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Time Dependent Perturbation Theory) Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai November 15, 2018

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Time Dependent Perturbation Theory) Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai November 15, 2018 Quantum Mechanics-II Approximation Methods : (Time Dependent Perturbation Theory) Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai November 15, 2018 1 Introduction Till now we have discussed systems where the Hamiltonian had no explicit time dependence. We will now consider situation where a time dependent perturbation V (t) is present in addition to the time independent Hamiltonian H0, the solution of the latter problem is known to us and its eigenstates are given by H0 j φni = En j φni (1) In the absence of the perturbation, if the system initially happened to be in a particular eigenstate |j ni of H0, it would continue to be in that state for ever, apart from picking up an unimportant phase factor eiEnt=~. A time dependent perturbation would change that and the system would no longer in that eigenstate but be found in some state j (t)i. Since any arbitrary state can always be expressed as a linear combination of the complete set of eigenstates fj φnig of H0, it implies that the perturbation induces transi- tion between different eigenstates. The time evolution operator is no longer exp iHt=~, as would be the case if H were independent of time. The question that we address is what is the probability of transition from the given initial state j ni to different eigenstates m of the Hamiltonian H0, where m 6= n ? 2 Schr¨odinger,Heisenberg and Interaction Pictures We start with a review of Schr¨odingerand Heisenberg pictures, which we had come across in QM-1 course. In the Schr¨odingerpicture, the state vectors describing the system are time independent, their evolution being governed by Schr¨odingerequation 1 d j (t)i i s = H j (t) (2) ~ dt s formal solution of the equation is given in terms of the time evolution operator ^ j s(t)i =U(t; t0i j s(t0)i ^ −iH(t−t0)=~ ≡ e j s(t0)i (3) ^ The operator U(t; t0) is unitary and satisfies the following identities U^(t; t) = 1 ^ y ^ ^ −1 U (t; t0) = U(t0; t) = U (t; t0) ^ ^ ^ U(t1; t2)U(t2; t3) = U(t1; t3) (4) It may be noted that the operators corresponding to the physical observables are time independent in the Schr¨odinger picture. In the Heisenberg picture, on the other hand, the state vectors are time independent while the operators depend on time. One can therefore, relate the state vectors of the two pictures by the fact that the Schr¨odingerstate at time t can be obtained from some initial state at t = 0 by −iHt=~ j s(t)i = e j s(0)i ≡| H i (5) where H are time independent. ^ The Heisenberg operators AH (t) satisfies the following equation of motion dA (t) i H = [A^ ; H] (6) ~ dt H where H is the Hamiltonian of the system, assumed to have no explicit time dependence. It may be observed that the expectation value of an operator is the same in both the pictures ^ ^ h s(t) j As j s(t)i = h h j AH (t) j H i (7) ^ iHt=~ ^ iHt=~ where AH (t) = e Ase . Let fj φnig denote a complete set of eigen- states of the Hamiltonian H. Consider j H i, the Heisenberg state of the system or equivalently, the Schr¨odngerstate of the system at t = 0 ). We can write X j H i = cn j φni (8) n 2 The coefficients cn are time independent since the left hand side of the above equation is so. The Schr¨odingerstate at time t can be written as −iHt=~ j s(t)i = e j H i −iHt=~ X = e cn j φni n X −iEnt=~ = cne j φni (9) n Note that the time dependence on the right is contained only in the expo- −iEnt=~ nential term e , while the coefficient cn are still time independent, a consequence of the fact that the Hamiltonian has no explicit time depen- dence. Suppose now, the Hamiltonian, in addition to the time independent part, which will henceforth denote by H0, has a time dependent part V (t) H = H0 + V (t); the