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Time Evolution Operator

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Time Evolution Operator Time evolution operator external constant magnetic field parallel to the z-axis, it In quantum mechanics will precess in the xy-plane: the probability for the result ¯h/2 in the measurement SGxˆ oscillates between 0 and 1 • unlike position, time is not an observable. as a function of time. In any case, the probability for the • there is no Hermitean operator whose eigenvalues result ¯h/2 or −¯h/2 remains always as the constant 1. were the time of the system. Generalizing, it is natural to require that X 2 X 2 • time appears only as a parameter, not as a |ca0 (t0)| = |ca0 (t)| . measurable quantity. a0 a0 So, contradictory to teachings of the relativity theory, In other words, the normalization of the states does not time and position are not on equal standing. In depend on time: relativistic quantum field theories the equality is restored hα, t |α, t i = hα, t ; t|α, t ; ti by degrading also the position down to the parameter 0 0 0 0 † level. = hα, t0|U (t, t0)U(t, t0)|α, t0i. We consider a system which at the moment t is in the 0 This is satisfied if we require U(t, t ) to be unitary, i.e. state |αi. When time goes on there is no reason to expect 0 † it to remain in this state. We suppose that at a later U (t, t0)U(t, t0) = 1. moment t the system is described by the state |α, t0; ti, (t > t0), 2. Composition property The evolution from the time t0 to a later time t2 should where the parameter t0 reminds us that exactly at that be equivalent to the evolution from the initial time t0 to moment the system was in the state |αi. Since the time is an intermediate time t1 followed by the evolution from t1 a continuous parameter we obviously have to the final time t2, i.e. lim |α, t0; ti = |αi, U(t2, t0) = U(t2, t1)U(t1, t0), (t2 > t1 > t0). t→t0 and can use the shorter notation Like in the case of the translation operator we will first look at the infinitesimal evolution |α, t0; t0i = |α, t0i. |α, t0; t0 + dti = U(t0 + dt, t0)|α, t0i. Let’s see, how state vectors evolve when time goes on: Due to the continuity condition evolution |α, t0i −→ |α, t0; ti. lim |α, t0; ti = |αi t→t0 We work like we did with translations. We define the we have time evolution operator U(t, t0): lim U(t0 + dt, t0) = 1. dt→0 |α, t0; ti = U(t, t0)|α, t0i, So, we can assume the deviations of the operator U(t + dt, t ) from the identity operator to be of the order which must satisfy physically relevant conditions. 0 0 dt. When we now set 1. Conservation of probability U(t0 + dt, t0) = 1 − iΩdt, We expand the state at the moment t0 with the help of the eigenstates of an observable A: where Ω is a Hermitean operator, we see that it satisfies the composition condition X 0 |α, t0i = ca0 (t0)|a i. a0 U(t2, t0) = U(t2, t1)U(t1, t0), (t2 > t1 > t0), At a later moment we get the expansion is unitary and deviates from the identity operator by the term O(dt). X 0 |α, t0; ti = ca0 (t)|a i. The physical meaning of Ω will be revealed when we a0 recall that in classical mechanics the Hamiltonian generates the time evolution. From the definition In general, we cannot expect the probability for the 0 system being in a specific state |a i to remain constant, U(t0 + dt, t0) = 1 − iΩdt i.e. in most cases we see that the dimension of Ω is frequency, so it must be |ca0 (t)| 6= |ca0 (t0)|. multiplied by a factor before associating it with the Hamiltonian operator H: 1 For example, when a spin 2 particle, which at the moment t0 is in the state |Sx; ↑i, is subjected to an H = ¯hΩ, or (ii) The Hamiltonain H depends on time but the operators iH dt H corresponding to different moments of time commute. U(t0 + dt, t0) = 1 − . ¯h 1 For example, a spin 2 particle in the magnetic field whose The factor ¯h here is not necessarily the same as the factor strength varies but direction remains constant as a ¯h in the case of translations. It turns out, however, that function of time. A formal solution of