Конспект лекций по квантовой физике
.pdf3.6.ˆá¯®«ì§ãï á®®â-®è¥-¨¥ -¥®¯à¥¤¥«¥--®á⥩, ®æ¥-¨âì í-¥à- £¨î ®á-®¢-®£® á®áâ®ï-¨ï ç áâ¨æë ¢ ¯®«¥ U (x) = jxj.
3.7.ˆá¯®«ì§ãï á®®â-®è¥-¨¥ -¥®¯à¥¤¥«¥--®á⥩, ®æ¥-¨âì £«ã¡¨-
-ã ã஢-ï ¢ ®¤-®¬¥à-®© ¬¥«ª®© ﬥ.
x4. Š®®à¤¨- â-®¥ ¨ ¨¬¯ã«ìá-®¥ ¯à¥¤áâ ¢«¥-¨ï.
Ž¯¥à â®àë 䨧¨ç¥áª¨å ¢¥«¨ç¨-
•«®â-®áâì ¢¥à®ïâ-®á⨠- ©â¨ ç áâ¨æã ¢ â®çª¥ x | ¢¥«¨ç¨- dW=dx
| ¯à®¯®à樮- «ì- j (x; t)j2. …᫨ |
(x; t) -®à¬¨à®¢ - ãá«®¢¨¥¬ |
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â® |
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dxj (x; t)j2 = 1; |
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dW (x; t) |
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= j |
(x; t)j2 : |
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dx |
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’®£¤ á।-¥¥ §- ç¥-¨¥ x à ¢-® |
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hxi = Z |
x dW = Z |
x j (x)j2 dx = Z |
dx (x) x (x) : |
А- «®£¨ç-®, á।-¥¥ §- ç¥-¨¥ «î¡®© äã-ªæ¨¨ F (x) à ¢-ï¥âáï
Z
hF (x)i = dx (x) F (x) (x) :
ɇǬ
Z
(x) = dk A(k) eikx ;
â® ¢¥à®ïâ-®áâì - ©â¨ ç áâ¨æã á ¨¬¯ã«ìᮬ p = hk ¯à®¯®à樮- «ì-
jA(k)j2, ¨«¨
dW (k) / jA(k)j2 : dk
ɇǬ
Z
dx j (x)j2 = 1;
â® ¨
Z
dk j'(k)j2 = 1 :
‡¤¥áì
A(k)
'(k) = p
2
| -®à¬¨à®¢ --ë© ”ãàì¥-®¡à § äã-ªæ¨¨ (x), â® ¥áâì
(x) = Z |
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eikx |
Z |
e−ikx |
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dk '(k) |
p |
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dx (x) p |
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(4:1) |
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•®í⮬ã |
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dW |
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= j'(k)j2 |
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¨ |
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dk |
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hF (k)i = Z |
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dk ' (k) F (k) '(k) : |
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‚ëà §¨¬ hpi ç¥à¥§ |
(x). •®¤áâ ¢«ïï ¢ á®®â-®è¥-¨¥ |
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hpi = Z |
dk ' (k) hk '(k) |
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¢ëà ¦¥-¨¥ '(k) ç¥à¥§ |
(x) ¨§ (4.1), ¯®«ã稬 |
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p |
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dk 8 |
dx0 |
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eikx0 |
9 hk 8 dx (x)e− |
ikx |
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h i |
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p2 |
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p2 |
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ˆá¯®«ì§ãï ⮦¤¥á⢮ |
