Chapter 16

Time-dependent Perturbation Theory

16.1  INTRODUCTION

In the previous Chapter, we discussed the effects of small time-independent perturbation Ĥ′(r) on the energy eigenvalues img and eigenstates equation of the unperturbed Hamiltonian Ĥ0. In Chapter 7 we learnt that the stationary eigenstates equation of a time-independent Hamiltonian Ĥ0 form a complete orthonormal set equation such that any state function ψ(r) in that space may be expressed as a linear combination of basis vectors equation For example, to find the perturbed eigenstates up to Ist order in perturbation, we started with the eigenstate equation and then to estimate the correction ψ(1) we expressed it in terms of the basis vectors equation like equation where (time-independent) coefficients were found in terms of the perturbation Ĥ′(r).

Now if we consider the perturbation to be time-dependent Ĥ′(t), the total Hamiltonian Ĥ0 + Ĥ′(t), becomes time-dependent. In absence of perturbation Ĥ′(t), the eigenstates of Ĥ0 are given by

equation

And if the unperturbed system is initially in a particular eigenstate equation it will stay there for all time. The only time-dependence of equation is of the form

equation

If now perturbation Ĥ′(t) is switched on, we expect the eigenstate to start changing. But just like in case of time-independent perturbation, it should be still possible to express the changed eigenstate as the linear combination of the basis states equation However, unlike time-independent case, here the expansion coefficients should be time-dependent, as the resultant eigenstate ψ(r, t) has to be time-dependent. So, an expansion like

equation

should be used. And to find out in what way the perturbation Ĥ′(t) changes the state of the system with time, we should find out the time-dependence of coefficients an (t).

16.2  TIME DEVELOPMENT OF STATES AND TRANSITION PROBABILITY

As discussed in the Introduction, let us consider a system with total Hamiltonian

equation

where unperturbed Hamiltonian Ĥ0 is time- independent and Ĥ′(t) is time-dependent perturbation. As in the case of time-independent perturbation theory, we have introduced the parameter λ (having value 0 to 1) to identify the different orders of the perturbation calculations. Let the Schrodinger equation corresponding to the unperturbed Hamiltonian Ĥ0 [Eq. (16.1)] be exactly solvable (i.e., the energy eigenvalues equation and eigenstates equation are known). Now for the time-dependent Hamiltonian Ĥ, we shall have to solve time-dependent Schrodinger equation

equation

Since Ĥ is time dependent, the energy of the system is not conserved. So, it is meaningless to seek corrections to the energy eigenvalues. Instead, the objective is to find out (approximately) the eigenstate ψ(r, t) in terms of the unperturbed eigenstates equation As indicated in the Introduction above, this shall be done by expanding eigenstate ψ(r, t) in terms of unperturbed states equation with time-dependent coefficients: an approach known as Dirac’s method of variation of constants. So, ψ(r, t) is expanded as

equation

which is the same as Eq. (16.3) with equation If ψ(r, t) is normalized to unity, we should have

equation

or

equation

Also we have

equation

We see that equation is the probability that at time t the system is found in the unperturbed state equation

We now proceed to develop a perturbation theory, where we can find out equations to calculate the unknown coefficients Cn(t) for the given perturbation Ĥ′(t). Let us substitute the expansion (16.6) into the time-dependent Schrodinger equation (16.5). We get

equation

or

equation

or

equation

Here Ċn(t) denotes dCn(t)/dt. On the R.H.S. of Eq. (16.9) we have used Eq. (16.1). We may easily note that the second term on L.H.S. is same as first term on R.H.S. So, we get

equation

Taking the scalar product of both sides of Eq. (16.10) with a function equation (which belongs to the set equation) and using the orthonormality relation

equation

one gets,

equation

where equation denotes the matrix element

equation

and

equation

It is clear that Eq. (16.12) is really representing a set of equations corresponding to various values of k. This set of simultaneous linear homogeneous differential equations may be put in matrix form as

equation

It may be noted that the set of equations represented by Eq. (16.15) [or Eq. (16.12)] are exact just like (exact) Schrodinger equation (16.5). In fact, no approximation has been made in arriving at Eq. (16.15) starting from Eq. (16.5). So Eq. (16.15) is simply Schrodinger equation (16.5) put in a different form. It can be easily noted that for any k, the time rate of change [dCk(t)/dt] depends upon all states equation connected with equation via perturbation matrix elements equation [given by Eq. (16.13)]. This may also be understood in terms of probability conservation expressed by Eq. (16.7). In fact, Eq. (16.7) implies that any change in one of the coefficients, say Ck(t), is accompanied by corresponding changes in other coefficients.

