Unformatted text preview: Then 215/2 1
(100W(t)l210) = 35 aDeEoe‘V (12)
To calculate the probability of transition to ﬁrst order in the perturbation we need the following
. t 1
021005) : %/ dtl 6Lw21t/<IOOIV(tI)I210> (13)
0
t 215/2 t ‘ ,
._ ~ )6
_ ‘E 35 aOeEo f0 dt’ ewe V (14)
1' 215/2 1 _ €(iw21—7jt
: _ E H 15
h 35 (106‘ 0 21.021 — ’7 ( )
where E 2
‘ E2  1 36
: m = I 16
(“21 ’1 80071 ( )
Therefore, the transition probability in the t —> oo limit is
215 (1 6E 2 1
2 O 0
= _ _ __ 17
C2ro(oc)l 310 ( h > wgl + 72 ( )
2. 15.2.
V1(m, t) = A (a: — 3) sin wt ' (18) (a) Lets calculate the matrix element of the perturbation between the ground state and the ﬁrst
excited state (111/112) = Asinwt<1l<r — 3) I2) : Asinwt(1xl2) (19)
2 a 2
2 Asin wt—/ dcr msin 7r_a: sin 2 (20)
a 0 a a
= isinwt/A dxx (GOSH? cos 3E) (21)
(L 0 (1 CL
, a a 3 .,
=§sinwt [m (gsinE—isiniﬂ) ~/ dr gsin7—rE—isillﬂ)] (22)
a 71’ a 37r a 0 0 7r (1 37r a
A 2 2 ‘ , a.
= —sinwt ices—7E — Leosﬂ (23)
(1 7r2 (1 97r2 a 0
1 A
= 7 967?ch sinwt . (24) Assuming the system is in the ground state at t : O, the transition amplitude to ﬁrst order in 7A ...
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 '08
 DAVIDA.HUSE
 mechanics

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