Κβαντική Μηχανική
201
description
Lecture Notes on Foundations of Quantum Mechanics (in Greek)
Transcript of Κβαντική Μηχανική
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1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 . . . . . . . . . . . . . 131.4.5 . . . . . . . . . . . . . . . . . . . . . . 151.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 . . . . . . . . . . . . . . . . . . . 18
2 A 212.1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 B 373.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.4 . . . . . . . . . . . . . . . . . . 46
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 . . . . . . . . . . . . 473.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.3 . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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4 574.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 . . . . . . 574.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.4 . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 775.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 785.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1 . . . . . . . . . . . . . . . . . . 875.5.2 . . . . . . . . . . . . . . . . . . . . . . . 895.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . 915.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.8.1 qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 1116.1 . . . . . . . . . . . . . . . . . . . . . . 111
6.1.1 . . . . . . . . . . . . . . . . . . . . . . . 1116.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 . . . . . . . . . . . . . . . . . . . 112
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6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.3 . . . . . . . . . . . . . . . . . . . . . . 1176.2.4 . . . . . . . . . . . . . . . . . . . . . 121
6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3.1 qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3.2 Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4 . . . . . . . . . . . . . . 1296.4.1 Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.4.2 Kennard-Robertson. . . . . . . . . . . . . . . . . . . . 1306.4.3 . . . . . . . . . . . . . . . . . . . . . . . 132
7 1377.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.5.1 qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.6 . . . . . . . . . . . . . . . . . . . . 1457.6.1 . . . . . . . . . . . . . . . . . . . . . . . . 1457.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.7 . . . . . . . . . . . . . . . . . . . . . . . . 1487.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 1538.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2.1 . . . . . . . . . . . . . . . . . . . . . . 1548.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 1558.2.3 . . . . . . . . . 1568.2.4 . . . . . . . . . . . . . . . . . . . 157
8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.3.1 . . . . . . . . . . . . . . . . 1598.3.2 . . . . . . . . . . . . . . . . . . . . . . . 161
8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.4.1 qubit . . . . . . . . . . . . . . . . . . . . . . . . . 1638.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1648.4.3 . . . . . . . . . . . . . . . . . . . . . 165
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9 1699.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1699.2 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1729.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1729.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.3.1 qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.5.1 - . . . . . . . . . . . . . . . . . . . . . . . . . 1829.5.2 - . . . . . . . . . . . . . . . . . . . . . . . . 1839.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
189.1 Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
.2.1 H . . . . . . . . . . . . . . . . . . 190.2.2 Laplace . . . . . . . . . . . . . . . . . . . . . 190
.3 Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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1
, , . . , , . , - .
. , , . , .
1.1 ,
1687, 20 . .
1. . t 2 R ( )
x = (x1; x2; x3) 2 R3; .
2. , ( -). N t N - ri, i = 1; 2; : : : ; N . , , N ri(t).
3. . , - ri(t) 2o
mid2ridt2
= Fi (1.1)
1
-
. ,
1.1:
mi i, ,
Fi i.
4. ,
Fi =Xj 6=i
Fi(rj); (1.2)
Fi(rj) j i. o
Fi(rj) = Fj(ri): (1.3)
N 3N - . , ri(0) _ri(0) ri(t) t 2 R.
(). t0 t ( ).
1.1. Laplace
. , -, , ,
2
-
1.
. .
P.S. de Laplace, (1801).
1.2 1.2.1
- . (, ) , , .
, . , - , .
N . 3 - qa, a = 1; 2; : : : ; 3N . qa Q . qa ,Q = R3N .
i pi = mi _ri. N 3 , pa. a , pa qa.
3N 3N , , . ,
= (q1; p1; : : : ; q3N ; p3N):
= R6N N ( ).
H : !R,
H(qa; pa) =3NXa=1
p2a2ma
+ V (q1; q2; : : : ; q3N); (1.4)
V : Q ! R . E . .
, . .
3
-
. ,
_qa =@H
@pa_pa = @H
@qa(1.5)
qa = @V@qa
; (1.6)
(1.1), Fa =@V /@qa. E . (1.5) H ., (1.4), .
1.2.2 Poisson N
, F : ! R, F (qa; pa).
_F =Xa
@F
@qa_qa +
@F
@pa_pa
=Xa
@F
@qa
@H
@qa @F@pa
@H
@qa
= fF;Hg; (1.7)
Poisson F G ,
fF;Gg :=Xa
@F
@qa
@G
@qa @F@pa
@G
@qa
: (1.8)
Poisson
fqa; qbg = 0 (1.9)fpa; pbg = 0 (1.10)fqa; pbg = ab; (1.11)
ab Kronecker
ab =
1 a = b0 a 6= b : (1.12)
Poisson
1. fG;Fg = fF;Gg ()2. fF;GHg = fF;GgH + fF;HgG ( Leibniz)3. ffF;Gg; Hg+ ffH;Fg; Gg+ ffG;Hg; Fg = 0 ( Jacobi),
F;G H . Poisson
.
4
-
1.
1.2.3 Liouville Liouville .
U , U . , , .
[U ] U
[U ] :=
ZU
d3Nqd3Np: (1.13)
. U t qa pa, t + t q0a = qa+ t@H/@pa p0a = pa t@H/@qa.
@(q0a; p0a)
@(qa; pa)= 1: (1.14)
o U t+ t
[U 0]t+t =ZU
d3Nq0d3Np0 =ZU
@(q0a; p0a)
@(qa; pa)d3Nqd3Np
=
ZU
d3Nqd3Np = [U ]t: (1.15)
, . Liouville.
Liouville . , .
, .
1.3 -
. 19 - , . Faraday Maxwell , , .
- , E B. q F (- Lorentz)
F = q(E+ v B); (1.16)
v .
5
-
. ,
- Maxwell
r E = (1.17)r B = 0 (1.18)r E = 1
c
@B@t
(1.19)
r B = 1c
@E@t
+ j; (1.20)
, j c . , = j = 0 Maxwell
r2E = 1c2@2E@t2
: (1.21)
E(x; t) = E0eikxi!t; (1.22)
E0 , k ! . - . (1.21)
! = cjkj: (1.23) (1.22) . (1.17) = 0 k E0 = 0. , . - E .
1.4 ,
. - . o , , .
1.4.1 . , -
- . ,
o Lorentz- Heavyside, : 0 = 0 = 1. e = 0; 54 1013m3/2kg1/2/s.
6
-
1.
, - .
. -, 1, 2, 3, 4, 5 6. A = B = - 5 6. f1g, f2g, f3g, f4g, f5g,f6g, A f1; 3; 5g B f5; 6g. f1; 2; 3; 4; 5; 6g.
. - . -. (
). . .
A A = A, A A [ A = A \ A = ;.
A B , A \ B = ;. . .
1.1 . - 36 1 6.
= f(1; 1); (1; 2); (1; 3); (1; 4); (1; 5); (1; 6); (2; 1); (2; 2); (2; 3); (2; 4); (2; 5); (2; 6);(3; 1); (3; 2); (3; 3); (3; 4); (3; 5); (3; 6); (4; 1); (4; 2); (4; 3); (4; 4); (4; 5); (4; 6);
(5; 1); (5; 2); (5; 3); (5; 4); (5; 5); (5; 6); (6; 1); (6; 2); (6; 3); (6; 4); (6; 5); (6; 6)g:
,
1. A = = f(6; 6)g (1 ).
2. B = = f(1; 2); (2; 1)g (2 )
3. C = = f(1; 5); (5; 1); (2; 4); (4; 2); (3; 3)g (5 )
4. D = = f(1; 6); (6; 1); (2; 5); (5; 2); (3; 4); (4; 3)g (6 )
5. ( ) - ( )
- (ne-grained) (coarse-grained). .
. , , 32 . , , =f1; 2; Xg.
7
-
. ,
1.2: .
1.4.2 , ( )
. Prob A Prob(A) [0; 1]. Prob(A) A.
.
Kolmogorov
1. A;B , A \B = ;, Prob(A [B) = Prob(A) + Prob(B):
2. Prob(;) = 0.3. Prob() = 1.
Prob( A) = 1 Prob(A); (1.24) A .
B A. B AB B[ (AB) = A. Kolmogorov
Prob(AB) = Prob(A) Prob(B) (1.25) B A.
,
Prob(A [B) = Prob(A) + Prob(B) Prob(A \B): (1.26)
8
-
1.
. A0 = A (A (A \B),B0 = A (A (A \B), C = A \B A0 [ B0 [ C = A [ B. 1 Kolmogorov, Prob(A [ B) =Prob(A (A \B)) + Prob(B (A \B)) + Prob(A \B), (1.26) (1.25).
= fx1; x2; : : : ; xng, n . - n pi = Prob(fxig), i = 1; : : : n, fxig.
nXi=1
pi = 1: (1.27)
n pi !w = (p1; p2; : : : ; pn): (1.28)
Kolmogorov - . ,
Prob(fx1; x3g) = Prob(fx1g) + Prob(fx3g) = p1 + p3: f : ! R
= fx1; x2; : : : ; xng: pi, ,
f : hfi :=Pni=1 pif(xi). n- f : hfni :=Pni=1 pif(xi)n. f f : (f)2 := hf 2i hfi2. 1 2, 1 = fx1; x2; : : : ; xng
2 = fy1; y2; : : : ; ymg. ( 1.1, - , 1 = 2 = f1; 2; 3; 4; 5; 6g.) (xi; ya), i = 1; : : : ; n a = 1; : : :m. O pia := Prob[f(xi; ya)g].
p1i p2a
p1i =mXa=1
pia = Prob(fxig 2) (1.29)
p2a =nXi=1
pia = Prob(1 fyag): (1.30)
p1i 1 p2a 2.
1 2 ,
1 2 .
1.2. - 1.1. p = 1/36 36
. A;B;C D 1.1
Prob(A) = 136; Prob(B) = 2
36=
1
18; Prob(C) = 5
36; Prob(D) = 6
36=
1
6:
9
-
. ,
. . .