As it is generally not possible to solve the set of time-dependent coupled differential equations (16.15) exactly, we shall now proceed to find their approximate solutions. Let us assume the perturbation λĤ to be weak. We may, therefore, expand the coefficients Ck(t) in powers of the parameter λ as

equation

Substituting this expansion in Eq. (16.12) and equating on both sides, the coefficients of equal powers of λ, we get

equation

In principle, for given perturbation matrix elements equation these equations may be integrated successively to any order in perturbation. Let us assume that the perturbation is switched on at time t = 0. And let the unperturbed system up to time t = 0 be in one of its eigenstate, say equation So, in the absence of perturbation, the coefficient Ca(t) is time-independent and is equal to unity.

According to expansion (16.16), the initial conditions of the system (i.e., the system in unperturbed state, at t = 0) are defined by equation Equation (16.17) simply confirms that the coefficients equation are time-independent. Thus we may write

equation

Now substituting Eq. (16.20) into Eq. (16.18), we have

equation

For k = a, Eq. (16.21) gives

equation

while for k ≠ a (16.21) gives

equation

In writing Eqs (16.22), we have used the above mentioned initial conditions, that is, equation and equation Equations (16.22) are the bases for the first-order time-dependent perturbation theory. To first-order in the perturbation, the transition probability corresponding to the transition equation (i.e., the probability that at time t, the system has made transition from its initial state equation to the state equation is

equation

Also, we may write the coefficient Ca(t) (t > 0) of the stateequation, to first order in the perturbation as

equation
16.3  CONSTANT PERTURBATION

Let us consider a small perturbation Ĥ′(t) that is zero before the initial time t = 0 and is a constant (independent of time) for t > 0. And as we considered in the previous section, let the unperturbed system at t = 0 be in eigenstate equation Then from Eq. (16.24) we have

equation

So that |Ca(t)|2 ≈ 1. Therefore, so far as initial state equation is concerned, the main effect of the first order perturbation is to change its phase. Looking at the expansion (16.6) of the perturbed state ψ(r, t), the contribution of the initial stateequation is

equation

The R.H.S. expression suggests that the energy of the perturbed state, in the first-order approximation is

equation

 

where equation This result is, in fact, in agreement with the first order correction to the energy eigenvalues in the time-independent perturbation theory.

Now the next step is to find approximately how the coefficients equation change with time (for t > 0) under the influence of a small perturbation equation switched on at t = 0. (It may be remembered here that at time t = 0, Ca = 1 and all other coefficients Ck are zero.) For at least some small period of time after t = 0, all the Ck’s (except Ca) are small, while Ca remains close to unity. Thus to a first approximation equation and for time-independent perturbation matrix element equation the equation becomes [Eq. (16.22b)].

equation

It may be noted here that if the perturbation Ĥ′ is switched off (say) at time t1, the amplitudes equation at a later time t > t1 is given by

equation

But the second integral above is zero, so

equation

So equation (for t > t1) remains constant at its value at t = t1. Now turning to Eq. (16.28), we can find the transition probability from (initial) state equation to state equation as

equation

where equation We show in Figure 16.1 the schematic plot of the transition probability equation(t, ωka), for a fixed value of t. We note the following features:

  1. The transition probability equation is sharply peaked about the value ωka = 0 with a width of approximately ∆ω ≈ (2π / t). It is clear that transition to those final states equation for which ωka (2π / t), will be strongly favoured. That simply means the transition will be mainly into those final states whose energy equation is located in energy width
    equation

    around the initial energy equation So the unperturbed energy is conserved within [ΔE (2πħ / t)]. This result may also be interpreted as the energy time uncertainty relation in a measurement process. In fact, the perturbation process is equivalent to the process of measurement of energy equation of the system by inducing transition equation For this measurement, we switch on perturbation for time duration t; the uncertainty related to this energy measurement should be of order ħ/t in agreement with Eq. (16.31).

  2. Let us, next, examine the transition probability as a function of t. We have equation(t, ωka) = 0 at t = 0 for all values of ωka. Plot of equation(t, ωka) as function of t for a fixed value of ωka = ω0
    Figure 16.1

    Figure 16.1 Schematic plot of transition probability equation as a function of ωka at a fixed value of time t

    Figure 16.2

    Figure 16.2equation as a function of time t for a fixed value of ωka = ω0

    (say), as shown in Figure 16.2, shows equation oscillating with period T = (2π / ω0) about the average value.

equation

For times t much smaller than the period of oscillation T = (2π/ω0), we have sin (ω0t / 2) (ω0t / 2) and, therefore, from Eq. (16.30).

equation

So, for small time t, the transition probability increases quadratically with time.