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ke−ikx = i |
d |
e−ikx |
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dx |
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¨ ¨-⥣à¨àãï ¯® x ¯® ç áâï¬, ¯®«ã稬 ®ª®-ç ⥫ì-® |
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p |
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dx |
(x) |
− |
ih |
d |
! |
(x) : |
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h i |
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dx |
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‡¤¥áì ¯à¨ ¨-⥣à¨à®¢ -¨¨ ¯® k ¨á¯®«ì§®¢ - |
¨§¢¥áâ- ï ä®à¬ã« |
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Z |
dk eik(x0 −x) = 2 (x0 − x) : |
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’ ª¨¬ ®¡à §®¬, ¯à¨ - 宦¤¥-¨¨ hpi ¬®¦-® ¯®«ì§®¢ âìáï ä®à¬ã«®©
hpi = Z |
dx (x) p^ (x) ; |
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£¤¥ ®¯¥à â®à |
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p^ = −ih |
d |
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(4:2) |
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dx |
‚ª¢ -⮢®© ¬¥å -¨ª¥ ¯®áâ㫨àã¥âáï, çâ® ¤¨- ¬¨ç¥áª¨¥ ¯¥à¥-
¬¥--ë¥ ®¯¨áë¢ îâáï ®¯¥à â®à ¬¨, â ª çâ® á।-¥¥ §- ç¥-¨¥ -¥- ª®â®à®© ¯¥à¥¬¥--®© A ¢ á®áâ®ï-¨¨ á § ¤ --®© ¢®«-®¢®© äã-ªæ¨¥©
(x) (¨«¨ '(p)) à ¢-®
Z Z
hAi = dx (x) A^ (x) = dp ' (p) A^ '(p):
‚ ç áâ-®áâ¨, ®¯¥à â®à ¨¬¯ã«ìá ¢ x-¯à®áâà -á⢥ ®¯à¥¤¥«ï¥âáï ä®à- ¬ã«®© (4.2), ¢ p-¯à®áâà -á⢥ | íâ® ¯à®áâ® ®¯¥à â®à ã¬-®¦¥-¨ï p^ = p. А- «®£¨ç-®, ®¯¥à â®à x^ = x ¢ x-¯à®áâà -á⢥ ¨
x^ = +ihdpd
12
¢ p-¯à®áâà -á⢥.
ˆ§ ®¯¥à â®à®¢ ^r ¨ p^ áâà®ïâáï ¢á¥ ¤¨- ¬¨ç¥áª¨¥ ¯¥à¥¬¥--ë¥. • - ¯à¨¬¥à, ®¯¥à â®à ¬®¬¥-â ¨¬¯ã«ìá
M^ = ^r p^ = −ihr r :
‚Ž••Ž‘› |
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4.1. „«ï ¯®â¥-æ¨ «ì-®£® ï騪 |
¢¨¤ |
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U(x) = |
8 |
1 |
¯à¨ x < 0 |
> |
0 |
¯à¨ 0 < x < a |
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< |
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¯à¨ x > a |
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- ©â¨ En ¨ n(x). Žæ¥-¨âì En ¤«ï
) ç áâ¨æë ¬ ááë m 1 £ ¢ ï騪¥ á a 1 á¬;
¡) ¬®«¥ªã«ë H2 ¢ ï騪¥ á a 1 á¬; - ©â¨ n, ᮮ⢥âáâ¢ãî騩 í-¥à£¨¨ En kT , £¤¥ T 300 Š; ®æ¥-¨âì (En − En−1)=En ¤«ï ¤ --®© í-¥à£¨¨;
¢) í«¥ªâà®- ¢ ï騪¥ á a 10−8 á¬.
‘à ¢-¨âì ª« áá¨ç¥áªãî ¯«®â-®áâì ¢¥à®ïâ-®áâ¨, ®¯à¥¤¥«¥--ãî
á®®â-®è¥-¨¥¬ |
dW (x)ª« áá |
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dx |
v(x)Tª« áá |
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£¤¥ Tª« áá | ª« áá¨ç¥áª¨© ¯¥à¨®¤ ª®«¥¡ -¨©, ¨ ª¢ -⮢ãî ¯«®â-®áâì ¢¥à®ïâ-®á⨠dW=dx = j n(x)j2 ¯à¨ n = 1 ¨ n 1. •à®¢¥á⨠⠪®¥ ¦¥ áà ¢-¥-¨¥ ¤«ï dW=dp | ¯«®â-®á⨠¢¥à®ïâ-®á⨠¢ ¨¬¯ã«ìá-®¬ ¯à®- áâà -á⢥.