A;B , A B, Prob(AjB),
Prob(AjB) = Prob(A \B)Prob(B) : (1.31)
1.3. 1.1 1.2. D = E = Prob(DjE). ,
E = f(3; 3); (3; 4); (3; 5); (3; 6); (4; 3); (4; 4); (4; 5); (4; 6);(5; 3); (5; 4); (5; 5); (5; 6); (6; 3); (6; 4); (6; 5); (6; 6)g:
E 16 . (3,4) (4,3). Prob(DjE) = 2/16 = 1/8. ,
D \ E = f(3; 4); (4; 3)g;
Prob(D\E) = 2/36 = 1/18. Prob(E) = 16/36 = 4/9. Prob(D\E)/Prob(E) =(1/18)/(4/9) = 1/8, .
P(AjB) = P(A), - : B A. P(A \B) = Prob(A)Prob(B).
1.4. ( ) 7. , ;. Ai 7 i , i = 1; 2; 3. 1.2, 7 16 , p(Ai) = 1 16 = 56 . L 7, L = A1\A2\A3. , Prob(L) = Prob(A1 \ A2 \ A3) = Prob(A1)Prob(A2)Prob(A3) = (5/6)3 = 125216 . W L, Prob(W ) = 1 Prob(L) =1 125216 = 91216 ' 0; 42.
1.5. , 1 p 0 q = 1 p. . N . Prob(n) 1 n ( 0 N n )
Prob(n) = pnqNn N !(N n)!n! : (1.32)
10
-
1.
1.6. 1.4, 1 : p
-
. ,
p(x; y) =R2, x y.
p1(x) =
Zdyp(x; y) p2(y) =
Zdxp(x; y) (1.38)
x y .
1.2.
. - Kolmogorov, . .
An , n = 1; : : : ;1, An \Am = ; n 6= m,
Prob([1n=1An) =1Xn=1
Prob(An):
, . R, U
[a; b]
. . - f : R ! R, .
, - , . . - , . 5 .
- , Kolmogorov, .
1.7. R .
pGauss(x) =1p22
e(xx0)2
22 ; (1.39)
, x0 . - Z 1
1dxx2neax
2=
r
a
1 3 : : : (2n 1)(2a)n
(1.40)Z 11
dxx2n+1eax2
= 0; (1.41)
n a > 0.
12
-
1.
1.3: () = . - .
(1.40, 1.41)
hxi = x0; x = : (1.42)
pLorentz(x) =1
2 + (x x0)2 ; (1.43)
x0 . hxi = x0. x (x =1 hx2i =1).
1.4.4
1.4.2. - . , , , , - .
, , .
1. . . 3 1, 1/(3 + 1) = 1/4. :
13
-
. ,
. .
2. , - . ( ) - . : , - . , . .
3. . , . . .
, : - . - .
1. . , - N . , - nA . A nA/N . , N -, nA/N Prob(A). ( .) . , .
2. - , . Prob(AjB), . A B Prob(AjB) = 1. B ! A. B,
14
-
1.
( ) A. B ! A , Prob(AjB) = 1 , 0 <
-
. ,
, , . - , . , , . ( ): , , 5m/s 10m/s.
=
=
=
: ! R+,
(q1; p1; q2; p2 : : : ; q3N ; p3N)d3Nqd3Np (1.44)
d3Nqd3Np. , ,Z
d3Nqd3Np (q1; p1; q2; p2 : : : ; q3N ; p3N) = 1: (1.45)
= (q1; p1; : : : ; q3N ; p3N) , (1.45)
Rd() = 1, d = d3Nqd3Np.
- ( ), - Liouville
@
@t= fH; g; (1.46)
H .. (1.45) t, d/dt = 0,
@
@t+Xa
@
@qa_qa +
@
@pa_pa
= 0: (1.47)
_qa; _pa (1.5) . (1.8) - . (1.46).
Gibbs , , T ,
can() = Z1e
H()kBT ; (1.48)
16
-
1.
kB Boltzmann, Z . - (1.45),
Z =
Zde
H()kBT : (1.49)
hEi =RdH()eH()RdeH()
= @ logZ@
; (1.50)
= 1/(kBT ).
1.4.6 ,
() . . . . - , 6 . 2 f1; 2; 3; 4; 5; 6g.
., . C .
1. C \ C0 = ;; 6= 0 ( )
2. [C = ( ).
. . - ( ) - . F : ! R,
F () =X
C(); x 2 ; (1.51)
C C 2 ., F : ! R, -
C,
C = f 2 jF () = g: (1.52)
: F .
17
-
. ,
1.4: C .
1.5 -
, - .
1. . . 2 , - (- ).
2. . F . (1.52), F .
3. . - . - U , Prob(U) =
Rd()U().
4. . , - (1.5) Liouville (1.46).
5. . , . (1.31).
6. . 1 1, 2 - 2, 1 2 1 2. .
( R6, - R6 R6 = R12.), , , , .
18
-
1.
1.
. . . . -
. .
2. , 14 , . . ;
3. , , - . - , . . ( .) ( ) ;
4. ( ) ( ). 5% - 5 95%. 5 0,05. - ; ( .)
5. -X .
() 95% X 300 400GeV .() X 30% 70% -
.() 2016 .
.
1.
( ). - (1,2) (2,1) . , .
19
-
. ,
2. : ( ). (1,2) (2,1) , (2,2) . , .
3. ; 1 2.
4. 6 50%, . (i) , (ii) , (iii) ;
5. . p > ; p(t), [t; t+ t].
6. ( Brown). . -. t t(x),
@t@t
=1
2D@2t@x2
; (1.53)
D .
() . (1.53)
t(x) =1p2Dt
Zdx0 exp
(x x0)
2
2Dt
0(x0); (1.54)
0 t = 0.() t = 0 x0 = 0,
hx(t)i x(t).
, . H. Goldstein,
(, 1980).
, . C.Anastopoulos, Parcle or Wave: The Evoluon of the Concept of Maer in Modern Physics (PrincetonUniversity Press, 2008), . 1.-3. , . R. Dugas,A History of Mechanics (Dover, 2011).
, . M. R. Spiegel, (,).
, I. Hacking, An Introduconto Probability and Inducve Logic (Cambridge University Press, 2001).
, . C. J. Isham,Lectures on Quantum Theory (Allied Publishers, 2001), . 4.
20
-
2
A
, - . , , . - , .
2.1 K
, , , . .C. N. Yang, Parcle Physics (Princeton University Press, 1961).
- : .
, -. , . . . . . -. , , , . .
, . - . . , . , . - : .
21
-
. ,
. .
! . !, hEi . V [!; !+!] g(V; !)!, g(V; !) . 1
g(V; !) =8V !2
2c3: (2.1)
(!)! V - [!; ! + !]
(!) = g(V; !)hEi: (2.2)
hEi m !
H =p2
2m+
1
2m!2q2: (2.3)
Z = 2!
, . (1.50)
hEi = 1 (2.4)
(!) =8V !2kBT
2c3: (2.5)
(2.5) Z 10
d!(!) Z 1
d!!2
. , . - (!) - . 2.1.
. - - . -.
(Max Planck) 1900. - (!), -. . .
: , . - !, , ,
22
-
2. A
2.1: - ( ) . , .
h! nh!, n = 0; 1; 2; : : :. h , h = 1:0545 1034m2kg/s.
, . , pn ! nh! (n )
pn =enh!
Z; (2.6)
Z =1Xn=0
enh! =1
1 eh!: (2.7)
(2.6) .
hEi =P1
n=0(nh!)enh!
Z= @
@logZ = h!
eh! 1: (2.8)
(!)
(!) =8V h!3
2c3(eh! 1); (2.9)
. . (2.9) . 2.2 . h!
-
. ,
2.2: . (2.9) . a b c.
2.2
(Albert Einstein), (1905) .
...[ ] . , , , , , -.
- . - , ; ( 103104 .) - , .
, - ! ,
E = h!: (2.10)
. k ,
p = hk: (2.11)
, . ( 1925) .
. ,
A. Einstein, Ann. Physik 17, 132 (1905).
24
-
2. A
. , 19 . , , 20 . - Maxwell.
Compton, (1923). , . - . (2.10) .(2.11), ( , . 3)
0 = 2hmec
(1 cos ); (2.12)
, 0 , ( ) me . . (2.12) , , - .
Compton, ., . 2.1.
2.1.
2.3.
M (down converter). - . (1012 1010) , . ( ) .
(beam splier). : 50% 50% - 90o.
D0; D1; D2 , - .
, D0 , D1 D2. , D1 D2, D1 D2 . - D0D1 D0D2, D1D2. D1D2
25
-
. ,
2.3: (. 2.1 ).
.
2.3
. 1895 Rntgen - , 1896 J. J. Thomson H. Bequerel -, 1900 . Rutherford F. Soddy (). , , , .
(Rutherford) . H. Geiger .Marsden, . 2.4. -, 90o. . , ( ) . .
, , Coulomb, -.
K - . Coulomb, . , . (. 4) 1011s - . .
(Niels Bohr), .
26
-
2. A
2.4: . . , , .
, .
. . ( ). , . , , . .
, , . , h,
L = nh; n = 1; 2; : : : (2.13)
n (2.13) . (2.13)
. 2 Coulomb mev2/r =e2/(4r2), r
v2 = e2/(4mer): (2.14)
mevr = nh, (2.14) - r
rn =4h2
mee2n2: (2.15)
E = 12mv2e2/(4r) (2.14) E =
e2/(8r). r . (2.15)
27
-
. ,
En = E1n2; (2.16)
E1 = e4me/(322h2) ' 13; 6eV - (n = 1). , (2.15) r1 = 0,
0 =4h2
mee2' 0; 5 1010m (2.17)
. (3.40) .
, - .
- Em En, n m n < m.
h!m;m = Em En = E1
1
n2 1m2
; (2.18)
Rydberg 1888.
, , - .
. , , ... . . - , . .
. . , . . - , . , - .
, - . - , :
(20.3.1913). N. Bohr, Essays 1958-1962 on Atomic Physics and HumanKnowledge (Vintage Books, New York, 1963).