the equation in order to recover Newton’s equations of motion in the ∂ classical limit both coefficients must be equal. i¯h U(t, t ) = HU(t, t ) Applying the composition property ∂t 0 0 U(t2, t0) = U(t2, t1)U(t1, t0), (t2 > t1 > t0) is now Z t we get i 0 0 U(t, t0) = exp − dt H(t ) , ¯h t0 U(t + dt, t0) = U(t + dt, t)U(t, t0) iH dt which, again, can be proved by expanding the exponential = 1 − U(t, t ), ¯h 0 function as the series. (iii) The operators H evaluated at different moments of where the time difference t − t0 does not need to be 1 time do not commute For example, a spin 2 particle in a infinitesimal. This can be written as magnetic field whose direction changes in the course of H time: H is proportional to the term S · B and if now, at U(t + dt, t0) − U(t, t0) = −i dt U(t, t0). ¯h the moment t = t1 the magnetic field is parallel to the x-axis and, at the moment t = t2 parallel to the y-axis, Expanding the left hand side as the Taylor series we end then H(t1) ∝ BSx and H(t2) ∝ BSy, or up with [H(t ),H(t )] ∝ B2[S ,S ] 6= 0. It can be shown that the ∂ 1 2 x y i¯h U(t, t ) = HU(t, t ). formal solution of the Schr¨odinger equation is now ∂t 0 0 This is the Schr¨odingerequation of the time evolution U(t, t0) = operator. Multiplying both sides by the state vector ∞ n X −i Z t Z t1 |α, t0i we get 1 + dt1 dt2 ··· ¯h t t ∂ n=1 0 0 t i¯h U(t, t0)|α, t0i = HU(t, t0)|α, t0i. Z n−1 ∂t dtn H(t1)H(t2) ··· H(tn). t0 Since the state |α, t0i is independent on the time t we can write the Schr¨odinger equation of the state vectors in the This expansion is called the Dyson series. We will assume form that our Hamiltonians are time independent until we ∂ i¯h |α, t0; ti = H|α, t0; ti. start working with the so called interaction picture. ∂t Suppose that A is an Hermitean operator and In fact, in most cases the state vector Schr¨odinger equation is unnecessary because all information about the [A, H] = 0. dynamics of the system is contained in the time evolution operator U(t, t0). When this operator is known the state Then the eigenstates of A are also eigenstates of H, called of the system at any moment is obtained by applying the energy eigenstates. Denoting corresponding eigenvalues of definition the Hamiltonian as Ea0 we have |α, t0; ti = U(t, t0)|α, t0i, 0 0 H|a i = E 0 |a i. We consider three cases: a (i) The Hamiltonian does not depend on time. For example, The time evolution operator can now be written with the a spin 1 particle in a time independent magnetic field 2 help of these eigenstates. Choosing t = 0 we get belongs to this category. The solution of the equation 0 iHt X X iHt ∂ exp − = |a00iha00| exp − |a0iha0| i¯h U(t, t0) = HU(t, t0) ¯h ¯h ∂t a0 a00 is iE 0 t X 0 a 0 iH(t − t0) = |a i exp − ha |. U(t, t ) = exp − 0 ¯h 0 ¯h a as can be shown by expanding the exponential function as Using this form for the time evolution operator we can the Taylor series and differentiating term by term with solve every intial value problem provided that we can respect to the time. Another way to get the solution is to expand the initial state with the set {|a0i}. If, for compose the finite evolution from the infinitesimal ones: example, the initial state can be expanded as N (iH/¯h(t − t ) iH(t − t ) X 0 0 X 0 lim 1 − 0 = exp − 0 . |α, t0 = 0i = |a iha |αi = ca0 |a i, N→∞ N ¯h a0 a0 we get in an eigenstate |a0i of an operator A commuting with the Hamiltonian H. Suppose, we are interested in the iHt expectation value of an operator B which does not |α, t0 = 0; ti = exp − |α, t0 = 0i ¯h necessarily commute either with A or with H. At the X iEa0 t moment t the system is in the state = |a0iha0|αi exp − . ¯h a0 0 0 |a , t0 = 0; ti = U(t, 0)|a i. In other words, the expansion coefficients evolve in the In this special case we have course of time as 0 † 0 iEa0 t hBi = ha |U (t, 0)BU(t, 0)|a i ca0 (t = 0) −→ ca0 (t) = ca0 (t = 0) exp − . ¯h iEa0 t iEa0 t = ha0| exp B exp − |a0i ¯h ¯h So, the absolute values of the coefficients remain = ha0|B|a0i, constant.
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