4.2.• ©â¨ ¨§¬¥-¥-¨¥ á â¥ç¥-¨¥¬ ¢à¥¬¥-¨ ¢®«-®¢®© äã-ªæ¨¨ -¥-
५ï⨢¨áâ᪮© ᢮¡®¤-®© ç áâ¨æë ¬ ááë m, ¥á«¨ ¢ - ç «ì-ë© ¬®-
¬¥-⠢६¥-¨
(r; 0) = A e−(r2=a2)+ibr :
4.3. • ©â¨ '(k) ¤«ï
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e−r=a |
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h2 |
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(r) = |
p |
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a = |
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= 0; 53 |
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mee2 |
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a3 |
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(®á-®¢-®¥ á®áâ®ï-¨¥ ⮬ ¢®¤®à®¤ ). •ãáâì ¤ -- ï ¢®«-®¢ ï äã-ª- æ¨ï ®¯¨áë¢ ¥â á®áâ®ï-¨¥ ᢮¡®¤-®£® í«¥ªâà®- ¯à¨ t = 0. Žæ¥-¨âì, - ª ª®¬ à ááâ®ï-¨¨ ®ª ¦¥âáï íâ®â í«¥ªâà®- ç¥à¥§ 1 á.
x5. Ž¯¥à â®à ƒ ¬¨«ìâ®- . “à ¢-¥-¨¥ ˜à¥¤¨-£¥à
Š« áá¨ç¥áª ï äã-ªæ¨ï ƒ ¬¨«ìâ®-
H = p2 + U(r)
2m
§ ¬¥-ï¥âáï ®¯¥à â®à®¬ ƒ ¬¨«ìâ®-
H^ = −2hm2 + U (r) :
‘®¡á⢥-- ï äã-ªæ¨ï í⮣® ®¯¥à â®à á ᮡá⢥--ë¬ §- ç¥-¨¥¬ En 㤮¢«¥â¢®àï¥â ãà ¢-¥-¨î ˜à¥¤¨-£¥à (“˜) (1925 £.)
H^ n(x) = En n(x) :
•¥и¥-¨п нв®£® га ¢-¥-¨¥п ¨йгвбп ¢ ª« бб¥ ®¤-®§- з-ле ¨ -¥¯а¥- ал¢-ле дг-ªж¨©. ‚ б«гз ¥ б¢п§ --ле б®бв®п-¨© нв¨ дг-ªж¨¨ -®а-
¬¨à㥬ë (¤«ï -¨å R dxj n(x)j2(=)1) ¨ ¯®í⮬ã |
n(x) ! 0 ¯à¨ x ! 1. |
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•®¢¥¤¥-¨¥ ¯à®¨§¢®¤-®© 0 x ®¯à¥¤¥«ï¥âáï ¢¨¤®¬ ¯®â¥-æ¨ « . ˆ-- |
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⥣à¨àãï “˜ ¢ ¬ «®© ®ªà¥áâ-®á⨠â®çª¨ x = a, ¯®«ãç ¥¬ |
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Zaa "" dx 00(x) = 0(a + ") − 0(a − ") = |
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+ |
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2m |
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Za−" |
dx [U(x) − E] (x) = |
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(a) |
Za−" |
dx U(x) ; |
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h2 |
h2 |
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â® ¥áâì |
0(x) -¥¯à¥àë¢- ¢ â®çª¥ x = a, ¥á«¨ ¯®â¥-æ¨ «ì- ï í-¥à£¨ï |
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U(x) -¥¯à¥àë¢- |
¢ í⮩ â®çª¥ ¨«¨ ¨¬¥¥â à §àë¢ 1-£® த |
(ª®-¥ç-ë© |
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᪠箪). |
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“ ¯®â¥-æ¨ «®¢, ¨¬¥îé¨å ᪠窨 2-£® த , |
0(x) ¬®¦¥â |
¨¬¥âì à §àë¢ë (á¬. ¯à¨¬¥à ¯®â¥-æ¨ «ì-®£® ï騪 ). „«ï U(x) = −G (x − a) ¨¬¥¥¬
0(a + ") − 0 |
(a − ") = − |
2mG |
(a) : |
(5:1) |
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h2 |
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„¨áªà¥â-ë¥ ã஢-¨ ¢ ®¤-®¬¥à-®© § ¤ ç¥ ¢á¥£¤ |
-¥¢ë஦¤¥- |
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-ë, â® ¥áâì ª ¦¤®¬ã ᮡá⢥-- |
®¬ã §- ç¥-¨î í-¥à£¨¨ ᮮ⢥âáâ¢ã- |
