《高等量子力学》课程参考教材：Advanced Quantum Mechanics - A practical guide（YULI V. NAZAROV、JEROEN DANON，2013）.pdf_P26-P30

15 1.6 Time-dependent perturbation theory We have to re-develop the perturbation theory. As a matter of fact, the resulting timedependent theory appears to be more logical and simpler, facilitating its application for complex systems with many degrees of freedom. To start, let us note that we could formally solve the Schrödinger equation (1.5) by introducing the time-evolution operator Uˆ defined as |ψ(t) = Uˆ (t, t )|ψ(t ), (1.43) assuming that t > t . That is, Uˆ (t, t ) takes the initial state |ψ(t ) as input and returns the final state |ψ(t). For a time-independent Hamiltonian, this operator takes the form of an operator exponent, Uˆ (t, t ) = e − i h¯ Hˆ (t−t ) , (1.44) where the exponent of an operator is defined by the Taylor series of the exponential function eAˆ ≡ 1 + Aˆ + AˆAˆ 2! + AˆAˆAˆ 3! + ... (1.45) What does Uˆ (t, t ) become in the case of a time-dependent Hˆ ? It becomes a time-ordered exponent, Uˆ (t, t ) = T exp − i h¯ t t dτ Hˆ (τ ) , (1.46) T indicating the time-ordering. To see the meaning of this expression, let us divide the time interval (t, t ) into many subintervals of (small) duration t, (tn, tn − t), where tn = t + nt. Since Uˆ (t, t ) is an evolution operator, it can be written as an operator product of the evolution operators within each subinterval, Uˆ (t, t ) = Uˆ (t, t − t) ··· Uˆ (tn, tn − t) ··· Uˆ (t1, t ). (1.47) If the subdivision is sufficiently fine, we can disregard the time-dependence of Hˆ within each subinterval so that we have a long product of corresponding elementary exponents Uˆ (t, t ) = e − i h¯ Hˆ (t)t ··· e − i h¯ Hˆ (tn)t ··· e − i h¯ Hˆ (t1)t . (1.48) The time-ordered exponent in (1.46) is the limit of this long product for t → 0. Timeordering therefore means that the elementary exponents corresponding to later times are placed in a product to the left from those at earlier times. Control question. Explain the difference between two exponents T exp − i h¯ t t dτ Hˆ (τ ) and exp − i h¯ t t dτ Hˆ (τ ) . Is there a difference if the operators Hˆ (t1) and Hˆ (t2) commute for all t1 and t2? Since (1.43) presents the formal solution of the Schrödinger equation, and the Heisenberg and Schrödinger pictures are equivalent, the same time-evolution operator

16 Elementary quantum mechanics helps to solve the Heisenberg equation (1.25) and to determine the evolution of operators in the Heisenberg picture, Aˆ H(t) = Uˆ (t , t)Aˆ H(t )Uˆ (t, t ). (1.49) Note that in the above equation the time indices in the evolution operators appear in a counter-intuitive order (from right to left: evolution from t to t, then an operator at time t , then evolution from t to t ). This is the consequence of the Heisenberg and Schrödinger pictures being complementary. The “true” time-evolution operator in the Heisenberg picture differs by permutation of the time indices from the operator U(t , t) in the Schrödinger picture. In simple words, where wave functions evolve forward, operators evolve backward. This permutation is actually equivalent to taking the Hermitian conjugate, Uˆ (t, t ) = Uˆ †(t , t). (1.50) Indeed, time reversal in the time-ordered exponent (1.46) amounts to switching the sign in front of the i/h¯. Let us also note that the product of two time-evolution operators with permuted indices, Uˆ (t , t)Uˆ (t, t ), brings the system back to its initial state at time t , and therefore must equal the identity operator, Uˆ (t , t)Uˆ (t, t ) = 1. Combining this with (1.50) proves that the time-evolution operator is unitary, Uˆ †(t, t )Uˆ (t, t ) = 1. Let us now return to perturbation theory and the Hamiltonian (1.42). The way we split it into two suggests that the interaction picture is a convenient framework to consider the perturbation in. Let us thus work in this picture, and pick Hˆ 0 to govern the time-evolution of the operators. As a result, the perturbation Hamiltonian Hˆ (t) acquires an extra timedependence,2 Hˆ I(t) = e i h¯ Hˆ 0t Hˆ (t)e − i h¯ Hˆ 0t . (1.51) Since the time-dependence of the wave functions is now governed by Hˆ I(t) (see (1.31)), the time-evolution operator for the wave functions in the interaction picture reads Uˆ I(t, t ) = T exp − i h¯ t t dτ Hˆ I(τ ) . (1.52) The time-evolution operator Uˆ (t, t ) in the Schrödinger picture now can be expressed in terms of Uˆ I(t, t ) as Uˆ (t, t ) = e − i h¯ Hˆ 0t Uˆ I(t, t )e i h¯ Hˆ 0t . (1.53) Control question. Do you see how the relation (1.53) follows from our definition |ψI(t) = e i h¯ Hˆ 0t |ψ? 2 We have made a choice here: at time t = 0 the operators in interaction and Schrödinger picture coincide. We could make any other choice of this peculiar moment of time; this would not affect any physical results.

