= ^ c n( t ) [ H aU ) + W {x , t ) ] f//0) (x , t )

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( x > o = ^ c„ o ) f f r( x , t ) $ о) ( x , t).
( x , t ) W ( x , t ) p f ( x , t ) d v . (878) 1 1

= ^ c n( t ) [ H aU ) + W {x , t ) ] f//0) (x , t )
(8.76)
y/f] ( x,t)
to‘ lqin funksiyalari (8.73) tenglamani qanoatlantirishi hisobga
olinsa, (8.76) tenglamaning chap va o ‘ ng tomondagi birinchi hadlari
aynan bir-biriga teng bo‘ ladi va ulami qisqartirib yuborish mumkin, u
holda quyidagi tenglamaga kelinadi:
( x > o = ' ^ c„ o ) f f r( x , t ) $ о) ( x , t).
(8 77)
n
n
Hosil bo‘ lgan (8.77)
tenglamaning ikkala tomonini chap
tomonidan
vll4
(>v)funksiyaga
ko‘paytirib, so‘ ngra butun fazo
bo‘yicha integrallanadi va quyidagi natijaga kelinadi:
^
^»0) (x’ d v =
и
= 2 c» W / ^ i 0)
( x , t ) W ( x , t ) p f ( x , t ) d v .
('8'78)
1
1
Olingan (8.78) dagi ifodaning o ‘ng tomonidagi y ig ‘ indida faqat bitta
n = m
ga tegishli boigan had saqlanib qoladi, chunki
J V i0)
( x ) v i 0) ( x ) d v =
j ^ ’
n
=
m
(8 79)
ifodalar o ‘rinlidir.
(8.78) tenglamaning o ‘ng tomoni esa quyidagicha yozilishi
mumkin:
w m„
( f ) =
f P f
(*> 0
W (x ,
Г V „(0)
{x , t ) d v =
= f / f (x )e ~ »EmW
(x,?V ,!0)
{ x )e ~ * B'‘d v = Wmne '"
.'
(8-80)
Bu yerda
Wmn ( t
) had orqali g ‘ alayonlanish operatorining matrik
elementi belgilanadi
va bu had g ‘alayonlanishning vaqtga b o g liq
ko‘paytuvchisini ham o ‘ z ichiga olgan matritsa elementidir. Shunday
qilib, (8.78) tenglamani quyidagi ko‘ rinishda yozish mumkin:
252

bunda
=
( 0 е*4"'
(8.81)
teng bo‘ lib, /Г; energiyaga ega bo'Igan statsionar holatdan
E"
statsionar
holatga o ‘ tishdagi chastotalarni ifodalaydi.
Olingan (8.81) dagi tenglamalar sistemasi aniq bo‘ lib„ o ‘zaro ta’ sir
tasavvurdagi Shredinger tenglamasi deyiladi.
Hozirgi vaqtgacha hech qanday yaqinlashishlar to'g'nsida gap
yuritilmagan edi. (8.81) dagi tenglamalar sistemasining yechimini hosil
qilish
matematik
nuqtayi
nazardan
berilgan
Shredinger
to‘ la
tenglamasini yechish bilan ekvivalentdir.
Faraz
qilaylik,
/ = 0
vaqt
momentida
g ‘ alayonlanmagan
sistemaning energiyasini o ‘ lchash natijasida

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