28
-
2. A
n .
( , ) - n . En = En+1 En .
EnEn
! 0; n!1; (2.19)
. - (2.19) (En/En n1) (En/En n1).
2.4 ; , ,
. 1924 , (Louis de Broglie).
. . -
E = h!, !. , - . .
[ ] , -. , .
: ( ) . . , , [..] , .
, .
k - ! (. 2.2)
vg =@!
@k : (2.20)
, m
v = pm
=@E
@p ; (2.21)
L. de Broglie, Nobel Lecture (12.12.1929)
29
-
. ,
E = p22m
. - , vg = v. E = h!, (2.20) (2.21)
p = hk: (2.22) (2.22) = 2/jkj
= 2h
jpj : (2.23)
.
2.2. .
vg . A(x; t)
A(x; t) =
ZdkA(k)eikxi!(k)t; (2.24)
! k. - , A(k) - k k0. Taylor,
!k = !0 + vg(k k0); (2.25)
!0 = !(k0) vg = (@!/@k)k=k0 . (2.24)
A(x; t) = ei(vgk0!0)tZdkA(k)eikxivgt; (2.26)
jA(x; t)j = jA(x vgt; 0)j: (2.27)
jA(x; t)j2, (2.27) vg.
- d - (). , , () (. 2.3).
2.3. .
, d . C. Davisson L. Germer 1927 ( !) (). -
30
-
2. A
d - (d ' 1010m.)
, 1929 I. Estermann O. Stern NaCl, (). 1940, E. O. Wollan C. Shull .
, ( o ) . 60 , Young . - ().
(), (), () ().
-
C48H26F24N8O8
114 1.298 (). ( ' 5pm) ( 1nm)!
. - , . mP =
phc/G ' 2 108kg -
(G ). , !
C. J. Davisson, Bell System Tech. J. 7 , 90 (1928)I. Estermann and O. Stern, Z. Phys. 61, 95 (1930); I. Estermann, R. Frisch, and O. Stern, Z.
Phys. 73, 348 (1931).C. Jnsson, Z. Phys. 161, 454474 (1961).A. Zeilinger, R. Ghler, C. G. Shull, W. Treimer, and W. Mampe, Rev. Mod. Phys. 60, 1067
(1988).D. W. Keith, M. L. Schaenburg, H. I. Smith, and D. E. Pritchard, Phys. Rev. Le. 61, 1580
(1988); O. Carnal and J. Mlynek, Phys. Rev. Le. 66, 2689(1991).M. Arndt, et al., Nature 401, 680 (1999)T. Jumann et al, Nature Nanotechnology 7, 297 (2012)
(. . 2.5), 19 (. 6). .
, - . , ( ). - , . . , . , - . , . .
31
-
. ,
2.5: . . . . - .).
. , . . 10.000 , 50.000 (. . 2.6). , . . , .
; ( ). , . - , : , , . , .
. , . - . . .
[A Tonomura et al, American Journal of Physics57, 117 (1989)] 1 103s 1; 5 108m/s. 150km 1; 5m. - , .
32
-
2. A
2.6: Tonomura. : ) 8 , ) 270 , ) 2000 , ) 60000 .
( ), , - . , . .
1. -
, 19 . ;
2. - . . , . ;
3. - . ;
1. L
( ) (/L)3 .
g(V; !) =8V !2
2c3; (2.28)
33
-
. ,
V = L3.
2. (2.9) !/T = b b.
3. . (2.12) - . - - E = c
pp2 +m2ec2.
4. q a E Larmor
dE
dt= 2q
2a2
3c3: (2.29)
r. E = e2/(8r). . (2.29)
dr
dt= e
4
3c3m2er2: (2.30)
r(0) = r0 (r = 0)
=m2ec
3r30e4
: (2.31)
r0 ' 1010m, .
5. (2.21)
E = cpp2 +m2c2 p = mv/
p1 v2/c2):
6. Young (. 2.7). A1(t; x) A2(t; x). d L -.
() kd sin , k .
() A, - I(y) = jA1 +A2j2
I(y) = 4jAj2 cos2kd sin
2
; (2.32)
y.()
sin ' y/L.
34
-
2. A
2.7: .
, . H. Kragh, (,
2004). . , (, ), .1 .
MaxJammer, The Conceptual Development of Quantum Mechanics, (2nd ed: New York: American Instuteof Physics, 1989).
B. L. van derWaerden, Sources of Quantum Mechanics (Dover, 2007).
, www.nobelprize.org/nobel_prizes/physics/laureates/1922/Bohr-lecture.html
, www.nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-lecture.html
- ,
P. Rodger, Physics World, September 2002 (. 15). M. Arndt and A. Zeilinger, Physics World, March 2005.
Hitachi :www.youtube.com/watch?v=PanqoHa_B6c
.www.youtube.com/watch?v=vCiOMQIRU7I.
35
-
. ,
36
-
3
B
, , . - , , . - .
, - : . , .
3.1
-. - , . . , . , .W. Bragg, 1922.
-. Bragg , . (Werner Heisenberg).
-
!nm = [E(n) E(m)]/h (3.1)
n m E(n) ( ). n. x(n; t).
The Robert Boyle Lecture 1921, Scienc Monthly 14, 158 (1922).W. Heisenberg, Zeit. Phys. 33, 879, 1925.
37
-
. ,
, x(n; t) Fourier
x(n; t) =1X
m=1xm(n; t)eim!(n)t; (3.2)
!(n) n T (n) n: !(n) = 2/T (n). x(n; t) , !n;m = m!(n) m n, . (3.2). Maxwell, . , (3.1). x(n; t) .
- . , - (3.1). .
1. , Anm m n.
Anm ! Anmei!nmt; (3.3) !nm (3.1).
2. Anm , . 1 . - a Anm, a2
Bnm =Xl
AnlAlm; (3.4)
(3.3). (3.4) - . . ( - ). , Anm . - .
3. Xnm x , Pnm p, X
l
(XnlPlm PnlXlm) = ihnm; (3.5)
X P
XP PX = ih1: (3.6)
38
-
3. B
- .
-, . , - .
. - , ( ) , . - .
, . , . , (Max Born) Pascuale Jordan (Paul Dirac) .
3.2 3.2.1
, - , . (Erwin Schrdinger), .
, - (t; x) = eikxi!t. E = h! p = hk E p ! k ,
ih @@t (t; x) (t; x)
E.
ih @@x
(t; x) p.
E = p22m
(-) m, (t; x)
ih@ (x; t)
@t= h
2
2m
@2 (x; t)
@x2: (3.7)
(3.7) . ; O . -
H =p2
2m+ V (x)
39
-
. ,
V (x) , h
2
2m@2
@x2 (3.7) ( p2
2m)
h2
2m@2
@x2+ V (x) .
( )
ih@ (x; t)
@t=
h
2
2m
@2
@x2+ V (x)
(x; t): (3.8)
. ,
ih@ (r; t)@t
=
h
2
2mr2 + V (r)
(r; t): (3.9)
! E = h!.
(r; t) = E(r)eiEt/h (3.10)
. (3.9) h
2
2mr2 + V (r)
E(r) = EE(r): (3.11)
, (x; t) = E(x)eiEt/h . (3.8), -
h2
2m
@2
@x2+ V (x)
E(x) = EE(x): (3.12)
3.2.2 -
m ! , . (3.12) V (x) = 1
2m!2x2,
h2
2m
@2
@x2+
1
2m!2x2
E(x) = EE(x): (3.13)
, , . , . (3.13) -
limx!1
(x) = 0: (3.14)
(3.13) m;! h, : x0 =
qhm!
. x = x/x0, .(3.13)
00E + (2 2)E = 0; (3.15)
40
-
3. B
0 ,
=E
h!: (3.16)
>> p, . (3.15)
00E 2E = 0: (3.17)
s = 122, . (3.17) d2E/ds2 + E = 0,
es es. . (3.17) e2/2, e2/2 (3.14).
E . (3.15) () = e2/2uE(). . (3.15)
u00E 2u0E + (2 1)uE = 0: (3.18)
. (3.18) Hermite. o 3.1, Hermite (3.14),
= n+1
2; n = 0; 1; 2; : : : : (3.19)
. (3.18) n - Hermite Hn(). Hermite,. .
n = 0; 1; 2; : : :
n(x) = Cn exphm!
2hx2iHn
rm!
hx
; (3.20)
Cn . Cn n(x) :
Rdxj
phin(x)j2 = 1. Hermite () jCnj22nn!
px0,
Cn =1p2nn!
m!h
1/4: (3.21)
n(x) 3.1. ( )
En = (n+1
2)h!: (3.22)
. 1
2h!
.
41
-
. ,
3.1: (3.20) n =0; 1; 2; : : : ; 7.
R11 dxjn(x)j2 = 1.
3.1. Hermite.
uE() = 0
uE() =1Xk=0
akk; (3.23)
ak. uE
u0E() =1Xk=0
ak+1(k + 1)k; u00E() =
1Xk=0
ak+2(k + 1)(k + 2)k: (3.24)
(3.18) - ak
ak+2 =2(k + 12 )(k + 1)(k + 2)
ak: (3.25)
, - . - (3.25) 2, a0 a1 - ak. a0 P1
l=0 a2l2l a1
P1l=0 a2l+1
2l+1 . 12 = n, n = 0; 1; 2; : : :, an+2 = 0 (3.23)
n . a0 = 1 a1 = 2 Hermite Hn(). E() = e
2/2uE() (3.14).
, (3.23) . k , ak+2/ak 2/k. , e2
P1l=0 a2l
2l a2l = 1/l!. a2l+2/a2l 1l , ak+2/ak 2/k,
42
-
3. B
(3.23). uE() e2 E()
e2/2. (3.14) . (3.14) 12 = n -
(3.22).
3.2.3 (3.11)
, V (r), r = jrj.
(r; ; ), r 2 [0;1), 2 [0; ] 2 [0; 2]. Laplace
r2f = 1r2
@
@r
r2@f
@r
+
1
r2 sin @
@
sin @f
@
+
1
r2 sin2 @2f
@2; (3.26)
f(r; ; ).