16.3.1   Transition to a Group of Final States: The Fermi Golden Rule

We have till now studied the transition process of the system from its initial state equation to a final state equation, when a constant perturbation is switched on at time t = 0. Instead, it is very often necessary to deal with transitions from initial state equation to a group of closely spaced final states equation, whose energy lies within a given interval equation centred about the value equation

Therefore, total (first-order) transition probability P(1)(t), that the system at time t has made a transition away from its initial state equation, is obtained by summing the transition probabilities (16.30) over all find states equation Such a situation arises when we have a large number of closely spaced discrete energy levels or the continuum of energy levels to which transition may take place; the example of the former case is the electronic energy levels in bands in solids and that of the latter is the scattering problem. In such situations, the distribution of energy levels may be described through the density of levels, denoted by equation and defined as the number of energy levels in the unit energy interval around equation So equation denotes the number of energy levels lying in the interval equation and equation

Now the expression for total transition probability P(1)(t), assuming that none of the final states equation is degenerate with the initial state equation, is given as

equation

For very closely spaced final energy levels, the summation in Eq. (16.34) may be converted into integration using the definition of the density of states equation. The probability that in time t, the system initially in state equation will make a transition to any of the final states is given as

equation

The integration is over the energy range of the final states considered. Now equation and equation are fairly constant over the range of energy Δ around equation, whereas the function equation is sharply peaked around equation so we may write P(1) (t) as

equation

 

In fact, the function equation has peak width of ~ (2πħ / t). And we take time t to be large enough such that Δ satisfies the condition

equation

So, it is clear that the integral in Eq. (16.36) will be small except for transitions which conserve energy [within δE = (2πħ/t)]. Keeping the condition (16.37) in mind, we may replace the limits of integral in Eq. (16.36) to get

equation

Now using the standard result

equation

We get

equation

The rate of change of transition probability or transition probability per unit time or the transition rate W(t) may be obtained as

equation

This formula of transition rate to first order in perturbation theory was obtained by Dirac and was termed as ‘Golden Rule’ of transition rate by Fermi.

16.3.2  The Degenerate Case

Let us consider the case when the initial state equation and stateequation are degenerate, that is, when equation equation In this situation, we have from Eq. (16.28),

equation

So the first-order transition probability for the degenerate case, under a constant perturbation switched on at time t = 0, is given by

equation

For large enough time, the transition probability given by Eq. (16.43a) may exceed unity, which is obviously not acceptable. Therefore, it may be easily concluded that the perturbation treatment developed in this section cannot be applied to systems involving transitions amongst degenerate states, perturbed over long period of time.

16.3.3  Conservation of Probability

From Eq. (16.34), we have the expression for the probability P(1) (t) at time t that the system has made transition away from its initial state equation,

equation

Also from Eq. (16.25), we have the expression for the probability that at time t, the system is still in its initial state equation (i.e., the survival probability of the initial state) as

 

Pa(t) = |Ca(t)|2 = 1       (16.43b)

 

In fact, the sum of two probabilities (the probability that the system has made a transition away from its initial state equation and the probability that the system is still in its initial state equation should be unity at any time t. However, this is not followed by the sum of two corresponding expressions (16.34) and (16.43b). That simply means, total probability is not conserved here. Something is wrong somewhere. Let us correct it.

From Eq. (16.18), we have

equation

Now putting value of equation from Eq. (16.28) [it is clear from Eqs (16.16) and (16.17) that equation we get

equation
(as ωak = –ωka)

 

Integrating both sides of Eq. (16.45) from 0 to t gives [Ca (0) = 1],

equation

or

equation

Finding mod square of both sides and retaining terms only up to equation we get

equation

Equation (16.46) gives correct expression for the survival probability of the initial state at time t. It can be easily checked from Eqs (16.34) and (16.46) that the sum of transition probability and the survival probability is unity, the required result.

16.4  THE ADIABATIC APPROXIMATION

Let us apply the perturbation theory developed in Chapter 15 and in this Chapter to an interesting case where perturbation is turned on very slowly. Suppose a system is described by the unperturbed time-independent Hamiltonian Ĥ0 for time t ≤ 0. At time t = 0, a perturbation is switched on very slowly. Let the perturbation be denoted as Ĥ′(t) = Vf(t), where f(t) is a slowly varying function as shown in Figure 16.3. Now, though the perturbation is varying very slowly with time, we are supposed to apply time dependent perturbation theory to find out perturbed eigenstates and eigenvalues. The results we shall get are very interesting.

Suppose the unperturbed system (up to time t = 0) is described by the Hamiltonian Ĥ0 and has exactly known energy eigenvalues equation and eigenstates equation:

equation

 

The slowly varying potential (switched on at t = 0) has value Ĥ′(t1) at time t = t1. We shall find that the eigenstate ψn(r) of the perturbed Hamiltonian Ĥ0 + Ĥ′(t1), in first-order perturbation, comes out to be

equation

which is precisely the first-order result of the time-independent perturbation theory corresponding to the perturbation Ĥ′(t1).

