6 Elementary quantum mechanics helps to solve the Heisenberg equation(1.25)and to determine the evolution of operators in the Heisenberg picture. An()=U(.DAn()U(.). (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. s is the co tures being comple The e”time-evy lution operat pic nberg pictur differs by permutation of the time indices from the operator U(.)in the Schrodinger pic ture.In simple words,where wave functions evolve forward,operators evolve backward. This permutation is actually equivalent to taking the Hermitian conjugate. 0u,=0d.0 (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,U(r,)0(t,r),brings the system back to its initial state at time r',and therefore must equal the identity.Combining this with (10) proves that the time-evolution operator is unitary,(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 Ho to govern the time-evolution operators.As a result,the perturbation Hamiltonian ()acquires an extra time 的和=e(t)eo (1.51)） Since the time-dependence of the wave functions is now governed by (t)(see (1.31)). the time-evolution operator for the wave functions in the interaction picture reads iu=T[ep-厂fr (1.52) The time-evolution operator (t)in the Schrodinger picture now can be expressed in terms of Ur(t.r')as i(t,)=e-o Un(t,teftor (1.53) Control question.Do you see how the relation (1.53)follows from our definition 10)=eo12 2 We have made a choi at time t =0 th in inte and Schrodinger16 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 pic￾tures 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 pic￾ture. 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 time￾dependence,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.