Time_evolution.nb: 9/26/04::22:57:04 1
Time evolution and quantum dynamics
Sakurai 2.1, 2.2 , Merzbacher 14, BJ 5.7, 5.8
ü Time evolution operator & the Hamiltonian
ü Basis kets and evolving states
Start by defining a basis i which spans the vector space of possible states. These could be eignstates of some operator, say A. It is often useful to consider eigenstates of the Hamiltonian ? H . These states are static. If defined in terms of an operator, and the operator is a function of time, then the definition of the basis should specify the time, in which case we may write i t . This does not imply time evolution, it is just a label for which set of states we are using as a basis. For the moment we will consider a basis that does not change with tim + /? e.
We have postulated that the state of the system can be expressed as a linear combination of the basis kets.
a = cai i ? ? Now, in general, the state will evolve in time, so we write
a, t = cai t i ? + / ? Note that the time dependence is all in the cai t . This is the so called "Schroedinger" picture. + /
ü U+t, t0/
Define the time evolution operator
U t, t0 a, t0 = a0, t + / ? ? The subscript 0 is intended to denote that the system was in the state a at time t0 , and then evolved to t. This is generally not the same as a system which is in state a at time t. ? ? We desire a few properties of the operator U
† + Unitary: If a, t0 a, t0 = 1, then a, t a, t = a, t0 U t, t0 U t, t0 a, t0 = 1, i.e, we want U to be unitary. 2 Note: this implies ; S« cai t? = 1 for all ; t. « ? ; + / + / ? i + /
+ Law of composition: U t2, t0 = U t2, t1 U t1, t0 , for t2 > t1 > t0 . This composition law is associative U t3, t2 U t2, t1 U t1, t0+ = U/ t3, t2+ U /t2, t+1 U /t1, t0 + + / + // + / + / + + / + // + Identity: U t1, t1 = 1. + / 1 † + If we add Inverse: U t, t0 = U t, t0 , then the time evolution operations form a group. We will discuss symmetry groups later on in connection with spatial translations and rotations.+ / + / Time_evolution.nb: 9/26/04::22:57:04 2
Unitarity suggests the expansion for infinitesimal time translations
U t0 + dt, t0 = 1 i W dt. + / The Hermitian operator W is the generator of time translations. The explicit factor of i is required to make the infiinitesimal translation Unitary, i.e. it is required to have U 1 = U † = 1 + i W dt, so that to first order
U † U = 1 + i W dt 1 i W dt = 1 + $ dt2 + / + / + / W has dimensions of frequency. The Hamiltonian is defined by H = W and has dimensions of energy, but is otherwise just the generator of time translations.
ü Schroedinger equations
The Schroedinger equation is a differential equation which describes the time evolution of a quantum system. The most basic version describes the time evolution of the time evolution operator itself. If this is known, it may be applied to determine the evolution of the system in any state.
H Consider U t + dt, t0 = U t + dt, t U t, t0 = 1 i ÅÅÅÅÅÅ dt U t, t0 . For small dt + / + / + / + / + / U+t+dt,t0/ U+t,t0/ ∑ HU t, t0 = i ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdt = i ÅÅÅÅÅÅ∑t U t, t0 + / + / ü For a?
Alternatively, one can derive an equation for the state vector directly.
∑ ∑ HU t, t0 a, t0 = i ÅÅÅÅ∑tÅÅ U t, t0 a, t0 or H a, t = i ÅÅÅÅ∑tÅÅ a0, t . Note the difference between the static state a, t0 and the evolving state + / ? a0, t+ . / ? ? ? ? ? ü Basic case, H is constant (no explicit time dependence)
One can solve the differential equation with boundary condition U t0, t0 = 1. + / ÅÅiÅÅ H t t U t, t0 = e + 0/ + / One may also consider the limit of many short evolution steps.