1.6Time-dependent thry The fact that P()is assumed small compared to Ho allows us to expand )in orders of A(r).For instance,up to third order this yields u=1-片解-一aTa创 (1.54 +话画西s+ 。 We are now dealing with rime-ordered products of operators Acorresponding to the time ordered exponent in(1.52).A time-ordered product is defined as follows:given a sequence of operatorsat different m ments of time.it re-orders the sequence in such a way that terms with an earlier time index are shifted to the right of terms with a later time index. Generally,it requires a permutation of the operators.For example,for two operators T[2)】= a2)m)f2>, (1.55) )u2)if1>2. There exists a simple way to explicitly include this time-ordering in the integrals.Let us illustrate this with the second order term.Using(1.55)we rewrite 2u,0=- /a-+9u- (1.56 where(denotes the Heaviside step function 1if1>0. 0={3ift=0. (1.57) 0 ift<0. To arrive at the last equality in(1.56).we exchanged the dummy integral variablesn in the second term,and combined the two terms.The upper limit of the second integral then effectively becomes2.the variable of the first integral.This automatically guarantees that 2>.which equals the effect of time-ordering.Since we combined two equal terms the factor2inhe nator is canceled.Gen ly.for the Nth order ne nu of!this always canceling the factor N!in the denominator coming from the expansion of the exponential. We have now shown how to write a perturbation expansion for the evolution operator U for an arbitrary perturbation A().Evaluating an evolution operator is,however,useless without specifying the initial conditions:the quantum state at some initial moment=o For practical would like to take something simple for this initial state for instance an eigenstate of Ho.This,however,would deliver a sensible answer only for a rather specific situation:for a perturbation that is switched on diabarically.Indeed,for this simple initial state to be relevant,we require that there is no perturbation before the moment .rafer the timethe perturbation rather suddenly jumps to a finite value This situation can be ealized if we ave means to change the

17 1.6 Time-dependent perturbation theory The fact that Hˆ (t) is assumed small compared to Hˆ 0 allows us to expand Uˆ I(t, t ) in orders of Hˆ (t). For instance, up to third order this yields Uˆ I(t, t ) = 1− i h¯ t t dt1Hˆ I(t1) − 1 h¯ 22! t t dt2 t t dt1T Hˆ I(t2)Hˆ I(t1) + i h¯ 33! t t dt3 t t dt2 t t dt1T Hˆ I(t3)Hˆ I(t2)Hˆ I(t1) + ... (1.54) We are now dealing with time-ordered products of operators Hˆ I corresponding to the timeordered exponent in (1.52). A time-ordered product is defined as follows: given a sequence of operators Hˆ I at different moments of time, it re-orders the sequence in such a way that terms with an earlier time index are shifted to the right of terms with a later time index. Generally, it requires a permutation of the operators. For example, for two operators T Hˆ I(t2)Hˆ I(t1) = Hˆ I (t2)Hˆ I (t1) if t2 > t1, Hˆ I (t1)Hˆ I (t2) if t1 > t2. (1.55) There exists a simple way to explicitly include this time-ordering in the integrals. Let us illustrate this with the second order term. Using (1.55) we rewrite Uˆ (2) I (t, t ) = − 1 h¯ 22! t t dt2 t t dt1 Hˆ I(t2)Hˆ I(t1)θ(t2 − t1) + Hˆ I(t1)Hˆ I(t2)θ(t1 − t2) = − 1 h¯ 2 t t dt2 t2 t dt1Hˆ I(t2)Hˆ I(t1), (1.56) where θ(t) denotes the Heaviside step function, θ(t) = ⎧ ⎪⎪⎨ ⎪⎪⎩ 1 if t > 0, 1 2 if t = 0, 0 if t < 0. (1.57) To arrive at the last equality in (1.56), we exchanged the dummy integral variables t1 ↔ t2 in the second term, and combined the two terms. The upper limit of the second integral then effectively becomes t2, the variable of the first integral. This automatically guarantees that t2 > t1, which equals the effect of time-ordering. Since we combined two equal terms, the factor 2 in the denominator is canceled. Generally, for the Nth order term, the number of equal terms is N!, this always canceling the factor N! in the denominator coming from the expansion of the exponential. We have now shown how to write a perturbation expansion for the evolution operator Uˆ I for an arbitrary perturbation Hˆ (t). Evaluating an evolution operator is, however, useless without specifying the initial conditions: the quantum state at some initial moment t = t0. For practical calculations, one would like to take something simple for this initial state, for instance an eigenstate of Hˆ 0. This, however, would deliver a sensible answer only for a rather specific situation: for a perturbation that is switched on diabatically. Indeed, for this simple initial state to be relevant, we require that there is no perturbation before the moment t0, i.e. Hˆ (t) = 0 for t < t0. Then, at or after the time t = t0 the perturbation rather suddenly jumps to a finite value. This situation can be realized if we have means to change the