E(r; ; ) = R(r)Ylm(; ); (3.27)
Y lm(; ) . R(r) . l m l = 0; 1; 2; : : : m =l;l + 1; : : : ; l 1; l.
r2r2Y lm(; ) =1
sin @
@
sin @Y
lm(; )
@
+
1
sin2 @2Y lm(; )
@2
= l(l + 1)Y lm(; ): (3.28) . (3.27) . (3.11) R(r)
h2
2mr2d
dr
r2dR(r)
dr
+
h2l(l + 1)
2mr2+ V (r)
R(r) = ER(r) (3.29)
. (3.29) u(r) = rR(r) R(r).
h22m
d2u(r)
dr2+
h2l(l + 1)
2mr2+ V (r)
u(r) = Eu(r) (3.30)
. (3.29) V (r). , . (3.12),
V (r) = V (r) +h2l(l + 1)
2mr2: (3.31)
(3.30) u(r) l , m.
( ) E(r) ! 0 ! 1. , E - r = 0. , . (3.30)
u(0) = 0 limr!1
u(r) = 0 (3.32)
43
-
. ,
Coulomb ,Z . , .
V (r) = Ze2
4r; (3.33)
Z . m me. ( , , . 2.)
. (3.30 (3.33) h;me e, , . . (2.17),
0 =4h2
mee2(3.34)
= Zr/0 r, . (3.30)
u00 ++
2
l(l + 1)
2
u = 0: (3.35)
0
=2meE
20
h2Z2: (3.36)
(3.32). ! 0, .(3.35) 2u00 = l(l+1), u = l+1 u = l. (3.32), . ! 1, . (3.35) u00 + u = 0, u = e
pjj
, < 0. u()
u() = l+1epjjf(); (3.37)
. (3.35)
f 00 + 2(l + 1pjj)f 0 + 2(1 (l + 1)
pjj)f = 0 (3.38)
. (3.38) (3.32) - n > l jj = n2. 3.2. . (3.38) (n l 1)- fnl(). - Laguerre. .
, - () n; lm, n = 1; 2; 3; : : :, 0 l < n l m l,
nlm(r; ; ) = Cn;l
Zr
0
le Zn0
rfn;l
Zr
0
Y ml (; ); (3.39)
Cn;l . n
En = Z2e4me
322h2n2; (3.40)
44
-
3. B
3.2: Rn;l(r) (3.39) , n l.
3.2. (3.38)
f() = 0
f() =1Xk=0
akk; (3.41)
ak.
f 00() =1Xk=0
ak+1(k + 1)kk; f 0() =
1Xk=0
akkk; f 0() =
1Xk=0
ak+1(k + 1)k:
(3.38) - ak
ak+1 =2[pjj(l + 1 + k) 1](k + 1)(k + 2l + 2)
ak: (3.42)
k = k 1 pjj(l+1+k) = 1, ak+1 = 0 (3.41) k . n = l+k+1 >l,
jj = n2; (3.43)
n l 1. (3.43), (3.42)
k ak+1/ak = 2pjj/k.
f() exp[2pjj]. u()
l+1epjj (3.32)
. (3.32) (3.43) (3.40).
45
-
. ,
Laguerre Ln , - fnl() = L2l+1nl1(/n)
a0 = L2l+1nl1(0) =
n+ l
n l 1: (3.44)
fn;l():
f1;0() = f2;0() = 2(1 /2) f2;1() = f3;0() = 3(1 2
3 +
2
272) f3;1() = 6(1 1
9) f3;2() = : (3.45)
, nlm -
Rr2 sin drddjnlmj2 = 1.
nlm(r; ; ) =
s2
n0
3(n l 1)!2n(n+ l)!
eZrn0
2r
n0
lL2l+1nl1
2Zr
n0
Y ml (; ): (3.46)
3.2.4 .
, , .
, N
i@
@t(r1; : : : ; rN) =
"
NXi=1
h2
2mir2i + V (r1; : : : ; rN)
# (r1; : : : ; rN); (3.47)
mi V (r1; : : : ; rN) . . .
. -. , .
. : . 3. .
- . .
1. i. (x; t), . .
46
-
3. B
2. N , - N . Q = R3N N . , - R3.
3. . (r; t) (3.9), (r;t) . , .
3.3.
io 1926 - . , - . , , - . .
. - ... . : , -. : , . - , , -, , .
W.Heisenberg,Quantum theory and its interpretaons, Quantumtheory and Measurement, eds. J. A. Wheeler and W. H. Zurek (Princeton UniversityPress 1983).
3.3 3.3.1
, -, . : -
N t = 0, t N(t) = N0et - , . - .
47
-
. ,
, . .
. n(x) i En. X^ P^ - (x)
X^ (x) = x (x); P^ (x) = i @@x
(x): (3.48)
, , (3.6) . E
Xmn =
Zdx m(x; t)x n(x; t) (3.49)
Pmn =
Zdx m(x; t)(i@/@x) n(x; t): (3.50)
(3.10), n(x; t) = n(x)eiEnt/h. Xmn Pmn
Xmn(t) = Xmn(0)ei(EmEn)t/h; Pmn(t) = Pmn(0)ei(EmEn)t/h; (3.51)
(3.3). , , ,
. .
3.3.2 .
. - . - , . , . , -
x w; (3.52) w .
. - . , p = 2h/. ,
p 2h: (3.53)
w 0; 47/NA NA . 2 1,45. w = 0; 3 . (3.54) xp > 2h.
48
-
3. B
3.3: . . , - . -. .
(3.52) (3.53)
xp (2w)h h: (3.54) (3.54) . -
. x p .
(3.54) , . . . t. , t. E = @E
@pp = vp, v .
t x = vt, . (3.54)
Et > h (3.55)
. ,
, (3.54) : x p , x - p . x p .
49
-
. ,
, , . , - .
, - , . , .
, . . - , . , , , 3.4.
3.4.
- (3.55). . T . , . , m1 m2 (1) (2) E = (m1 m2)c2. , E -. t . Et .
. L. Rosenfeld :
... . , - , . . . , , , .... .
, , , . - , . E m : E = mc2. F = mg , m = F/g. - pz ( -) F = p/T , T . E = pzc2/(Tg).
,
W. Heisenberg, Physics and Philosophy (Prometheus Books, New York, 1999).
50
-
3. B
. , z z+z, T T +t , t/T = gz/c2T . z , t .
Et pzc2
Tg
gzT
c2= pzz > h; (3.56)
- . - ().
Albert Einstein: Philosopher-Scienst, P.A.Schilp editor (Evanston 1949). Quantum theory and Measurement,edited by J. A. Wheeler and W. H. Zurek. , . A. C. de laTorre et al, Eur. J. Phys. 21, 253 (2000).
3.3.3
. , - . , .
: (x; t) (x; t) = j (x; t)j2 x t. , U R3 , Prob(U; t) U t
Prob(U; t) =ZU
d3xj (x; t)j2: (3.57)
j (r; t)j2 .Z
d3xj (x; t)j2 = 1: (3.58)
(5.2) (3.9). ,
@
@t=
@
@t+
@
@t= i
h h
2
2mr2 + V
+
i
h
h
2
2mr2 + V
=
ih
2m( r2 r2 ); (3.59)
. (3.9). (3.59)
@
@t+r JS = 0; (3.60)
JS
JS =ih
2m( r r ) : (3.61)
51
-
. ,
U
@
@tProb(U; t) =
I@U
d2 JS; (3.62)
@U U . U = R3 @U , ( ).
@
@t
Zd3xj (x; t)j2 = 0; (3.63)
(5.2) .
, ( ) .
j (x; t)j2 , t. . Prob(U; t) . - . , t , - j (x; t)j2. .
3.4
. ( ) 5 Solvay , 1927, .
, : . - . . , , , - 1927. M , . , .
Solvay . - . . . .
, - . , . .
52
-
3. B
3.4: . W. Heisenberg, The Physical Principles of the Quantum Theory (Dover, New York, 1949).
: . . , . - .
. - -. . - - .
, -, . . (J. vonNeumann), . , , (Hilbert) - .
- . , - , , . , .
P. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 1930). J. von Neumann, MathemascheGrundlagen der Quantenmechanik (Berlin: Julius Springer, 1932).
53
-
. ,
6 . - . 1.5. 3.1. 6 . , . , -. , . .
3.1:
H H
1.
.
2. .
3. x p - .
4. - , - .
5. JS ;
1.
m V (r) = 12m!2r2.
2. . (3.47) m M r1 r2 . V (r1 r2) . rc m +M
54
-
3. B
x = r1 r2 V (x),
=Mm
m+M: (3.64)
.
3. m ! . -
(x; t) = C(t) exphm!
2h[x q(t)]2 + ip(t)x/h
i: (3.65)
(3.65) q(t) p(t) .
4. ( ) ear2 a > 0; , ( ) 1/ cosh(bx), b > 0;
. . 2.
,. . , , . 5-6.
, . . , (, 1978).
J.A. Wheeler and W. H. Zurek, Quantum Theory and Measurement (Princeton University Press, 1984).
, http://www.nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-lecture.html
, http://www.nobelprize.org/nobel_prizes/physics/laureates/1933/schrodinger-lecture.html
, http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.html
55
-
. ,
56
-
4
- . . . , - . .
h = c = 1.
4.1 4.1.1
.
1. . -: 1(r; t) 2(r; t) c1 1(r; t) + c2 2(r; t) , c1 c2. - . i -.
2. - , - .
3. (3.50)
Rdx (x)(x) (x)
(x). j (x)j2 . .
, . .
57
-
. ,
4.1.2 ,
. 1. V C V -
: ; 2 V + 2 V , : 2 C 2 V 2 V ,
.1. ; ; 2 V (+ ) + = + (+ ).2. ; 2 V + = + .3. 0 2 V + 0 = , 2 V .4. 2 V , 2 V , + () = 0.5. 1; 2 2 C, 1(2) = (12).6. 2 V , 1 = .7. 2 C ; 2 V (+ ) = + .8. 1; 2 2 C (1 + 2) = 1+ 2.