¥â ¥¤¨-á⢥-- ï ᮡá⢥-- ï äã-ªæ¨ï. „®¯ãá⨬ ®¡à â-®¥: ¯ãáâì
14
1(x) ¨ 2(x) | ¤¢¥ à §-ë¥ á®¡á⢥--ë¥ äã-ªæ¨¨ H^ , ®â¢¥ç î騥 ®¤-®¬ã §- ç¥-¨î E. ’®£¤
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2m |
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(U − E) = |
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¨«¨ |
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d |
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00 |
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00 = 0 = |
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0 2 |
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1 0 ): |
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Žâáî¤ á«¥¤ã¥â, çâ® |
0 |
2 |
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0 = const. |
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«¥¥, const = 0 ¨§-§ |
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¯®¢¥¤¥-¨ï n(x) - |
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¡¥áª®-¥ç-®áâ¨. ‚ ¨â®£¥, 1 = C 2. |
‚®¤-®¬¥à-®© § ¤ ç¥ ¤¨áªà¥â-ë¥ ã஢-¨ ç¥â-®£® £ ¬¨«ìâ®-¨-
-, H^ (−x) = H^ (x), ¨¬¥îâ ®¯à¥¤¥«¥--ãî ç¥â-®áâì, â® ¥áâì «¨¡®
n(−x) = + n(x), «¨¡® |
n(−x) = − n(x). „¥©á⢨⥫ì-®, ¤«ï â ª®£® |
£ ¬¨«ìâ®-¨ - äã-ªæ¨¨ |
n(x) ¨ n(−x) п¢«повбп а¥и¥-¨п¬¨, ®в¢¥- |
ç î騬¨ ®¤-®¬ã ¨ ⮬㠦¥ §- ç¥-¨î En, â® ¥áâì, ¯® ¯à¥¤ë¤ã饬ã
ã⢥ত¥-¨î, n(x) = C |
n(−x). ‘¤¥« ¢ ¥é¥ ®¤-® ®âà ¦¥-¨¥ ª®®à- |
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¤¨- â, ¯®«ã稬 |
n(−x) = C n(x) = C2 |
n (−x), ®âªã¤ C = 1. |
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•à¨¬¥à |
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•àאַ㣮«ì- ï ¯®â¥-æ¨ «ì- ï ï¬ |
£«ã¡¨-®î V ¨ è¨à¨-®î 2a, â® |
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¥áâì |
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U(x) = 8 |
−V |
¯à¨ jxj < a |
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¯à¨ x |
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j j |
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ç ¥â í-¥à£¨ï |
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‘¢ï§ --ë¬ á®áâ®ï-¨ï¬ ®â¢¥: |
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E < , ¯à¨ í⮬ “˜ ¨¬¥¥â |
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¢¨¤ |
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= q |
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+ k2 |
= 0 |
¯à¨ jxj < a; |
hk |
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2m(V − jEj) |
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00 |
− 2 |
= 0 |
¯à¨ jxj > a; |
h = q |
2mjEj |
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ˆé¥¬ à¥è¥-¨ï â ª¨¥, ç⮡ë (x) ¨ 0(x) ¡ë«¨ -¥¯à¥àë¢-ë, ç⮡ë (x) ! 0 ¯à¨ x ! 1 ¨ ç⮡ë (x) ¡ë« «¨¡® ç¥â-®©, «¨¡® -¥ç¥â-
-®© äã-ªæ¨¥©, â ª ª ª H^ (−x) = H^ (x).