n i i Dt ÅÅÅÅ H+t t0/ U t, t0 = Lim 1 ÅÅÅÅ H ÅÅÅÅÅÅÅ = e nض n + / + / Alternatively, consider a single energy eigenstate
a, t = ci t i ? + / ? ∑ i ÅÅÅÅ∑tÅÅ ci t i = H ci t i = Ei ci t i or, since i is just constant and not changing with time, we have + / ? + / ? + / ? ? i w t t ci t = e i+ 0/ ci t0 with wi = Ei + / + / s And for a more general state use Time_evolution.nb: 9/26/04::22:57:04 3
i i i e ÅÅÅÅ H t 1 = S e ÅÅÅÅ Ht i? ;i = S i? e ÅÅÅÅ Ei t ;i i i
or for a state a0, t = S ca j t j? j ? + / i w t i w t a0, t = U t, t0 a, t0? = S ca j t0 S i? e i i j = S cai t0 e i i? j i i ? + / + / ; « ? + /
ü Eigenvectors are fixed, but energies vary with time
b) H varies with time, but H t1 , H t2 = 0. This corresponds to the case where the eigenvectors remain the same, but 1 the energy eigenvalues change with time. An example of this would be a spin # + / + /' ÅÅ2ÅÅ particle propagating through a magnetic field where B changes in magnitude, but not direction. ¹¶ ÅÅÅÅi t H t' dt' ¼t + / U t, t0 = e 0 + / and similarly for ci .
ü Eigenvectors are changing in time
1 c) H t1 , H t2 ∫ 0 This case is non-trivial. An example would be the s = ÅÅÅÅ case just mentioned, but where B 2 ¹¶ changes direction. A formal solution is (see Sakurai)# + / + /'
i n t t1 tn 1 U t, t0 = 1 + S ÅÅÅÅÅÅÅ dt1 dt2 ... dtn H t1 H t2 ... H tn n t0 t0 t0 + / + / ¼ ¼ ¼ + / + / + / This series is related to formal perturbation theory.
ü Transition Amplitudes
Time evolution can be considered as a "transition amplitude". At t0 the system starts in state a . At time t, system has evolved to a0, t = U t, t0 a, t0 . Then, the amplitude to be in a state b at time t is given by ? ? + / ? ? Aab t = b, t a0, t = b, t U t, t0 a, t0 . + / ; « ? ; « + / « ? The probability for a system initially in state a to evolve to state b is just the square of this amplitude. ? ? 2 Pab t = Aab t + / + / Note that the inner product is only defined at equal times - b, t a, t0 is not defined. To compare two states from different times, one must time evolve one of the states to the time of the other.; « ?
ü Example for two state system
Consider a two state system which, at t = 0, is in the state
a ª a, 0 = c1 1 + c2 2 . ? ? ? ? where 1 and 2 are energy eigenstates. Then, at some later time t, the system will evolve to ? ? Time_evolution.nb: 9/26/04::22:57:04 4
iw t iw t a0, t = c1 e 1 1 + c2 e 1 2 ? ? ? The amplitude for the state to still be in state a at time t is ? Aaa t = a U t, 0 a * * iw1 t iw1 t + / =; «1 c+1 + /2« c?2 c1 e 1 + c2 e 2 2 iw t 2 iw t = +c;1 e 1; + c/2+ e 2 ? ?/ The probability to remain in state a is ? 2 Paa t = Aaa t 4 4 2 2 + / = c1 + /+ c2 + 2 c1 c2 cos wt + / where the oscillation frequency is w = w1 w2 .
Here are three oscillation plots. Note the fully mixed case in black. The small mixing case in red. The blue curve has intermediate amount of mixing, and is set to oscillate at twice the frequency. This is a description of neutrino oscillation
with two flavors. Start out in state of ne and oscillate into another "flavor". A detector at different distances from the source will measure different fluxes of n? e . ?
paa #c1_, c2_, w_, t_ ' : Abs #c1 '^4 Г Abs #c2 '^4 Г 2 Abs #c1 '^2 Abs #c2 '^2 Cos #w t'; Plot #paa #Sqrt #.5 ', Sqrt #.5 ', 1, t', paa #Sqrt #.01 ', Sqrt #.99 ', 1, t', paa #Sqrt #.2 ', Sqrt #.8 ', 2, t' , t, 0, 10 ';
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You may wish to consider the difference between "appearance" and "disappearance" experiments, especially in the case where wt + 1.
