i,j)
,
q
l(i,j)
k l
x ω
0
⊂ ω
0
ω
0
y x
T [T ]
T !x A [T ], x
T !x A
0
n ∃T : [T ] = n &T !n
T
0
n T
0
!n ∃T : n = [T ] &T 6!n
A
0
A
0
T
0
T
0
![T
0
] T
0
6![T
0
] A
A
0
n
T x
n
f
T
(n) T
n f
T
T
F
F