2. V V .
, . 3. V . 4. V ; 2 H (; ) .
2 C ; ; 2 H, ( + ; ) = (; ) + (; ). ( ).
; 2 H, ( ; ) = (; ). () 2 H, (; ) 0. (; ) = 0 = 0. () , .
1. : (; + ) = (; ) + (; ). 5. kk 2 V : jjjj =p(; ) 6. D(; ) ; 2 V :D(; ) = k k.
(0) V .
58
-
4.
4.1.3 V -
. V , . V Cn n.
2 Cn
=
0BB@12: : :n
1CCA (4.1) , ; 2 Cn
(; ) =nXi=0
ii : (4.2)
, . 2, , (x). .
7. V -, .
, . 4.1. , - V n V , V . . - .
4.1.4 1. l2 = fi; i = 1; 2; : : :g
P1i=1 jij2 0 N n;m > N kn mk < .
Cauchy V . , V - R. n(x) = tanh(nx)/(1+x2) - , sgn(x)/(1 + x2) x = 0, V .
.
60
-
4.
10. V , fn 2 V jn = 1; 2; : : :g Cauchy, 2 V .
, , .
11. fnjn = 1; 2; : : :g V V . 2 V > 0, N n > N j(n; ) (; )j < .
. . , n(x) = sin(nx) L2([0; 2])
[0; 2]
(; ) =
Z 20
dx(x)(x): (4.6)
n n ! 1. - (x) limn!1(n; ) = 0. fng 0
w-limn!1n = 0: (4.7)
4.2 . -
.
1. , .
. , . .
. 6 1 , .
4.2.1 1, .
, ; . ,
. , , , .
. , - , . .
61
-
. ,
, (J.W. Gibbs). , () , . . , -.
- . : - . -. . 1 2. , 1 - , 2 ( ). - .
: - .
- . - . , - , .
- . , . - - . .
4.2.2
. H. ; 2 H - , + ( 2 C) . - - , , .
- , + . - ,
62
-
4.
4.1: Mach-Zehnder ( ). ! - ! + ! " ".
, , , . : .
+ . E1 E2, + E1 E2. + E1 E2 .
Mach-Zehnder
- . , , . Mach-Zehnder, 4.1.
Mach-Zehnder (beam splier). ( 50%) . - , - . . - ( ) . - .
! () " (). ( 45o) 90o ( Snell). i .
63
-
. ,
! ! i "; " ! i !: (4.8)
, (4.8).
! ! 1p2(i " + !); " ! 1p
2( " + i !); (4.9)
( !; !) = ( "; ") = 1 ( !; ") = 0. 1/p2
. U ! -
! . ( U .) , 4.1 !, - , 1p
2( !+!) ip2( !!).
U : ! ! ! Mach-Zehnder .
- . , - . , - .
.
. , - . , - . 5e = 5 7e = 7 .
. 5e+ 7e, 5e 7e. , . . , . : . . 7 .
.
E. Schrdinger, Proceedings of the American Philosophical Society, 124, 323 (1935). QuantumTheory and Measurement, J.A. Wheeler and W.H. Zurek, eds. (Princeton University Press, New Jersey 1983).
64
-
4.
( ): , 12 , . , , . , . ( ) .
, ....
. (.. ) - , . . . -.
- , - , - .
-, Schrdinger . (. . 4.2), - , . , - . . . , , - . , .
- . . 8, , - , - . () , .
. - , N . - , (1015Kg) 1012amu .
65
-
. ,
4.2: - , . () , - . . () . , , . () . , , ....(;)
. . - - : . - 4.2.
4.2. .
-.
1. 1996, S. Haroche cole Normale Suprieure (), - = c1+c2, - . 5-10.
2. 1996, D. J. Wineland Naonal Instute of Standards andTechnology () Be+ (x) = [ (x x1) + (x x2)]/
p2,
x = 0. jx1 x2j ' 80nm, -
- , 4.2.
66
-
4.
' 7nm, ( 0; 1nm). .
3. 2000 2002 , Del () StonyBrooke (), - -. A . - 1010 , 103 ().
M. Brune et al, Phys. Rev. Le. 77, 4887 (1996).C. Monroe et al, Science 272, 1131 (1996).C. H. van der Wal et al, Science 290, 773 (2000)J. R. Friedman et al, Nature 406, 43 (2002).J. I. Korsbakken et al, EPL 89, 30003 (2010).
, . : - ( ) , , . 6, -.
4.3
, : , - , .
4.3.1 ; H.
2. ( ) (; ) = 0, k+ k2 = kk2 + k k2: (4.10)
. k+ k2 = (+ ; + ) = (; ) + ( ; ) + (; ) + (; ) = kk2 + k k2. 3. ( Schwarz) j(; ) kkk k, = , 2 C.. = 0, . 6= 0 = (; ) ; . (; ) = (; ) (; )( ; )/( ; ) = 0,
kk2 = (; )( ; )
2 + kk2 = j(; )j2k k2 + kk2 j(; )j2k k2 : kk2k k2 j(; )j2 Schwarz. h = 0, = = (; )/( ; ) 2 C.
Schwarz 1 (; )kkk k 1, .
67
-
. ,
12. (; ) ;
cos (; ) = j(; )jkkk k : (4.11)
4. ( ) jkk k kj k+ k kk+ k k.. (kk + k k)2 k + k2 = kk2 + k k2 + 2kk k k kk2 k k2 (; ) ( ; ) =2kk k k
1 Re(; )kkk k
2kk k k
1 j(; )jkkk k
0,
Schwarz. k+ k kk+ k k. .
5. ( )
(; ) =1
4
k+ k2 k k2 + ik+ i k2 ik i k2 : (4.12). .
.
4.3.2 13. feig - (ei; ej) = ij .
14. feig H - .
(Cn), o - . - .
15. .
- . . - , .
-. , i feig . (i 2 f1; 2; : : : ; ng) (i 2 f1; 2; : : :g). , .
Pi
Pni=1
P1i=1.
.
68
-
4.
6. ( ) feig H, and =
Pi ciei, ci 2 C. ci = (; ei).
. =Pi ciei ej . , (; ej) =
Pi ci(ei; ej) =
Pi ciij = cj .
7. ( Parseval) feig H 2 H, kk2 =Pi j(ei; )j2.. =Pi ciei,
kk2 = (; ) =Xi
Xj
ci cj(ei; ej) =Xi
Xj
ci cjij
=Xi
jcij2 =Xi
j(ei; )j2;
6.
6
=Xi
(; ei)ei: (4.13)
2 H feig . . (4.13) .
feig fdag H.
Uia = (ei; da): (4.14)
- .
1. ei =P
a Uiada, da =P
i Uiaei.
2.P
a UiaUja = ij ,
Pi UiaU
ib = ab. ( Uia .)
4.3.3 H1 H2.
16. F : H1 ! H2
1. -- .
2. : F (+ ) = F () + F ( ).
3. : kF ()k = kk 2 H1.
, , :
(F (); F ( ))H2 = (; )H1 : (4.15)
69
-
. ,
17. . .
H n fe1; : : : ; eng . - 2 H = Pni=1 ciei H Cn, F () = (c1; : : : ; cn). Parseval . n Cn.
H, fe1; e2; : : :g
F : H ! l2 ,
F () = fc1; c2; c3; : : :g (4.16) = P1i=1 ciei. Parseval . l2, .
. . (-) . , - .
( ) - . - , , , .
4.3.4 .
H1 H2,
=
0BB@c1c2: : :cn
1CCA 2 H1 =0BB@
d1d2: : :dm
1CCA 2 H2: (4.17) H1 H2
=
0BBBBBB@c1: : :cnd1: : :dm
1CCCCCCA (4.18)
-. K .
70
-
4.
8. Cn Cm = Cn+m. 9. 0+ 0 = 2 H1 2 H2. H1 H2 . 10. ( .) feig H1 fojg H2, fei 0; 0 ojg H1 H2.
, . 18. H1 H2 H1 H2, 2 H1 H2, (1 1) + (2 2) =(1 + 2) ( 1 + 2) (1 1; 2 2) = (1; 2) + ( 1; 2).
4.3.5 19. V H H . ; 2 V 2 C, + 2 V .
. - V , n 2 V , - V . V .
V . . - -. , -. 20. Span(S) S H H Pa caa ( ), ca 2 C a 2 S. 21. V ? V H 2 H (; ) = 0 2 V .
21 V ?? = V . , 2 V 2 V ? = 0. 11. ( ) V H. 2 H = V + V ? , V 2 V V ? 2 V?. V V .. feig V . H fwkg V ?. = Pi(; ei)ei +Pk(;wk)wk. V = =P
i(; ei)ei V ? =P
k(;wk)wk, = V + V ? . 0V 0V ? = 0V + 0V ? .
V 0V = V ? 0V ? . V V ?, V 0V =V ? 0V ? = 0.
10 11 H = V V ?.
71
-
. ,
4.3.6 -
. H
=
0BB@12: : :n
1CCA 2 H; (4.19) (4.2).
O H H = (1; 2; : : : ; n) : (4.20)
(; )H =Xi
ii : (4.21)
.1. 2 H : H ! C,
2 H, (f) =Pi ii 2 H. H H.
2. 2 H . (4.19) = (1; 2; : : : ; n) 2 H. !
(; )H = ( ; )H; (4.22) ; 2 H. H H ! . , ,H = H. Riesz.
. - . 4.3, .
4.3.
22. H H - H, : H ! C,
(+ ) = () + ( ); (4.23)
; 2 H 2 C. H (1+2)() =1()+2() 1; 2 2 H, 2 H 2 C.
kk = sup2H
j()jp(; )
: (4.24)
72
-
4.
, H ( 5).
12. ( Riesz) 2 H, 2 H () = (; ) 2 H.. feig H. i = (ei) i, =
Pi
i ei. 2 H, (; ) =
Pi i(; ei) =
Pi (ei)(; ei) =
(P
i(; ei)ei) = !(). . 0 () = (; 0) 2 H, (; 0) = 0, 2 H. = 0, k 0k = 0, , = 0.