—¥â-ë¥ à¥è¥-¨ï ¨¬¥îâ ¢¨¤
(x) = |
8 |
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¯à¨ |
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< a; |
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Be− j j |
¯à¨ |
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ˆ§ -¥¯à¥àë¢-®á⨠0 |
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x)= |
(x) ¢ â®çª¥ x = a ¯®«ãç ¥¬ ãà ¢-¥-¨¥ |
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= v |
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tg ka = |
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2mV |
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15
¤î饥 ¤¨áªà¥â-ë© àï¤ §- ç¥-¨© kn ¨«¨ En (í-¥à£¨ï ª¢ -âã¥âáï).
•©¤¨â¥ -¥ç¥â-ë¥ à¥è¥-¨ï ¨ ¯®ª ¦¨â¥, çâ® ç¥â-ë¥ ¨ -¥ç¥â-ë¥
га®¢-¨ з¥а¥¤говбп.
•®ª ¦¨â¥, çâ® ¢ ¬¥«ª®© ﬥ, V h2=(ma2), áãé¥áâ¢ã¥â «¨èì ®¤-
-® á¢ï§ --®¥ á®áâ®ï-¨¥ á í-¥à£¨¥© |
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h2 2 |
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2aV m |
E0 = − |
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h2 |
¨ ¢®«-®¢®© äã-ªæ¨¥©
0(x) p 0 e− 0jxj :
Žæ¥-¨â¥ x ¨ p ¤«ï â ª®© ï¬ë. •®ª ¦¨â¥, ¨á¯®«ì§ãï ãá«®¢¨¥ (5.1), çâ® ¯®â¥-æ¨ «ì-®© í-¥à£¨¨ U (x) = −G (x) ᮮ⢥áâ¢ã¥â ¬¥«ª ï ï¬
á
0 = mG : h2
Žá樫«ï樮-- ï ⥮६
‚®«-®¢ ï äã-ªæ¨ï ¤¨áªà¥â-®£® ᯥªâà n(x), ᮮ⢥âáâ¢ãîé ï (n+ 1)-¬ã ¯® ¢¥«¨ç¨-¥ ᮡá⢥--®¬ã §- ç¥-¨î En, ®¡à é ¥âáï ¢ -ã«ì (¯à¨ ª®-¥ç-ëå x) n à § (á¬. ¯à¨¬¥àë ¯®â¥-æ¨ «ì-®£® ï騪 , ®á- 樫«ïâ®à ¨ â.¤.).
‚Ž••Ž‘›:
5.1. • ©â¨ En ¨ n(x) ¤«ï ¯®«ï
U(x) = |
8 |
1 |
¯à¨ x < 0 |
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V |
¯à¨ 0 < x < a : |
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− |
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¯à¨ x > a |
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5.2. • ©â¨ ã஢-¨ í-¥à£¨¨ ¨ ¢®«-®¢ë¥ äã-ªæ¨¨ á¢ï§ --ëå á®áâ®- ï-¨© ç áâ¨æë ¢ ¯®«¥ ¤¢ãå -ï¬ U(x) = −G (x + a) − G (x − a) ¯à¨ ãá«®¢¨¨ a h2=(mG). ˆáá«¥¤®¢ âì § ¢¨á¨¬®áâì ã஢-¥© í-¥à£¨¨
®â a. |
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5.3. „«ï ¯®«ï, ®¯¨á --®£® ¢ ¯à¥¤ë¤ã饩 § ¤ ç¥, ®¯à¥¤¥«¨âì |
(x; t), |
¥á«¨ ¯à¨ t < 0 ¬¥¦¤ã ï¬ ¬¨ ¡ë« -¥¯à®-¨æ ¥¬ ï ¯¥à¥£®à®¤ª |
¨ ç - |
áâ¨æ - 室¨« áì ¢ áâ 樮- à-®¬ á¢ï§ --®¬ á®áâ®ï-¨¨ ¢¡«¨§¨ «¥- ¢®© ï¬ë.
