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# ps5b - Then 215/2 1(100W(t)l210 = 35 aDeEoe‘V(12 To...

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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(1|xl2) (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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