ü comment on correlation amplitude
ü Evolution of expectation values & Conserved quantities
ü expectation values
Consider the evolution of the expectation value for an operator O. If at time t = 0 Time_evolution.nb: 9/26/04::22:57:04 5
O t=0 = a O a ; ? ; « « ? Then at some later time
i i † ÅÅÅÅ Ht ÅÅÅÅ Ht O t = a, t O a, t = a U t, 0 OU t, 0 a = a e O e a . ; ? ; ? ; + / + / ? ; ? ü Conservation and commutation with H
i ÅÅÅÅ Ht ∑ If O, H = 0, then O, e = 0, and O t = O t=0 , or alternatively, ÅÅÅÅ∑tÅÅ O = 0. If O commutes with H then O corresponds to a conserved quantity, but not otherwise.# ' $ ( ; ? ; ? ; ?
ü The two state system again
For example, consider the two state system from above.
* iw1 t * iw2 t iw1 t iw2 t a0, t O a0, t = 1 c1 e + 2 c2 e O c1 e 1 + c2 e 2 ; ? +; ; / + ? ?/ If O, H = 0 then the energy eigenstates are also eigenstates of the operator O, so O i = oi i and # ' ? ? * iw1 t * iw2 t iw1 t iw2 t a0, t O a0, t = 1 c1 e + 2 c2 e c1 e o1 1 + c2 e o2 2 2 2 ; = c1 ? o1 +; c 2 o2, ; / + ? ?/ O is independent of time and is conserved. ; ? On the other hand, if O, H ∫ 0then one must consider a more general O which mixes energy eigenstates. Adopting o o # 11 '12 * matrix notation O = . If O corresponds to an observable it is Hermitian, and o21 = o12 . Then L o21 o22 \ M ] N ^ iw t o11 o12 c1 e 1 * iw1 t * iw2 t c1 e c2 e iw t = L o21 o22 \ L c e 2 \ + / M ] M 2 ] N N iw1 t iw2 t ^o11 c1 e ^ + o12 c2 e * iw1 t * iw2 t 2 2 * iwt c1 e c2 e = c1 o1 + c2 o2 + 2 Re c1 c2 e o12 L o c e iw1 t + o c e iw2 t \ + / M 21 1 22 2 ] + / N ^ ü spin precesion
e 1 0 As an example, consider spin precesion. The Hamiltonian is H = ÅÅÅÅ sz , where sz = , e = w and state 1 2 L 0 1 \ M ] ? corresponds to the z+ polarization state. Precession is described by the evolution of N ^ 0 1 ? 0 i sx = , sy = and sz . First consider the case where the system starts in a spin up state, L 1 0 \ L i 0 \ ; ? >M ]B ; ? >M ]B ; ? c1 = 1, cN2 = 0. ^ Then sx =N sy =^ 0since for these operators o11 = 0. Also, sz t = 1. Next consider the case where ` 1 1 the system starts polarized in the ; ? ; ?x direction, c1 = c2 = 1 2 . Now, sz t ;= ÅÅ?ÅÅ+ / ÅÅÅÅ = 0. More interesting is r 2 2 t ; ? + / 1 iwt sx t = 0 + 0 + 2 Re ÅÅ2ÅÅ e = cos wt , and 1 iwt ;sy? +t/ = 0 + 0 + 2 Re + ÅÅ2ÅÅ e / i = sin + w/t . ; ? + / + / + / The spin precesses, as it should. Time_evolution.nb: 9/26/04::22:57:04 6
ü "Pictures"
Consider an operator O with no explicit time dependence, which is defined for some vector space spanned by the basis kets i . Then O is defined by the matrix elements i O j . Next, consider the evaluation of O between two states defined at time ? t = 0, b, 0 O a, 0 , and how this quantity evolves with time.; ? ; ? † b0, t O a0, t = b0 U t, 0 OU t, 0 a0 ª b0 O t a0 . ; ? ; + / + / ? ; + / ? The last step defines an operator O t . These are three ways of writing the same thing. The first puts the time dependence into the state evolution. The third puts the time dependence into the operators. In+ / the middle, both the states and the operators are static and the time dependence is accounted for by inclusion of time evolution operators. The first is called the Schroedinger picture, the third is called the Heisenberg picture. There is a bit of relativity here - which is moving the system or the operator? or should I say observer? I don't know of a name for the middle notation where only U knows about time.