8 Elementary quantum mechanics Hamiltonian at our will.Typically,however,this is not so,and most perturbations persist at all times.thusaso beforeIn this case.the perturbation inevitably affects the we need to know the igenstate the same corrections that we actually hoped to find from this perturbation theory.The latter therefore seems incomplete and void. The way out is to implement adiabatic.that is.very slow switching on of the pertur bation A'(t).We asst e that at t=-oo the pertu tion vanishes and the system is in an eigenstate of Ho.We also assum that the perturbation evolves further so slowly tha the quantum states remain close to the eigenstates of the Hamiltonian,that is,follow the Hamiltonian adiabatically.A handy way to implement adiabatic switching is to include an exponential time-dependence. 0→e严讯， (1.58) where n is infinitesimally small but positive,and the perturbation is assumed constant, H'(t)=H'.We see that now the perturbation indeed vanishes at t =-oo and coincides with the original perturbation at=0.At sufficiently small n the system evolves adiabat- ically,so that if far in the t was in an state of Ao will find it al cinstats of the chanted operators we see in the npext secto pas on,this require n<AE/h,with AE being the minimal separation of energy levels.In this case,an ini- tial eigenstate of Ho at t =-oo evolves to an eigenstate of Ao+A'at t =0.Thus,the modified perturbation(1.58)supplies us the desired corrections to the eigenstates of o at time.If we use this method to comute the perturbation toa )at we reproduce the results of the time-independent perturbation theory of Section 1.5 for the corresponding state n). The time-dependent perturbation theory presented by(1.54)is much simpler in structure and therefore more transparent than the time-independent theory.We see that it is straight- forward to write down a term of arbitrarily high order:the expressions do not increase in complexity like thet of the tim indepe ory do.Besides,at erm of an can be computed separately from the other orders.This is why time-dependent perturbatio theory is widely applied for systems with infinitely many degrees of freedom and particles, systems of advanced quantum mechanics. 1.6.1 Fermi's golden rule Quantum mechanics teaches us that a system remains in a given eigenstate forever,while practice shows the opposite:there are transitions between different states.For instance.a system in an excited state sooner or later gets to the ground state losing its excess energy We discuss transitions in detail in later chapters.Here we illustrate the power of time ry by deriving Fermi's golden rule,which is an indispensable Let us assume that the transitions between the eigenstates of an unperturbed Hamiltonian Ao are caused by a small perturbation.We use the first-order term of the perturbation