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x6. •à¬¨â®¢ë ®¯¥à â®àë
• §®¢¥¬ ®¯¥à â®à B^ íନ⮢® ᮯà殮--ë¬ ª ®¯¥à â®àã A^, ¥á«¨
¤«ï «î¡ëå ¤¢ãå äã-ªæ¨© |
1 ¨ |
2 á¯à ¢¥¤«¨¢® á®®â-®è¥-¨¥ |
Z |
1 |
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dx A 2 = dx (B^ 1) 2 : |
’ ª®© ®¯¥à â®à ®¡®§- 稬 B^ = A^+: …᫨ A^ = A^+, â® ¥áâì ®¯¥à - â®à ᮢ¯ ¤ ¥â ᮠ᢮¨¬ íନ⮢® ᮯà殮--ë¬, - §®¢¥¬ ¥£® íନ- â®¢ë¬ (¨«¨ á ¬®á®¯à殮--ë¬). „«ï íନ⮢ ®¯¥à â®à
Z |
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‘®¡á⢥--ë¥ §- ç¥-¨ï íନ⮢ ®¯¥à â®à |
¢¥é¥á⢥--ë: |
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A^ = |
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Z |
dx |
A^ = Z |
dx (A^ ) |
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Žâáî¤ á«¥¤ã¥â, çâ® = :
А- «®£¨ç-® ¯®ª §ë¢ ¥âáï, çâ® á।-¥¥ §- ç¥-¨¥ íନ⮢ ®¯¥à -
â®à dx |
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{ ¢¥é¥á⢥--®¥ |
A ¢ ª ª®¬-«¨¡® ª¢ -⮢®¬ á®áâ®ï-¨¨ |
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ç¨á«®.R ‚ᥠ|
®¯¥à â®àë 䨧¨ç¥áª¨å ¢¥«¨ç¨- íନ⮢ë. |
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‘®¡á⢥-- |
ë¥ äã-ªæ¨¨, ®â¢¥ç î騥 à §«¨ç-ë¬ á®¡á⢥--ë¬ §- - |
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ç¥-¨ï¬ íନ⮢ |
®¯¥à â®à , ¢§ ¨¬-® ®à⮣®- «ì-ë. „¥©á⢨⥫ì- |
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-®, ¤®¬-®¦¨¢ A^ |
= - , (A^ ) = - |
, ¨ ¯à®¨-⥣à¨- |
஢ ¢, ¯®«ã稬 |
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â® ¥áâì |
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‚ á«ãç ¥ ¢ë஦¤¥-¨ï ¬®¦-® ¢ë¡à âì ᮡá⢥--ë¥ äã-ªæ¨¨ ®àâ®- £®- «ì-묨 ¨, ᮮ⢥âá⢥--®, ¨á¯®«ì§®¢ âì ®àâ®-®à¬¨à®¢ --ãî
á¨á⥬ã äã-ªæ¨©
Z
dx n = mn :
m
•®«-®â á¨á⥬ë ᮡá⢥--ëå äã-ªæ¨© íନ⮢®£® ®¯¥à â®à :
f (x) = |
n an n(x); an = Z |
dx0 |
n (x0)f (x0) ; |
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dx0 f (x0) n |
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n(x0) : |
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Žâáî¤ |
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„¨à ª®¢áª¨¥ ®¡®§- ç¥-¨ï. Œ âà¨ç-ë© í«¥¬¥-â
Z
Af i = dx f (x) A^ i(x) = hf jA^jii :
‚ íâ¨å ®¡®§- ç¥-¨ïå íନ⮢®áâì ¨¬¥¥â ¢¨¤