form of the Schr¨odinger state at time t, instead of (5) can be written as X −iEnt=~ j s(t)i = cn(t)e j φni (10) n where the coefficients cn(t) now depend on time. We now define what is known as interaction picture or Dirac picture which is intermediate between the two pictures defined above. A state in the interaction picture is defined by iH0t=~ j I (t)i = e j s(t)i (11) Note that though at t = 0, the state of the system coincides with that of Schr¨odingerpicture, the time evolution of the state from t = 0 is governed by the time-independent Hamiltonian H0 and not by the full Hamiltonian H, as would be the case for the Schr¨odingerstate. Using the definition (11), we have @ @ i j (t)i = −H eiH0t=~ j (t)i + eiH0t=~(i ) j (t)i ~@t I 0 s ~ @t s iH0t=~ iH0t=~ = −H0e j s(t)i + e H j s(t)i iH0t=~ iH0t=~ −iH0t=~ iH0t=~ = −H0e j s(t)i + e He e j s(t)i iH0t=~ iH0t=~ −iH0t=~ = −H0e j s(t)i + e He j I (t)i ≡ VI (t) j I (t)i (12) 3 where, we have defined iH0t=~ −iH0t=~ VI (t) = e (H − H0)e (13) which is very similar to the way in which Heisenberg operators are connected to theSchr¨odingeroperators, with H0 replacing H in the time evolution op- erator. ^ Clearly, the equation of motion for the operator AI (t) in the interaction picture is given by dA (t) i I = [A^ (t); H ] (14) ~ dt I 0 It may be noted that the matrix elements of the operators in both the rep- resentations (and hence by (7) in all the three pictures) remain the same, iH0t=~ −iH0t=~ h s(t) j As j s(t)i = h I (t) j e Ase j I (t)i = h I (t) j AI (t) j I (t)i (15) 3 First Order Perturbation We will now obtain a solution of equation (12) governing the time develop- ment of the wavefunction in the interaction picture, @ i j (t)i = V (t) j (t)i ~@t I I I Let us define time evolution operator by U(t) j I (t = 0)i =j (t)i (16) The operator U(t) is unitary and we have @ @ i (U(t) j (0)i) = j (t)i ~@t I @t I = VI (t) j I (t)i = VI (t)U(t) j (t = 0)i (17) Thus the equation satisfied by the evolution operator is @ i U(t) = V (t)U(t) (18) ~@t I 4 with U(0) = 1. To first order i perturbation, the solution of the above is Z t i 0 0 U(t) = 1 − Vi(t )dt (19) ~ 0 Let us expand j I (t)i in terms of the complete eigenstates of the unperturbed Hamiltonian H 0 X j I (t)i = cn(t) j φni (20) n where H0 j φni = En j φni. The coefficients cn in (20) are the same as in (10) corresponding to the Hamiltonian H0 , as can be seen by operating both sides of the latter by eiH0t=~, iH0t=~ X iEnt=~ iH0t=~ e j s(t)i = cn(t)e e j φni n X = cn(t) j φni n In terms of these coefficients, (17) can be written as @ X X i c (t) j φ i = V (t) c (t) j φ i ~@t n n I m m n m Multiplying both sides of above by hφk j and introducing a complete set of eigenstates, we have @ X X i c (t)hφ j φ i = c (t)hφ j V (t) j φ ihφ j φ i ~@t n k n m k I l l m n m;l Using the orthogonality property of the eigen states, we get @ X i c (t) = c (t)hφ j V (t) j φ ic (t) (21) ~@t k m k I m m m But iH0t=~ −iH0t=~ hφk j VI (t) j φmi = hφk j e V (t)e j φmi i(Ek−Em)t=~ = e hφk j V (t) j φmi i(Ek−Em)t=~ i!kmt = Vkme ≡ Vkme (22) 5 where Ek − Em !km = = −!mk ~ The equation (21) then becomes @ X i c (t) = V ei!kmtc (t) (23) ~@t k km m m This equation is still exact but it leads to coupled differential equations for the coefficients ck(t). We will return to the solution of (23) a little later. Presently, let us assume that the system is initially in a particular eigenstate j φni. Thus cn(t = 0) = 1 and all other coefficients are initially zero. Thus j I (t)i = U(t) j I (t = 0)i = U(t) j φni i Z t = 1 − VI (t)dt j φni (24) ~ 0 The coefficients cm(t) are thus given by cm(t) = hφm j I (t)i (25) i Z t = − hφm j VI (t) j φnidt ~ 0 Z t i i!mnt = − e hφm j V (t) j φni (26) ~ 0 The transition probability from the initial state j φni to a state j φmi is given by 2 Pn!m = jcmj (27) 3.1 Constant perturbation If V does not depend on time but is switched on at time t = 0 for a time t, one can find out the transition rate easily.
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