18 Elementary quantum mechanics Hamiltonian at our will. Typically, however, this is not so, and most perturbations persist at all times, thus also before t = t0. In this case, the perturbation inevitably affects the eigenstates at t0. To choose a reasonable initial condition, we need to know the eigenstates of Hˆ +Hˆ (t), or in other words, to apply the perturbation theory correctly, we need to know the same corrections that we actually hoped to find from this perturbation theory. The latter therefore seems incomplete and void. The way out is to implement adiabatic, that is, very slow switching on of the perturbation Hˆ (t). We assume that at t = −∞ the perturbation vanishes and the system is in an eigenstate of Hˆ 0. We also assume that the perturbation evolves further so slowly that the quantum states remain close to the eigenstates of the Hamiltonian, that is, follow the Hamiltonian adiabatically. A handy way to implement adiabatic switching is to include an exponential time-dependence, Hˆ (t) → eηt Hˆ , (1.58) where η is infinitesimally small but positive, and the perturbation is assumed constant, H (t) = H . We see that now the perturbation indeed vanishes at t = −∞ and coincides with the original perturbation at t = 0. At sufficiently small η, the system evolves adiabatically, so that if far in the past it was in an eigenstate of Hˆ 0, we will find it always in an eigenstate of the changed operator Hˆ 0 + eηt Hˆ . As we see in the next section, this requires η E/h¯, with E being the minimal separation of energy levels. In this case, an initial eigenstate of Hˆ 0 at t = −∞ evolves to an eigenstate of Hˆ 0 + Hˆ at t = 0. Thus, the modified perturbation (1.58) supplies us the desired corrections to the eigenstates of Hˆ 0 at time t = 0. If we use this method to compute the perturbation correction to an eigenstate |n(0) at t = −∞, we reproduce the results of the time-independent perturbation theory of Section 1.5 for the corresponding state |n. The time-dependent perturbation theory presented by (1.54) is much simpler in structure and therefore more transparent than the time-independent theory. We see that it is straightforward to write down a term of arbitrarily high order: the expressions do not increase in complexity like the terms of the time-independent theory do. Besides, a term of any order can be computed separately from the other orders. This is why time-dependent perturbation theory is widely applied for systems with infinitely many degrees of freedom and particles, systems of advanced quantum mechanics. 1.6.1 Fermi’s golden rule Quantum mechanics teaches us that a system remains in a given eigenstate forever, while practice shows the opposite: there are transitions between different states. For instance, a system in an excited state sooner or later gets to the ground state losing its excess energy. We discuss transitions in detail in later chapters. Here we illustrate the power of timedependent perturbation theory by deriving Fermi’s golden rule, which is an indispensable tool to understand transitions. Let us assume that the transitions between the eigenstates of an unperturbed Hamiltonian Hˆ 0 are caused by a small perturbation. We use the first-order term of the perturbation

19 1.6 Time-dependent perturbation theory theory derived above to calculate transition rates between different eigenstates of Hˆ 0. For the sake of simplicity we concentrate on two states and evaluate the transition rate from the initial state |i to the final state |f. We assume a time-independent Hˆ , and implement adiabatic switching as in (1.58). The initial condition we use is that the system is in |i at time t = −∞. The probability of finding the system in the state |f at time t can then be expressed in terms of the time-evolution operator Uˆ I(t, −∞) in the interaction picture,3 Pf(t) = |f |Uˆ I(t, −∞)|i |2. We need to evaluate this matrix element in the lowest nonvanishing order, which is the first order in Hˆ . The first order correction to the evolution operator reads Uˆ (1) I (t, −∞) = − i h¯ t −∞ dt1Hˆ I(t1), with Hˆ I(t1) = e i h¯ Hˆ 0t1 eηt1Hˆ e − i h¯ Hˆ 0t1 . (1.59) Defining H fi as the matrix element f |Hˆ |i, we can write f |Uˆ (1) I (t, −∞)|i=− i h¯ H fi t −∞ dt1e i h¯ (Ef −Ei)t1+ηt1 , (1.60) where we make use of Ei and Ef , the energy levels of the states |i and |f respectively, so that e i h¯ Hˆ 0t |i, f = e i h¯ Ei,f t |i, f. Note that this transition is not characterized by the probability Pf(t), but rather by the corresponding rate, i.e. the change in the probability per unit of time. Therefore we should evaluate the time derivative of the probability dPf /dt at t = 0, which yields i→f = dPf dt = 2Re ∂ ∂t f |Uˆ (1) I (0, −∞)|i f |Uˆ (1) I (0, −∞)|i ∗ = 2 h¯ 2 |H fi| 2Re 0 −∞ dt1e − i h¯ (Ef −Ei)t1+ηt1 = 2 h¯ |H fi| 2 ηh¯ (Ei − Ef)2 + (ηh¯)2 . (1.61) We immediately see that the transition rate vanishes in the limit η → 0 provided that the difference Ei − Ef remains finite. This proves the adiabaticity criterion mentioned above: transition rates disappear as long as η |Ei − Ef |. Control question. Do you understand why this disappearing of transition rates guarantees adiabatic evolution of the system? Note, however, that the rate expression (1.61) can be explicitly evaluated in the limit η → 0, using the relation limy→0+ y/(x2+y2) = πδ(x), where δ(x) is Dirac’s delta function. We should remember that the delta function has the strange property that it is equal to zero for all x, except at x = 0 where it becomes infinite in such a way that dx δ(x) = 1. Control question. Evaluate dx f(x)δ(x − a) for any function f(x). 3 It differs from the evolution operator Uˆ in the Schrödinger picture only by a phase factor not affecting the probability, see (1.53).