hf jA^jii = hijA^jf i ;
®àâ®-®à¬¨à㥬®áâì ®§- ç ¥â
hf jii = f i ;
¯®«-®â |
X jni hnj = 1 :
n
‚Ž••Ž‘›
6.1. • ©â¨ ®¯¥à â®àë, ᮯà殮--ë¥ ª ®¯¥à â®à ¬
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d |
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C^ = m!x + h |
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6.2. „«ï ®¯¥à â®à |
C^, ®¯à¥¤¥«¥--®£® ¢ ¯à¥¤ë¤ã饩 § ¤ ç¥, - ©- |
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⨠ᮡá⢥--ë¥ äã-ªæ¨¨ ¨ ᮡá⢥ |
--ë¥ §- ç¥-¨ï. •à®¢¥à¨âì, çâ® |
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ᮡá⢥-- |
ë¥ §- ç¥-¨ï í⮣® ®¯¥à â®à ¬®£ãâ ¡ëâì ª®¬¯«¥ªá-묨, |
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ᮡá⢥-- |
ë¥ äã-ªæ¨¨, ®â¢¥ç î騥 à §«¨ç-ë¬ á®¡á⢥ ë¬ §- --- |
ç¥-¨ï¬, -¥ ®¡ï§ ⥫ì-® ®à⮣®- «ì-ë.
6.3.•ãáâì A^ | íନ⮢ ®¯¥à â®à, A^ = A^+. •®ª ¦¨â¥, çâ® á।-¥¥
§- ç¥-¨¥ ª¢ ¤à â í⮣® ®¯¥à â®à -¥®âà¨æ ⥫ì-® h j A^2 j i 0.
6.4. • ©â¨ ᮡá⢥--ë¥ äã-ªæ¨¨ ®¯¥à â®à x^ ¢ x- ¨ p-¯à¥¤áâ ¢«¥-¨ïå. ’® ¦¥ ¤«ï ®¯¥à â®à p^.
6.5. |
• ©â¨ ¢¨¤ ®¯¥à â®à A^ = 1=r ¢ ¨¬¯ã«ìá-®¬ ¯à®áâà -á⢥ |
(§ ¤ ç |
1.47 ƒŠŠ). |
x7. ‹¨-¥©-ë© ®á樫«ïâ®à U(x) = 1m!2x2
2
“஢-¨ í-¥à£¨¨ ¨ ¢®«-®¢ë¥ äã-ªæ¨¨
‚ í⮩ § ¤ ç¥ ¥áâ¥á⢥-- ï á¨á⥬ ¥¤¨-¨æ ¢ª«îç ¥â h, m, !. ˆ§ -¨å
q
áâநâáï ¥¤¨-¨æ ¤«¨-ë ` = h=(m!), í-¥à£¨¨ h! ¨ â.¤. (- ©¤¨â¥
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¥¤¨-¨æë ¢à¥¬¥-¨, ᪮à®áâ¨, ¨¬¯ã«ìá , ᨫë). •¥à¥©¤¥¬ ª ¡¥§à §-
¬¥à-ë¬ ¢¥«¨ç¨- ¬
x0 = x` ; E0 = h!E ;
¯à¨ í⮬ ¢®«-®¢ ï äã-ªæ¨ï (x) á¢ï§ - |
á ¡¥§à §¬¥à-®© 0(x0) á®®â- |
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’®£¤ ¬ë ¯®«ã稬 “˜ ¢ ¢¨¤¥ |
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¢ ¤ «ì-¥©è¥¬ èâà¨å¨ ®¯ã᪠¥¬. |
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•à¨ x ! 1 ¨¬¥¥¬ d2 =dx2 = x2 , â® ¥áâì ! e x2=2: •®í⮬ã |
¨é¥¬ -®à¬¨à㥬ë¥, ã¡ë¢ î騥 - ¡¥áª®-¥ç-®á⨠à¥è¥-¨ï ¢ ¢¨¤¥
= e−x2=2v(x) ;
£¤¥
ˆé¥¬ v ¢ ¢¨¤¥ àï¤ |
v00 − 2xv0 + (2E − 1)v = 0 : |
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‚®§-¨ª î饥 â ª¨¬ ®¡à §®¬ ãà ¢-¥-¨¥ |
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1 |
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2n) a |
+ (n + 1)(n + 2) a |
+2] = 0 |