A common notation is to denote states and operators in the Schroedinger picture by a subscript S , and corresponding quantities in the Heisenberg picture by a subscript H . Then the time evolution of the matrix element above is
† bS, t OS aS, t = bS, 0 U t, 0 OS U t, 0 aS, 0 ª bH OH t aH , ; ? ; + / + / ? ; + / ? Note that the matrix elements are the same in either picture - the physics should not depend on the coordinate system.
In the case where H is constant in time, the Heisenberg operator is given by
ÅÅiÅÅ Ht ÅÅiÅÅ Ht OH = e OS e
If OS, H = 0, then OH = OS . # ' ° Again taking H = 0, the Schroedinger states have explicit time dependence, but the Heisenberg states do not,
† aH = aS, 0 = U t, 0 aS, t ? ? + / ? Compared to the Schroedinger state, the corresponding Heisenberg state appears to be drifting backward in time. In fact, the Heisenberg state is static.
ü Example in the Schroedinger picture
Consider our friend the two state system, and for simplicity take b = a . In the Schroedinger picture, we apply U to the states ? ?
;a U † t, 0 OU t, 0 a? = . i w t i w t iw t + / + /e 1 0 o11 o12 e 1 0 c1 o11 o12 c1 e 1 c* c* * iw1 t * iw2 t 1 2 i w t i w t = c1 e c2 e iw t L 0 e 2 \ L o21 o22 \ L 0 e 2 \ L c2 \ L o21 o22 \ L c e 2 \ + / M ] M ] M ] M ] + / M ] M 2 ] N ^ N ^ N ^ N ^ N ^ N ^ ü Example in the Heisenberg picture
i i In the Heisenberg picture, we combine the operators first, O t = U † t, 0 OU t, 0 = e ÅÅÅÅ H t O e ÅÅÅÅ H t + / + / + / Time_evolution.nb: 9/26/04::22:57:04 7
ei w1 t 0 o o e i w1 t 0 c o o e iwt c c* c* 11 12 1 c* c* 11 12 1 1 2 i w t i w t = 1 2 iwt , where L 0 e 2 \ L o21 o22 \ L 0 e 2 \ L c2 \ L o e o \ L c2 \ + / M ] M ] M ] M ] + / M 21 22 ] M ] w = w1 wN 2 . ^ N ^ N ^ N ^ N ^ N ^
ü Equation of motion for operators in the Heisenberg picture.
The time evolution of O t = U † OU is given by + / dO+t/ dU† † dU i † † i † † ÅÅÅÅÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅÅÅÅÅÅÅÅdt OU + U O ÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅ U HUU OU ÅÅÅÅ U OUU HU = i † † i i ÅÅÅÅ U HUO t O t U HU = ÅÅÅÅ HO t O t H = ÅÅÅÅ H, O t + + / + / / + + / + / / # + /' where we used the Schr. eq for U , put in 1 = UU † , converted back to O t , and used U † HU = H for a Hamiltonian which is independent of time. Cutting out the middle steps + /
dO+t/ i ÅÅÅÅÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅ H, O t . # + /' Note the correspondence to Poisson brackets.
ü Interaction picture (Dirac)
The Heisenberg and Schroedinger pictures are prescriptions for how to treat the time evolution of the phases in a quantum system. They have different value for treating different problems. The Schroedinger picture gives an intuitive picture of how the state of a system evolves. It has the drawback that if the system evolves for a long time that the calculation of those phases may be annoying. In the Heisenberg picture the states are static for a time independent Hamiltonian, and so the evolution of the system is almost trivial. It has the drawback that the evolution of operators is not as intuitive a concept. In addition, if H is not constant in time that the Heisenberg states have non-trivial time evolution. A third method for allocating the time dependence of a system, the "interaction picture", is often used for calculating transition amplitudes.