4.4 Riesz ,
. , 2 H ket j i. ket
, .. , , . ,
j 4 i ket. ket
1p2j,i+ 1p
2j/i
4.2.
H bra hj. Riesz, bra ket. h j H ket j i.
Riesz H braket: hj i.
ketbra . (jihj) j i = hj iji. .
ket = = n 1. bra = = 1 n. braket = ( 1 n) ( n 1) = 1 1 = . ketbra = ( n 1) ( 1 n) = n n = .
feng jni, n = 1; 2; : : : , -
ket. , (3.39) Schrdinger jn; l;mi.
73
-
. ,
j i jni j i =Pn jnihnjfi.
1 =Xn
jnihnj: (4.25)
, o - bra ket. -, j iH Hhj.
!! To (; ) - h ji , .
, 5.6 . - , - . 5.6 .
1. ;
2. 1 . - 100 . . , 1 1023 -. ;
3. . ;
4. : , , . -;
5. R; (i) ex2/x, (ii) 1/px2 + 1,
(iii) eax, (iv)sin(x)/x, (v) 1/pjxj.
1. 0
2. ( ! ei) ;
2. qp(x) = C exp(x q)2/(22) + ipx L2(R), C -
.
() C, qp .
74
-
4.
() ( qp; q0p0).
3. p(x) = 1p2eipx 1
2x2 . p -
> 0, = 0. ( p; p0). ! 0;
Hilbert , . C. J. Isham, Lectures on Quantum Theory
(Allied Publishers, 2001), . 2.
, . L. E. Ballenne, Quantum Mechanics: aModern Development (World Scienc, 1998), . 8.
, . V. Scarani, Quantum Physics: A FirstEncounter: Interference, Entanglement, and Reality (Oxford University Press 2006).
, . . , (, , 2008), . 1.
N. Young, An Introducon toHilbert Space (Cambridge University Press, 1988) . 1-6 D. W. Cohen An Introducon to HilbertSpace and Quantum Logic (Springer, 1989) . 1-4.
, . A. J. Legge,Phys. Scr. 2002, 69 (2002) M. Arndt and K. Horberger, Nature Physics 10, 271 (2014).
D. J. Wineland, http://www.nobelprize.org/nobel_prizes/physics/laureates/2012/wineland-lecture.html
S. Haroche, http://www.nobelprize.org/nobel_prizes/physics/laureates/2012/haroche-lecture.html
75
-
. ,
76
-
5
, . : ,, . , - . .
5.1 5.1.1
.
23. A^ H A^ : H ! H, 2 H A^ 2 H, A^(+ ) = A^+ A^ , ; 2 H 2 C.
. .
H 1. . (A^+ B^) := A^+ B^, 2 H.2. . (A^) = A^, 2 C 2 H.3. . (A^B^) := A^(B^), 2 H. .
1. H , 1.
2. A^; B^ C^, (A^B^)C^ = A^(B^C^). ()
3. A^; B^ C^ 2 C, A^(B^ + C^) = A^B^ + A^C^ (B^ + C^)A^ = B^A^+ C^A^. ().
4. 1^ 1^ = 2 H, 1^A^ = A^1^ = A^ A^. ().
77
-
. ,
5. 0^ 0^ = 0 2 H, 0^A^ = A^0^ = 0^ A^+ 0^ = A^, A^
.
A^. B^, A^B^ = B^A^ = 1^, B^ A^ A^1.
13. ( .) A^ B^ , (A^B^)1 =B^1A^1. .
24. A^y A^, (A^; ) = (; A^y ), ; 2 H.
(cA^ + B^)y = cA^y + B^y, A^ B^ c 2 C. 14. ( ). (A^B^)y = B^yA^y.. (A^B^; ) = (B^; A^y ) = (; B^yA^y ). (A^B^; ) = (; (A^B^)y ), (; (A^B^)y ) = (; B^yA^y ) ; 2 H.
feng H A^ - Amn = (A^en; em). =
Pn cnen.
A^ =Xn
cnA^en =Xn;m
cn(A^en; em)em =Xm
(Xn
Amncn)em; (5.1)
. ! A^ cm ! c0m
PnAmncn -
feng. cn , Amn .
Amn = hmjA^jni. ket, bra ket. , (A^; B^C^ ) (C^yB^yA^; ) braket h jC^yB^yA^ji.
5.1.2 A^ H.
25. ( ). 2 H A^ = a , a 2 C, A^ A^.
A^ , c1 + c2 , c1; c2 2 H A^ . a Va . a.
A^ a , - . , Va . Va a.
78
-
5.
26. (kernel) A^ a = 0. ker(A^).
A^ . ( A^f = 0, A^1A^f = 0, f = 0). ker(A^) = f0g. 15. ( ). A^ A^ =a , A^n A^n = an , n = 1; 2; 3; : : :.. n = 2, A^2 = A^(a ) = aA^ = a2 . n.
5.1.3 27. kA^k A^
kA^k = sup 2H
kA^ kk k : (5.2)
(supremum) . (5.2) ., (k k)
. . . -
.
I
A^ B^ H .1. kA^k = 0, A^ = 0.2. kA^k = jjkA^k, 2 C.3. kA^ k kA^k k k, 2 H.4. kA^+ B^k kA^k+ kB^k.
. 2 H k(A^+B^) k = kA^ +B^ k kA^ k+kB^ k (kA^k+kB^k)k k. (5.2) .
5. ( Cauchy-Schwartz.) kA^B^k kA^k kB^k.. kA^B^ k kA^kkB^ k kA^kkB^kk k, 2 H. (5.2) .
1. (H = Cn), - n n. . Aij , i; j = 1; 2; : : : ; n .
2. - . , - x^ (x) = x (x) L2(R) -. 0(x) = ex2/2 2 L2(R). L(x) = 0(xL) L.
kx^ Lk/k Xk =rX2 +
1
2
79
-
. ,
, L. x^ . - . . , x^ L2(R) 1/
px2 + 1
x/px2 + 1, .
3. DA^ - A^ - H A^ . - A^ . A^ - (A^; ), ; 2 H. , x^ 0(x) = 1/
px2 + 1, -
= ex2/px2 + 1, > 0.
(x^ 0; ) lim!0(x^ ; ).
4. - .
5. H B(H).
5.2
5.2.1 28. (normal) A^, A^A^y =A^yA^.
16. ( ). A^ , A^ A^y .. A^ = a . k(A^ya1^) k2 = ((A^ya1^) ; (A^ya1^) ) = ((A^a1^)(A^ya1^) ; ) = ((A^y a1^)(A^ a1^) ; ) = 0, (A^ a1^) = 0. A^y = a .
17. ( ) A^ , kA^k = supi jaij ai A^.. kA^ k2 = (A^yA^ ; ). A^ A^yA^ ( ). 16, A^yA^ jaij2 ai A^. .
O . - , .
5.2.2 29. A^ A^ = A^y.
80
-
5.
18. ( ) -.. A^ = a (A^ ; ) = a( ; ) ( ; A^ ) = a( ; ). ( ; A^ ) =(A^ ; ), a = a. 19. ( ) A^ , A^ = a ; A^ = a0, a 6= a0, (; ) = 0.. (A^ ; ) = a( ; ) (A^; ) = a0(; ). (; A^ ) = a0(; ) a0 , (a a0)(; ) = 0. a 6= a0, .
19 , feng .
- .
, Va. -
. , . . Cn, A - det(A^ a1) = 0, a. n, n . n . Cn.
5.2.3 30. (unitary) U^ U^ y = U^1. 20. () U^ jj = 1.. U^ = k k = 1. 1 = U^ y U^ y . o U^ , U^ y . = 1 jj2 = 1.
17, kU^k = 1 P^ . 21. ( ) feig fe0ig. - U^ e0i = U^ei, i.. U^
U^ =X
( ; ei)e0i:
U^ , ,(U^ yU^ ; ) = (U^ ; U^) =
Xi;j
( ; ei)(; ej)(e0i; e
0j) =
Xi
( ; ei)(ej ; ) = ( ; )
U^ yU^ = 1^. U^ U^ y = 1^
22. ( ) .. U^1U^2(U^1U^2)y = U^1U^2U^ y2 U^
y1 = U^1U^
y1 = 1^.
H , H U(H). Un = U(Cn) U1 = U(l2).
81
-
. ,
5.2.4 31. P^ P^ 2 = P^ . 23. 0 1.. 15 , P^ = , 2 = , = 0 = 1.
17, kP^k = 1 P^ . H (k k = 1),
P^ P^ = (; ) 2 H . P^ = j ih j. P^ .
n S = fe1; : : : ; eng.. P^S P^S =
Pni=1(; ei)ei 2 H. P^S
. P^S =Pn
i=1 jiihij . -
. . 24. ( ) ( ) V H P^V , o. ,
V P^V P^V = V , V 2 H V 11. V ? P^V ? = 1^ P^V .
P^ V1 1 P^ .
5.1. 5.1.
, , - ( ) - .
, / . , (, , ) -. A . . (1.36) . ,
1. A. A = 1 A.2. A \B. A\B = A B .3. A, B , A \ B = ;.
A B = 0.4. A A[B. A[B = A+B
A B .5. A B, A B = A.
A;B A;B .
,
82
-
5.
. A . , (
) . (), - (, , ).
, . .
. : ( C) = (A ) ( C).
. - . . .
( ) /, .
1. V , - V ?. - P^V ? = 1^ P^V .
2. VA VB , A VA VB . P^VA P^VA = P^VB P^VA = P^VA .
3. A A . VA V ?B . , VA VB 0 .
4. VA VB : P^VA P^VB = P^VB P^VA .
P^V1 P^V2 P^V1 + P^V2 P^V1 P^V2 -
.
, .
G. Birkho and J. vonNeumann,Annals of Mathemacs, 37, 823 (1936).
5.2.5 32. A^ (A^ ; ) 0 2 H. A^ , A^ 0. 25. ( ) -.. A^ a < 0, (A^ ; ) =a( ; ) < 0. .