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¯à¨¢®¤¨â ª ४ãàà¥-â-®¬ã á®®â-®è¥-¨î ¤«ï ª®íää¨æ¨¥-⮢
2n + 1 − 2E an+2 = (n + 1)(n + 2) an :
Ž-® ®§- ç ¥â, ¢ ç áâ-®áâ¨, çâ® äã-ªæ¨ï v(x) ᮤ¥à¦¨â á« £ ¥¬ë¥ ®¤¨- ª®¢®© ç¥â-®áâ¨. “á«®¢¨¥
an+2 = 2 ! 0
an n
¯à¨ ¢á¥å x, -® v(x) ! ex2 ¯à¨ x ! 0 ¯à¨ x ! 1, -¥®¡å®¤¨¬® àï¤ ¤«ï
2E = 2n + 1:
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‚ ¨â®£¥ ¯®«ãç ¥¬ ã஢-¨ í-¥à£¨¨ ¨ -®à¬¨à®¢ --ë¥ ¢®«-®¢ë¥ äã-ª- 樨
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En = n + 2; |
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‡¤¥áì Hn { ¯®«¨-®¬ë •à¬¨â : |
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H0(x) = 1; H1(x) = 2x; Hn+1(x) = 2xHn(x) − 2nHn−1(x) : |
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Žâ¬¥â¨¬, çâ® |
n(−x) = (−1)n |
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Ž¯¥à â®àë ஦¤¥-¨ï ¨ ã-¨ç⮦¥-¨ï |
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‚¢¥¤¥¬ ®¯¥à â®àë |
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a^ = |
2 (x + ip^) ; a^+ = |
2 (x − ip^) ; |
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ç¥à¥§ ª®â®àë¥ £ ¬¨«ìâ®-¨ - § ¯¨áë¢ ¥âáï ¢ ¢¨¤¥ |
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H^ = 2 (^a+a^ + a^a^+) = a^+a^ + 2 |
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•¥âàã¤-® ¯®ª § âì, çâ® |
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H^ a^+ = a^+ H^ + 1 ; H^a^ = a^ H^ − 1 : |
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•ãáâì jni | -®à¬¨à®¢ --®¥ á®áâ®ï-¨¥ á í-¥à£¨¥© En = n + |
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¥áâì
H^ jni = En jni = (n + 1=2) jni :
’®£¤ a^+ jni ¨ a^ jni | á®áâ®ï-¨ï (-¥-®à¬¨à®¢ --ë¥) á í-¥à£¨¥© En +1 ¨ En − 1 ᮮ⢥âá⢥--®. „¥©á⢨⥫ì-®, ¨§ (7.1) á«¥¤ã¥â, çâ®
H^ a^+ jni = a^+(H^ + 1) jni = (En + 1) ^a+ jni;
â ª¦¥ - «®£¨ç-®¥ ãà ¢-¥-¨¥ ¤«ï a^ jni:
H^ a^ jni = (En − 1) a^ jni :
’ ª¨¬ ®¡à §®¬, ¤¥©á⢨¥ ®¯¥à â®à a^+ - á®áâ®ï-¨¥ jni ¯¥à¥¢®¤¨â ¥£® ¢ á®áâ®ï-¨¥ jn + 1 i, â® ¥áâì ¯®¢ëè ¥â í-¥à£¨î á®áâ®ï-¨ï - 1 (- h! ¢ ®¡ëç-ëå ¥¤¨-¨æ å),
a^+ jn i = cn jn + 1 i ; |
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