Suppose that the Hamiltonian of a system is given by H = H0 + H1 t , where H0 is time independent but H1 is not. Typically H0 has known eigenstates, labeled as n . A common case is that + / H1 t is only non-zero for an interval of time ti < t < t f . In this case, it would be useful to consider the system in the Heisenberg picture f ? + / or t < ti and for t > t f , but in a Schroedinger-like manner for the time when H1 is active. Suppose the system starts in a particular state, say i . It remains in that state until t = ti at which time it starts to evolve under the influence of H1 until t f , at which point the ? system stops evolving (at least in the Heisenberg picture). The time evolution of the system is captured by calculating the transition amplitudes n i from state i to any other state n . ; « ? ? ? In such a case it is convenient to separate out the effects of the time independent H0 , and define the operator in the interaction picture
ÅÅiÅÅ H t ÅÅiÅÅ H t OI t = e 0 OS e 0 + / Note that in this example the operator OS typically has explicit time dependence. In particular, the time evolution of a state in the interaction picture is determined by
ÅÅiÅÅ H t ÅÅiÅÅ H t H1 I t = e 0 H1 t e 0 + / + / Time_evolution.nb: 9/26/04::22:57:04 8
Schroedinger Wave Equation and correspondence to classical mechanics
For this course it is assumed that the student is familiar with correspondence principles. The discussion below focuses on deriving the Schroedinger wave equation from the more formal discussions already given, rather than taking it as an ad-hoc description of nature.
ü Schroedinger Wave Equation
ü Time dependent version in one dimension
Consider ya x, t = x a, t where x is just a basis ket in the continuum x basis, i.e. it is not a time dependent state. p2 We want to know how + / ; «y evolves in time under the Hamiltonian ? ? H = ÅÅÅÅÅÅÅÅÅ2 m +V? X . The equation of motion is + / ∑ i ÅÅÅÅ∑tÅÅ a, t = H a, t ? ? Acting on the left with x yields the usual Schroedinger equation for the wave function ya x, t . For the LHS there is no time dependence to ;x , and so + / ? ∑ ∑ ∑ x i ÅÅÅÅ∑tÅÅ a, t = i ÅÅÅÅ∑tÅÅ x a, t = i ÅÅÅÅ∑tÅÅ ya x, t ; ? ; « ? + / On the RHS
p2 x H a, t = x ÅÅÅÅÅÅÅÅÅ2 m + V X a, t ; ? ; p2 + / ? = x ÅÅÅÅÅÅÅÅÅ2 m a, t + x V X a, t ; ? ; + / ? For the potential term V X acts to the left, so x V X a, t = V x x a, t = V x ya x, t . For the momentum term, ∑ ∑ use p a, t = i ÅÅÅÅ∑xÅÅÅ a,+t /= i a', t , where ;a', t+ is defined by / ? + /x; a« ', t ?ª ÅÅÅÅ∑xÅÅÅ+ y/a x,+ t . Although this is a bit glib, it / is good enough for the current discussion. We will discuss the relation of ? ? ? ? ; « x and ? p in more detail later under the topic of + / p2 2 '' translational symmetry. For now this allows x ÅÅÅÅÅÅÅÅÅ2 m a, t = ÅÅÅÅÅÅÅÅÅ2 m ya x, t . Combining the left and right hand sides of the time evolution equation gives ; ? + /
∑ 2 '' i ÅÅÅÅÅÅ∑t ya x, t = ÅÅÅÅÅÅÅÅÅ2 m ya x, t + V x ya x, t + / + / + / + / ü Time dependent version in 3-d
p2+p2+p2 Allowing for 3 spatial dimensions, the Hamiltonian is H = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅx y z + V X , which leads to 2 m ¹¹¶ + / ∑ 2 ∑2 ∑2 ∑2 i ÅÅÅÅÅÅ∑t ya x, t = ÅÅÅÅÅÅÅÅÅ2 m ÅÅÅÅÅÅÅÅÅ∑x2 + ÅÅÅÅÅÅÅÅÅ∑y2 + ÅÅÅÅÅÅÅÅÅ∑z2 ya x, t + V x ya x, t ¶ ¶ ¶ ¶ + /2 2 , 0 + / + / + / = ÅÅÅÅÅÅÅÅÅ2 m “ ya x, t + V x ya x, t +¶ / +¶/ +¶ / ü comments
ü continuity equation, probability conservation Time_evolution.nb: 9/26/04::22:57:04 9
ü Classical limit