83
-
. ,
26. () > 0 A^ 0; B^ 0, A^+ B^ 0.. .
33. ( ) A^ B^, A^ B^, A^ B^ 0. 27. A^, A^yA^ 0.. , (A^yA^; ) = (A^; A^) = kA^k2 0.
5.3 A^,
A^n = A^A^ : : : A^; n : (5.3)
hn(x) =Pn
k=0 ckxk, n , ck,
hn(A^). -
. - , . , . , - f(x) = ex hn(x) =
Pnk=1 x
k/k!, n ! 1. f(A^) f : R! R.
. -, fn(x) = (1 + tanh(nx))/2 n ! 0 -
(x) =
0 x < 01 x > 0
: (5.4)
,
gn(x) =1
2(fn(x a) + fn(b x)) (5.5)
[a;b](x) [a; b] R. -
f(A^) f :R! C . . .
: eA^ =P1n=1 1n!A^n. : U(A^) -
U R. A^: n
pA^.
84
-
5.
27, A^, A^yA^ 0. - jA^j :=
pA^yA^.
15 hn(x), hn(A^) =hn(a) , A^ = a . .
28. ( ) A^ - A^ = a , f(A^) f
f(A^) = f(a) : (5.6)
A^ -. R f(x) = g(x)h(x),
f(A^) = g(A^)h(A^): (5.7)
, .
eixeix = 1 (eiA^)y = eiA^ A^ , eiA^ .
2U = U , U(A^) , A^ .
, f(x) = g(x) , - : f(A^) = g(A^).
1. (1 x)1 = 1 + x + x2 + x3 + : : : jxj < 1, A^, jjA^jj < 1,
(1 A^)1 = 1 + A^+ A^2 + A^3 + : : : : (5.8)
2. eix = lim!0R1+i1+i
deix A^,
eiA^ = lim!0
Z 1+i1+i
dei( A^)1: (5.9)
( A^)1 . (5.9) (resolvent) - A^.
5.4 34. A^ B^ [A^; B^] := A^B^ B^A^. 29. ( ) A^ B^ , C^ = i[A^; B^] .. C^y = i(A^B^ B^A^)y = i(B^A^ A^B^) = i[B^; A^] = C^.
85
-
. ,
30. ( ) .
1. [A^; 1^] = 0.
2. [A^; B^] = [B^; A^].3. [A^; B^ + C^] = [A^; B^] + [A^; C^].
4. [A^; B^C^] = [A^; B^]C^ + B^[A^; C^].
5. [A^; [B^; C^]] + [C^; [A^; B^]] + [B^; [C^; A^]] = 0. ( Jacobi)
31. ( BakerCampbellHausdor.)
eA^eB^ = expA^+ B^ +
1
2[A^; B^] +
1
12[A^; [A^; B^]] 1
12[B^; [A^; B^]] + : : :
:
. C^ = log[eA^eB^]. eA^eB^ = (1+ A^+ 12A^2+ 13!A^3 : : :)(1+ B^+ 12B^2+ 13!B^3+ : : :) =1 + (A^ + B^) + 12(A^
2 + B^2 + 2A^B^) + 16(A^3 + 3A^2B^ + 3A^B^2 + B^3) + : : :.
log(1 + x) = x 12x2 + 13x3 + : : : C^. A^ B^ . .
32. ( )
eA^B^eA^ = B^ + [A^; B^] +1
2![A^; [A^; B^]] +
1
3![A^; [A^; [A^; B^]]] + : : : :
. esA^B^esA^ Taylor s = 0. s = 1. Taylor B^. [A^; ]. .
x^ p^ - (3.6), [X^; P^ ] = i. , o L2(R; dx x^ (x) = x (x) p^ = i@ (x)/@x, 2 L2(R; dx).
[x^n; p^] = inX^n1: (5.10)
f(x) Taylor
[f(x^); P^ ] = if 0(x^): (5.11)
[x^; f(p^)] = if 0(p^).
5.5
. - , - . - .
86
-
5.
, (). . , , .
5.5.1 -
. - .
A^ CN . - an, n = 1; 2; : : : ; N . jni, jni ! eijni.
N jni CN , 1^ =PNn=1 jnihnj. ,
A^ =NXn=1
anjnihnj: (5.12)
Aij = hijA^jji, jni (un)i = hijni, (5.12)
Aij =NXn=1
an(un)i(un)j ; (5.13)
. A^ CN
. K , K < N . 5.1.2, an,n = 1; 2; : : : ; K Vn, an. Dn Vn - ,
PKn=1Dn = K.
P^n Vn - A^.
33. ( .) P^n A^,
1.PK
n=1 P^n = 1 ( )
2. P^nP^m = P^nnm ( ).
. n, Dn jn; ini, in = 1; 2; : : : ; Dn Vn, hn; injn; jni = injn . Vn X
in
jn; inihn; inj = P^n: (5.14)
19, hn; injm; jmi = 0 n 6= m. P^nP^n = 0 n 6= m, .
87
-
. ,
jn; ini CN : PKn=1Dn = N .
KXn=1
DnXin=1
jn; inihn; inj = 1^; (5.15)
. (5.14) .
- (. . 1.4.6), . 5.7. ( ) .
, (5.13)
A^ =KXn=1
DnXin=1
anjn; inihn; inj =KXn=1
anP^n: (5.16)
28 f ,
f(A^) =KXn=1
f(an)P^n: (5.17)
. 34. ( ) - A^ an - P^n,
1. O .
2. . (5.16).
3. f : R! R . (5.17).
1. C3, . . 5.8.1.
2. C3 /
A^ =
0@ 2 0 00 1 3i0 3i 1
1A : det(A^ 1^) = 0,
(+ 2)(2 2 8) = 0; = 2 = 4. = 4
j4i = 1p2
0@ 0i1
1A88
-
5.
= 2 0@ c1c2ic2
1A ; c1; c2 2 C, V2. V2
j 2; ai =0@ 10
0
1A j 2; bi = 1p2
0@ 01i
1A : = 4
P^4 = j4ih4j = 12
0@ 0i1
1A 0 i 1 = 12
0@ 0 0 00 1 i0 i 1
1A : = 2
P^2 = j 2; aih2; aj+ j 2; bih2; bj
=
0@ 100
1A 1 0 0 + 12
0@ 01i
1A 0 1 i
=
0@ 1 0 00 12 i20 i2
12
1A : A^
A^ = (2)0@ 1 0 00 12 i2
0 i212
1A+ (4)12
0@ 0 0 00 1 i0 i 1
1A : eiA^x . (5.17)
eiA^x = e2ix
0@ 1 0 00 12 i20 i2
12
1A+ e4ix 12
0@ 0 0 00 1 i0 i 1
1A :5.5.2
. , , .
p^ = i@/@x L2(R; dx). k(x) = eikx p^k = kk, - , . - . , . ket jki p^jki = kjki, .
89
-
. ,
ket jki. , -, ket jki .
1. , (A^) A^. (A^) A^ ( ) ( ). ket jki , - . 5.2.
2. , ket . 5.3.
-, , - , p^ = i@/@x L2(R; dx) .
5.2.
35. (A^) A^ H A^ 1^ .
.1. A^ (A^). , A^ 1^
, .2. x^ L2(R; dx). x^ 1^ -
(x). (x) (x )1. , x = . , = R + iI ,
k(x^ )1 k2 =Zdx
j (x)j2(x R)2 + 2I
1jI j2Zdxj (x)j2 (5.18)
(x) x = R. jj(x^ )1jj = jI j1, (x^ )1 I = 0. (x^) = R.
3. H^ = @2x L2(R; dx). (k) Fourier (x). H^ (x) =
Rdkeikxk2 (k), (H^ 1)1 (x) =R
dkeikx(k2 )1 (k). 0 k . , (H^ 1^)1 - 0, (H^) = R+.
1. A^ p(A^) (A^), -
(point spectrum) . 2 p(A^), (A^ 1^)1 2 V, V o A^.
2. (absolutely connuous spectrum) c(A^) A^ (A^) (A^
90
-
5.
1^) . 2 c(A^) (A^ 1^)1 . V H (A^1^)1 2 V . . 2 , (x )1 (x) 2 R () 6= 0. , (x) 0(x) 0() = 0 (x) = 0(x) (x) [ /2; + /2] . kk2 = R dxj(x)j2 < c2, c j(x)j. 0 k 0k2 < , (x)1 0(x) . (x )1 0(x) . x^ .
3. (singular spectrum) s(A^) . , -.
4. (A^) = p(A^) [ c(A^) [ s(A^). . , A^
H, HpHcHs A^p, A^c A^s , .
5.5.3
. p^ = i@x H = L2(R; dx).
fk;(x) =1p2
eikx12x2 ; (5.19)
> 0. , ketjk; i. ket jki
hkj i = lim!0hk; j i = lim
!0
Zdxp2
eikx12x2 (x) (5.20)
j i 2 H. , braket hkj i - (x): (k) = 1p
2
Rdxeikx (x).
(5.20) . lim!0(i@xfk; kfk;) = 0.
lim!0
(p^ k1^)jk; i = 0; (5.21)
p^jki = kjki: (5.22) . (5.22)
, . (5.21).
91
-
. ,
, hk; jk0; i = (k k0),
() =
r1
4e
2
4 : (5.23)
()
1. () = () 0.2. lim!0 () = 0 6= 0. lim!0 (0) =1.3.Rd() = 1. lim!0
Rdn() = 0, n.
4. f(),
lim!0
Zdf()() = f(0): (5.24)
To () ! 0, () . i , - , .
36. (x), Zdxf(x)(x) = f(0) (5.25)
f 0. (5.25) .
(5.23). (5.23) , . , .
lim
!0
(2 + x2)= (x): (5.26)
, (5.24)
35. ( .)
1. (x) = (x)..
Rdx(x)f(x) = R dy(y)f(y) = (0) = R dx(x)f(x).
f , .
