19
Schrödinger Hamilton 是时Goldstein , Classical Mechanics, 9.6 . , , ()= /~ (1) , , . 子作用于, 使始时 =0 , |(0)⟩−→|()= () |(0)= /~ |(0). 得到 |(), 的导, ~ d |()d = |()(2) Schrödinger . . , Schrödinger , Heisenberg , Dirac . Dirac , . (2) , 明描Hilbert , Schrödinger . , —— , 定的, (2) —— 分方分方. 1

Schrödinger 程 - USTC

  • Upload
    others

  • View
    15

  • Download
    0

Embed Size (px)

Citation preview

, ,
, = 0 ,
|(0) −→ |() = () |(0) = −/~ |(0) .
|() , ,
~ d |()
Dirac , .
Schrödinger . , —— ,
, (2) ——
.
1
. , , Schrödinger
, (1),,, (). ,
mechanics 72 .
.
~ d()
d = [,()].
C. = ∑
||. , H { |
} .
= ∑
() |. Schrödinger
~
1()
2() ...
()
. , .
, (). , ,
= ∑
||
, |. , | ,
. Schrödinger .
() = (0)−/~
, () (0) (),
() = ()(0) †().
= diag ( −1/~, −2/~, · · · , −/~
) |(0) , , |(0) = |(0),
|() = −/~ |(0) = −/~ |(0)
, |() = |(), , ,
, , (stationary state).
|(0) , , ,
|(0) = ∑
(0) | .
|()|2 = |−/~(0)|2 = |(0)|2
. , .
| = ∑
|, : |0|2 , |1|2 , . , ,
= ∑
3

(0) = ||(0)|2
() = ||()|2
2 () |(0), |. , ,
. , , .
, . |().
() = ()||().
d
, , . ,
1/2 , () , () ; ,
1 2 ()22, .
(3) .
d d




|(), . , |()
|() = ∑
() = ∑ ,
(0)*(0) | | (−)/~
, | | . () ,
= −
~ .
, , . Bohr .
| | ( = ) , .
, . , | | ( = ) , ,
[,] = 0, .
.
| | , [,] = 0, . |() “” . |() , (
|) () = | |() |2 =

. , , ——
.
.
=
=
= ||~ 2
2, , ~ 2 ,
=
·
,
2 ~0 = 0 = 0 · .
±1 2 ~0. |−, |+. |± ,
.
|(0) =
2
6
+0 2
() = − 0 2 = 1 cos
0
1(0) + 2(0) + 3(0) ()−−−−→
] +
= ~ 2
(4)
Bloch , . 0, .
, (4) , , ——
.
, Larmor , 0 Larmor
.


Schrödinger .
. () , ,
. , .
() ,
() = exp
() = †()() () = 1
2 ~1 +
~ d |()
d = ~
|()
†(), (), () ,
,
= √ 2 1 + 2, = (sin , 0, cos )
sin = 1√
2 1 + 2
eff() = exp
(5) (),
() .
:
|(0) = |0, +. (
), −1 () ?
, +1→−1.
+1→−1 = | 1|() |2
9

= 2
2 sin2
,
=
, Bloch .
10
|0 , Bloch .
. ,
.
, . , = 0 , Bloch
(0) , , .
11

, = 0.1. , , Bloch
, . , Bloch
.
() |(), = 0, 1,
() |() = () |()
= () |() . |()
|() = ∑
() |()
.
|0() Bloch .
Schrödinger ?
, , ,
~ d |()
.
() = |()|(0)
2 (6)
: () |(0), |(). Schrödinger .
: (6) , () |,
()| = |()
(7) | .
() (6) , |()|(0) . |() | , |() ; |(0) |() ,
|() | .
, .
() = †()() (8)
, . C2 .
= 1 2 ~, () = (), () , ,
. |.
, Schrödinger , ,
, , () , () , = 0
(0) ().
,, : ()
. () ,
(0) = (0) −→ () = ()()
() = −/~, () = †()(), Heisenberg .
() , () . , d d ()
, d()
d †()()
()
d () () †() .
, , (8) ,
() = †()()() (10)
(). d()
d d†()
~ d |()
d = |()
~ d()
14
() 1, , †() . (11) ,
d()
()
1
()
, †()() †()()(),
,
, () = .
.
,
= 0, .
= 0,
[ , ] = 0 =⇒
, , () = , [(), ] = 0 ,
[(), ] = [ †()(), ] = †()[,]()
() () = −/~ [(), ] = 0, ,
()
Schrödinger . ,

, . Noether
15

2 cos
2 |0 +
(0) = 1

†() sin cos+ () †() sin sin+ ()
†() cos ]
() †() =
⇓ () =
1
2 (1 + sin cos(+ ) + sin sin(+ ) + cos )
(0) = (
() = (
sin cos(+ ) sin sin(+ ) cos )
, , () †(), .
, . , Bloch
, , Pauli ,
—— .
() = 1
16
|(0) = ∑
Pauli .
(13) , ,,() ,, () ,
,,() = ()|,, |(). Bloch ()
()
()
() = (0)| †() () |(0) = (0)|()|(0)
, Heisenberg , () = †() (). ()
()
()
, .
| 1|2 | , arg 1|2 .
, 1|2 = 0, . 2S. Pancharatnam, Generalized theory of interference, and its applications. Proc. Indian Acad. Sci. 44, 247-262
(1956).
17
2 |1, () = −
2 ,
|() = () |(0) = − 2 cos


2 ,

= 0 |(0) |(), () = arg (0)|(). , , .
, , ().
() ()?
, + d. |() −→ |(+ d).
d = arg ()|(+ d)

()|(+ d) = 1 + ()|() d (17)
()|() ,
) ≈ − ()|() d,
= − ()|()
() = − ∫
() = () − () (18)
|() = () |(0) = − 2 cos

, (cyclic evolution), = 2,
(0) = |(0)(0)| = |( )( )| = ( ),
|( ) = − |(0), ( ) = , ( ) = − cos ,
( ) = (1 + cos ) mod 2.
, . , ,
, “” .