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solution9.3 - Quantum Physics 2008/09 Solutions to Tutorial...

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Quantum Physics 2008/09 Solutions to Tutorial Sheet 7: Time Dependent Perturbations and Radiative Transitions For the first two problems, you may assume that the hydrogen eigenfunctions are: u 100 = ( πa 3 0 ) - 1 / 2 exp( - r/a 0 ) u 211 = - πa 3 0 · - 1 / 2 r 8 a 0 sin θ exp ( ) exp ( - r/ 2 a 0 ) u 210 = 8 πa 3 0 · - 1 / 2 r 2 a 0 cos θ exp ( - r/ 2 a 0 ) u 21 - 1 = πa 3 0 · - 1 / 2 r 8 a 0 sin θ exp ( - ) exp ( - r/ 2 a 0 ) and Z 0 exp( - br ) r n d r = n ! /b n +1 , n > - 1 1. A hydrogen atom is placed in a uniform but time-dependent electric field of magnitude E = 0 for t < 0 , E = E 0 exp( - t/τ ) for t 0 ( τ > 0) where E 0 is a constant. At time t = 0 , the atom is in the ground ( 1 s ) state. Show that the probability, to lowest order in perturbation theory, that as t → ∞ , the atom is in the 2 p state in which the component of the orbital angular momentum in the direction of the field is zero, is given by p 1 s 2 p = | c ( ) | 2 = 2 15 3 10 ( e E 0 a 0 ) 2 ( E 2 p - E 1 s ) 2 + (¯ h/τ ) 2 [ Hint: take the field direction to be the z -direction. Write down the potential energy of the electron in the given field and treat as a time-dependent perturbation]. We will use time-dependent perturbation theory. Since the field decays away exponen- tially there is only a finite probability that the transition will occur. The approximation from TDPT is that there is no probablity of transition to 2p and back again. We take the direction of the electric field to be the z -direction, as suggested. Then the perturbation for t 0 is ˆ H ( t ) = - e ( -E 0 z exp( - t/τ )) and the transition probability amplitude is c 1 s 2 p ( t ) = ( i ¯ h ) - 1 t Z 0 210 | ˆ H ( t ) | 100 exp( iωt ) d t where ω = ( E 2 p - E 1 s ) / ¯ h .
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Using the given eigenfunctions, the required matrix element is 210 | ˆ H | 100 = e E 0 exp( - t/τ ) 210 | r cos θ | 100 = e E 0 exp( - t/τ ) πa 4 0 4 2 2 π Z φ =0 π Z θ =0 Z r =0 r 4 exp( - 3 r/ 2 a 0 ) d r cos 2 θ sin θ d θ d φ The φ integration just gives 2 π , whilst the θ integration yields +1 Z - 1 cos 2 θ d(cos θ ) = " cos 3 θ 3 # +1 - 1 = 2 3 The radial integral is Z 0 r 4 exp( - 3 r/ 2 a 0 ) d r = 2 a 0 3 5 4! from the given integral. Putting it all together 210 | ˆ H ( t ) | 100 = A exp( - t/τ ) where A = 2 15 / 2 3 5 e E 0 a 0 Thus c 1 s 2 p ( t ) = A i ¯ h t Z 0 exp( - t /τ + iωt ) d t = A i ¯ h " exp( iωt - t/τ ) - 1 - 1 # Thus the desired probability is p 1 s 2 p = | c ( ) | 2 = 2 15 3 10 ( e E 0 a 0 ) 2 ( E 2 p - E 1 s ) 2 + (¯ h/τ ) 2 If we had numbers, for we should now check that this probability is indeed small - if so then we justify our use of TDPT, if not then we know we should solve the problem exactly. What is the probability that it is in the 2 s -state? We can write: c 1 s 2 s ( t ) = ( i ¯ h ) - 1 t Z 0 200 | ˆ H ( t ) | 100 exp( iωt ) d t However, we note that the integral over space has an odd function as its integrand: 200 | z | 100 . It is therefore zero, and there is no probability of a 1 s 2 s transition to first order. Remark: As we saw in the Stark effect, a constant field induces a dipole on the atom.
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