2. (g(x)) =P
i(xxi)jg0(xi)j , xi g(xi) = 0.
. RU1dx[g(x)]f(x) U1 x1
y = g(x) 1-1 . y, dy = g0(x)dx = g0(g1(y))dx. Z
U1
dy(y)f(g1(y))jg0(g1(y))j =
f(g1(0))jg0(g1(0))j =
f(x1)
jg0(x1)j =1
jg0(x1)jZUdx(x x1)f(x):
92
-
5.
x = xi Zdx[g(x)]f(x) =
Xi
jg0(xi)j1Zdx(x xi)f(x)
f , .
3. (ax) = 1jaj(x).. 2 f(x) = ax.
4.R11 dke
ikx = 2(x)..
R11 dke
ikx 12k2 =
pe
k2
4 = 2(x), - . (5.23). ! 0.
, o hk; jk0; i ! 0,
hkjk0i = (k k0): (5.27)
5.3.
. 4.3.6 H - H C. Riesz 1-1 .
, . .4.1.3 1-1 . . H H , H . Riesz H = H
H : (5.28)
jki . , H = L2(R) , jxja a > 0. (x) ,
f 2 !Zdx(x)f(x) = f(0); (5.29)
C. .
(;H;) (rigged Hilbert space).
.
36. ( .) j i; ji 2 L2(R), Zdkknhjkihkj i = hjp^nj i; (5.30)
93
-
. ,
n = 0; 1; 2; : : :, jki (5.20).. . (5.20) hkj i = (2)1/2 R dxeikx (x).
I =
Zdkknhjkihkj i =
Zdkdxdx0kn(x0)eik(x
0x) (x)
=
Zdk
2dxdx0kneik(x
0x)(x0) (x):
35.4, Zdk
2kneikx = (i@x)n
Zdk
2eikx = (i@x)n(x):
I =Rdxdx0(i@x0)n(x0 x)(x0) (x). ,
I =
Zdxdx0(x x0)(i@x0)n(x0) (x) =
Zdx(i@x)n(x) (x)
=
Zdx(x)(i@x)n (x) = hjp^nj i:
(5.30) j i; ji 2 L2(R), Zdkknjkihkj = p^n; n = 0; 1; 2; : : : : (5.31)
(5.31) 1. n = 0, :
Rdkjkihkj = 1^.
2. n = 1 p^:Rdkkjkihkj = p^.
3. hn(k),Rdkhn(k)jkihkj = hn(p^)
4. f
Rdkf(k)jkihkj = f(p^)
34, - . jpihpj , 36.
f U U R, -
P^U = U(p^) =
ZU
dkjkihkj; (5.32)
( ) U . P^U 1. P^U1P^U2 = P^U1\U2 .
2. U1 \ U2 = 0, P^U1 + P^U2 = P^U1[U2 .3. P^R = 1^. P^U U R -
(Projector-Valued-Measure, PVM).
94
-
5.
5.5.4 .
A^ . -: H Hc Hd , Hc Hd .
A^ 36 .
-, . , .
c(A^) R, A^. - U c(A^) P^U = U(A^), .
1. P^U1P^U2 = P^U1\U2 .
2. U1 \ U2 = 0, P^U1 + P^U2 = P^U1[U2 .3. P^R = 1^.
P^U . (5.32).
P^, ,
P^ = P^[;+] (5.33)
! 0. P^ , jkihkj , .
1^ =
ZdP^ (5.34)
A^ =
ZP^d (5.35)
f(A^) =
Zf()P^d: (5.36)
1. x^ L2(R; dx)
x^ =
Zdxxjxihxj; (5.37)
ket jxi hxjx0i = (x x0): (5.38)
x ket jxi x = (x x). (x)
(x) = hxj i: (5.39)
95
-
. ,
jki
hxjki = 1p2
eikx: (5.40)
2. ket j i - (x) = hxj i, - (k) = hkj i. , p^ (x): p^ (x) =i@ @x ,
p^j i =Zdkkhkj ijki; (5.41)
p^ (k) : p^ (k) = k (k). - . - . , .
3. h^ = p^2 L2(R; dx) h^ =R10 dP^,
P^ =1
2p(jkihkj+ j kihkj) ; (5.42)
k =p jki ket .
4. L2(R3; dx1dx2dx3) jx1; x2; x3i,
(x1; x2; x3) = hx1; x2; x3j i: (5.43) hx1; x2; x3jx01; x02; x03i = (x1 x01)(x2 x02)(x3 x03).
x^i =
Zd3xxijx1; x2; x3ihx1; x2; x3j: (5.44)
P^x1 x^1
P^x1 =
Zdx2dx3jx1; x2; x3ihx1; x2; x3j: (5.45)
P^x2 P^x3 . o . - r^ , r^jri = rjri, hrjr0i = 3(r r0).
5.6 37. A^ H
TrA^ =Xn
hnjA^jni; (5.46)
n H.
96
-
5.
37. .. jn0i Tr0 .
Tr0A^ =Xn0hn0jA^jn0i =
Xn0
Xm;n
hn0jnihnjA^jmihmjn0i
=Xm;n
hmjnihnjA^jmi =Xn
Xn
hnjA^jni = TrA^:
38. 1. Tr(A^+ B^) = TrA^+ B^, 2 C.
. .
2. Tr(A^B^) = Tr(B^A^)..
Tr(A^B^) =Xn
hnjA^B^jni =Xn
Xm
hnjA^jmihmjB^jni
=Xm
hmjB^A^jmi = Tr(B^A^):
3. Tr(j ihj) = hj i.. Tr(j ihj) =Pnhnj ihjni = hj1^j i = hj i:
4. P^ N - TrP^ = N .. P^ PNi=1 jiihij, jii . TrP^ =
PNi=1hijii = N .
5. A^ =P
n anP^n, TrA^ =P
n anDn, Dn =TrP^n.. .
. . , - P^U =
RUdkjkihkj. ,
TrP^U =
ZU
dkhkjki =ZU
dk(0) =1:
5.4. Hilbert-Schmidt
HHS - H. , .
38. ( Hilbert-Schmidt) A^; B^
hA^; B^yi = Tr(A^B^y): (5.47)
jjA^jjHS =qTr(A^yA^): (5.48)
jjA^jjHS
-
. ,
Schwarz Hilbert-Schmidt
jTr(A^B^y)j2 Tr(A^yA^)Tr(B^yB^): (5.49)
A^ B^ Hilbert-Schmidt, .
A^ A^ =P
n anP^n,
kA^kHS =sX
n
Dnjanj2; (5.50)
Dn = TrP^n . 17 Hilbert-Schmidt
(. 21) kA^kHS kA^k. , ,
kA^ktr = TrjA^j; (5.51)
jA^j A^. kA^ktr 1 .
A^,
kA^ktr =Xn
Dnjanj: (5.52)
xi x21 + x22 + : : : x2n (jx1j+jx2j+ : : : jxnj)2,
kA^ktr kA^kHS kA^k (5.53)
5.7 , -
, . 1.4.6.
C, - . := C C X
= 1: (5.54)
f : ! R,
f(x) =X
(x): x 2 : (5.55)
(5.54) (5.55) .
1. A^ H$ f : ! R.2. A^$ f .
98
-
5.
3. A^$ C .4. H$ . .
2. A^ H - . , - A^, o A^.
, - (ne-grained). , . - (coarse-grained).
A^1 -. , A^2; A^3 , ([A^a; A^b] =0, a; b = 0; 1; 2; : : : ; N ), jni
A^ajni = a;njni: (5.56) jni N - (1;n; 2;n; : : : ; N;n). .
, . - . , -.
, . , x^ L2(R; dx), (x^) = R, n ,
n = [(n 12); (n+
1
2)]; (5.57)
n 2 Z. n xn = n. .
P^n =
Zn
dxjxihxj (5.58)
Xn
P^n =
Zn
dxjxihxj =Z[nn
dxjxihxj =ZRdxjxihxj = 1^; (5.59)
P^nP^m = mnP^n: (5.60)
99
-
. ,
2 . . , , .
, , . A^ A^ = Pn anP^n, P^n . , - Q^, jqi. P^n jqi + jq0i q 6= q0. [Q^; P^n] = 0 [Q^; A^] = 0. Q^ - .
. ; ; , -. , - : . .
. - . - , , , -. . , - , - .
5.8 5.8.1 qubit
o C2 - . C2 . , .
.
1. - j0i j1i. j0i; j1i .
100
-
5.
2. C2 ( ).
3. . - , R L. jLi, jRi , .
qubit, , . qubit .
Pauli
C2
A^ = a01 +Xi
ai^i; (5.61)
i Pauli,
^1 =
0 11 0
; ^2 =
0 ii 0
; ^3 =
1 00 1
: (5.62)
Pauli
1. (^i)2 = 1^.
2. ^1^2 = i3, ^2^1 = i3.
3. ^2^3 = i1, ^3^2 = i1.
4. ^3^1 = i2, ^1^3 = i2.
^i^j = 1^ij + i3X
k=1
ijk^k; (5.63)
ijk
ijk =
8
-
. ,
A^ = a01 +P
i ai^i a+ = a0 + jaj a = a0 jaj, jaj = pa a.
ja+i = 1p2jaj(jaj a3)
a1 ia2jaj a3
;
jai = 1p2jaj(jaj a3)
jaj+ a3a1 ia2
: (5.65)
P^ = jaihaj = 12
1
Xi
aijaj ^i
!: (5.66)
B qubit
n = ajaj : (5.67)
R3, S2. a0 - , jaj . qubit -
Pni=1 ni^i, n
Pauli = (^1; ^2; ^3), (5.61)
A^ = a01^ + a ; (5.68)
(5.66)
P^ =1
2(1 n ) : (5.69)
, (5.63)
(a )(b ) = a b+ i(a b) ; (5.70)
a b =3Xi=1
aibi (5.71)
(a b)k =3Xi=1
3Xi=1
ijkaibj
= (a2b3 a3b2; a3b1 a1b3; a1b2 a2b1): (5.72)
102
-
5.
qubit, 01
;
10
