Time Evolution Operator

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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 Schrödinger equation is now ∂t 0 0 This is the Schrödingerequation 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 Schrödinger 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 Schrödinger 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. The relative phase between different components will, however, change in the course of time that is, the expectation value does not depend on time. because the oscillation frequencies of different For this reason the energy eigenstates are usually called components differ from each other. stationary states As a special case we consider an initial state consisting of We now look at the expectation value in a superposition a single eigenstate: of energy eigenstates, in a non stationary state 0 |α, t0 = 0i = |a i. X 0 |α, t0 = 0i = ca0 |a i. At some later moment this state has evolved to the state a0 It is easy to see, that the expectation value of B is now 0 iEa0 t |α, t0 = 0; ti = |a i exp − . ¯h X X ∗ 0 00 i(Ea00 − Ea0 )t hBi = c 0 c 00 ha |B|a i exp − . a a ¯h Hence, if the system originally is in an eigenstate of the a0 a00 Hamiltonian H and the operator A it stays there forever. Only the phase factor exp(−iEa0 t/¯h) can vary. In this This time the expectation value consists of terms which sense the observables whose corresponding operators oscillate with frequences determind by the Bohr commute with the Hamiltonian, are constants of motion. frequency condition Observables (or operators) associated with mutually Ea00 − Ea0 commuting operators are called compatible. As mentioned ω 00 0 = . a a ¯h before, the treatment of a physical problem can in many cases be reduced to the search for a maximal set of 1 As an application we look at how spin 2 particles behave compatible operators. If the operators A, B, C, . . . belong in a constant magnetic field. When we assume the to this set, i.e. magnetic moments of the particles to be e¯h/2mec (like electrons), the Hamiltonian is [A, B] = [B,C] = [A, C] = ··· = 0, e and if, furthermore, H = − S · B. mec [A, H] = [B,H] = [C,H] = ··· = 0, If we choose B k zˆ, we have that is, also the Hamiltonian is compatible with other eB operators, then the time evolution operator can be H = − S . m c z written as e

The operators H and Sz differ only by a constant factor, iHt X iEK0 t exp − = |K0i exp − hK0|. so they obviously commute and the eigenstates of S are ¯h ¯h z K0 also energy eigenstates with energies Here K0 stands for the collective index: e¯hB E = − for state |S ; ↑i ↑ 2m c z A|K0i = a0|K0i,B|K0i = b0|K0i,C|K0i = c0|K0i,... e e¯hB E↓ = + for state |Sz; ↓i. Thus, the quantum dynamics is completely solved (when 2mec H does not depend on time) if we only can find a We define the cyclotron frequency ω so that the energy maximal set of compatible operators commuting also with c difference between the states ishω ¯ : the Hamiltonian. c Let’s now look at the expectation value of an operator. |e|B We first assume, that at the moment t = 0 the system is ωc ≡ . mec The Hamiltonian H can now be written as Lastly we look at how the statevectors corresponding to different times are correlated. Suppose that at the H = ωcSz, moment t = 0 the system is described by the state vector |αi, which in the course of time evolves to the state when we assume that e < 0. |α, t0 = 0; ti. We define the correlation amplitude C(t) as All information about the evolution of the system is contained in the operator C(t) = hα|α, t0 = 0; ti iω S t = hα|U(t, 0)|αi. U(t, 0) = exp − c z . ¯h The absolute value of the correlation amplitude tells us If at the moment t = 0 the system is in the state how much the states associated with different moments of time resemble each other.

|αi = c↑|Sz; ↑i + c↓|Sz; ↓i, In particular, if the initial state is an energy eigenstate |a0i, then it is easy to see that at the moment t it is in the state iEa0 t C(t) = exp − , ¯h iωct |α, t0 = 0; ti = c↑ exp − |Sz; ↑i and the absolute value of the correlation amplitude is 1 at 2 all times. When the initial state is a superposition of iω t +c exp + c |S ; ↓i. energy eigenstates we get ↓ 2 z X 2 iEa0 t C(t) = |c 0 | exp − . If the initial state happens to be |Sz; ↑i, meaning that in a ¯h the previous equation a0 When t is relatively large the terms in the sum oscillate c = 1, c = 0, ↑ ↓ rapidly with different frequencies and hence most we see that the system will stay in this state at all times. probably cancel each other. Thus we expect the This was to be expected because the state is stationary. correlation amplitude decreasing rather rapidly from its initial value 1 at the moment t = 0. We now assume that the initial state is |Sx; ↑i. From the relation We can estimate the value of the expression 1 1 |S ; ↑i = √ |S ; ↑i + √ |S ; ↓i x z z X 2 iEa0 t 2 2 C(t) = |c 0 | exp − a ¯h we see that a0 1 c↑ = c↓ = √ . more concretely when we suppose that the statevectors of 2 the system comprise so many, nearly degenerate, energy For the probabilities that at the moment t the system is eigenvectors that we can think them almost to form a in eigenstates of Sx we get continuum. Then the summation can be replaced by the ω t integration |hS ; ↑ |α, t = 0; ti|2 = cos2 c x 0 2 X Z 2 2 ωct −→ dE ρ(E), ca0 −→ g(E) , |hSx; ↓ |α, t0 = 0; ti| = sin . 2 a0 E≈Ea0 Even if the spin originally were parallel to the positive where ρ(E) is the density of the energy eigenstates. The x-axis a magnetic field parallel to the z-axis makes the expression direction of the spin to rotate. There is a finite probability for finding the system at some later moment X 2 iEa0 t C(t) = |ca0 | exp − in the state |Sx; ↓i. The sum of probabilities ¯h corresponding to different orientations is 1. a0 It is easy to see that the expectation values of the can now be written as operator S satisfy Z 2 iEt ¯h C(t) = dE |g(E)| ρ(E) exp − , hS i = cos ω t ¯h x 2 c ¯h which must satisfy the normalization condition hS i = sin ω t y 2 c Z dE |g(E)|2ρ(E) = 1. hSzi = 0.

Physically this means that the spin precesses in the In many realistic physical cases |g(E)|2ρ(E) is xy-plane. concentrated into a small neighborhood (size ∆E) of a point E = E0. Rewriting the integral representation as iE t C(t) = exp − 0 ¯h Z i(E − E )t × dE |g(E)|2ρ(E) exp − 0 , ¯h we see that when t increases the integrand oscillates very rapidly except when the energy interval |E − E0| is small as compared withh/t ¯ . If the interval, which satisfies |E − E0| ≈ ¯h/t, is much shorter than ∆E —the interval from which the integral picks up its contribution—, the correlation amplitudes practically vanishes. The characteristic time, after which the absolute value of the correlation amplitude deviates significantly from its initial value 1, is ¯h t ≈ . ∆E Although this equation was derived for a quasi continuous energy spectrum it is also valid for the two state system in our spin precession example: the initial state |Sx; ↑i starts to lose its identity after the time ≈ 1/ωc =h/ ¯ (E↑ − E↓) as we can see from the equation ω t |hS ; ↑ |α, t = 0; ti|2 = cos2 c . x 0 2 As a summary we can say that due to the evolution the state vector describing the initial state of the system will not any more describe it after a time interval of order ¯h/∆E. This property is often called the time and energy uncertainty relation. Note, however, that this relation is of completely different character than the uncertainty relation concerning position and momentum because time is not a quantum mechanical observable.