Let us proceed to find the result. We use the results of time-dependent perturbation theory with the time-dependent perturbation Ĥ′(t) = Vf(t). Equation (16.22b) gives the time-development of the expansion coefficient equation as

equation

where equation Integrating Eq. (16.49) by parts, we get

equation
Figure 16.3

Figure 16.3 Adiabatic perturbation

If Ĥ′(t) [= Vf(t) is slowly varying, the second term is negligible compared to the first one and equation may be approximated by

equation

It may be noted here that in finding expression of expansion coefficient equation, we have presumed that the system at time t = 0 was in eigenstate equation of Ĥ0. Now the perturbed eigenstate ψ(r, t) [Eq. (16.6)] up to first order in perturbation is given as

equation

As the perturbation is slowly varying and the energy eigenvalue of the system at t = 0 is equation, we may write

equation

Putting Eqs (16.50) and (16.52) in Eq. (16.51), we get

equation

This is the same result as obtained by first-order time-independent perturbation theory [Eq. (15.13b)]. In case of time-independent perturbation theory, Eq. (15.13b) represents the perturbed eigenstate ψη of the total Hamiltonian Ĥ0 + Ĥ′; in absence of the perturbation Ĥ′, the eigenstate was equation. Therefore, Eq. (16.48) represents the perturbed eigenstate ψ(r, t) of Ĥ0 + Ĥ′(t), to first order in Ĥ′. It may be noted here that the system at t = 0 in the eigenstate equation develops to the eigenstate ψ(r, t) of Ĥ0 + Ĥ′(t) at time t. In other words, it may be stated that the system originally (i.e., at t = 0) in the nth state of the Hamiltonian Ĥ0, is found in the nth state of the new Hamiltonian Ĥ0 + Ĥ′(t), at time t. So, we may say, the system remains in the nth state throughout the duration of the adiabatic perturbation: the only thing is that this nth state evolves slowly with time from equation (at t = 0) to (say)equation (at time t).

EXERCISES

Exercise 16.1

A system is having discrete eigenstates equation and eigenenergies equation. The system is exposed to the perturbation

 

H′(t) = Ae–t22

 

The perturbation is turned on at t = –∞, when the unperturbed system is in its ground state equation. Find the probability that the system makes a transition to a state equation

Exercise 16.2

A hydrogen atom in the ground state is put in a homogeneous electric field which is pointing in z-direction with its magnitude varying with time as

 

E(t) = E0 e–t/τ

 

This field is turned on at t = 0. Find the probability that the atom is excited to 2 p state at time t (> > τ).

Exercise 16.3

In Exercise 16.2, if the electric field in z-direction is switched on at t = –∞ and varies with time as

equation

find the probability of excitation of hydrogen atom from its ground state to 2 p state for t (> > τ).

Exercise 16.4

Consider a particle of mass m in the ground state of the one-dimensional potential well with its walls at x = 0, and x = L. At time t = 0, a potential V (x) is switched on such that

equation

 

Here V0 is less than equation the ground state energy of the unperturbed well. The perturbation V′(x) is switched off at t = T. Find the probability (in first order perturbation theory) that at t = T, the particle is in first excited state (i.e., it is with energy equation).

Exercise 16.5

A two-level system is represented by the Hamiltonian

equation

Now a time-dependent perturbation

equation

is switched on. At t = 0, the system is in its ground state equation Using first-order time-dependent perturbation theory (and assuming equation is not close to ±ħω), find the probability that the system has made a transition to excited state equation at time t.

SOLUTIONS

Solution 16.1

From Eq. (16.22b), we have the expression of amplitude Ck(t) of the kth state at time t and |Ck(t)|2 gives the probability that the system makes a transition to state equation at time t

equation
equation

We have

equation

so

equation

Solution 16.2

The given electric field produces the perturbation

 

Ĥ″ = |e| E(t) z = | e | E0r cosθ e–t/τ
= Ĥ′e–t/τ

 

where Ĥ′ is same as that in Eq. (15.58). From Eq. (16.22b), we can write the amplitude of the 2 p state at time t as

equation

where

equation

Now from Eq. (15.69), the above matrix element is non-zero only if m = 0. So

equation

which can be easily evaluated; let us say it has value A.

so

equation

Therefore, the probability that the atom is excited to 2 p state at time t, is

equation

Solution 16.3

Just like in the above exercise, we have

equation

The integral may be solved by going to complex t-plane, where the poles occur at t = ± iτ. And we get

equation

So, we get

equation

And the probability of excitation to the 2 p state is

equation

Solution 16.4

From Eq. (16.22b)

equation

Here

equation

So

equation

and the transition probability at time T is

equation

Solution 16.5

Amplitude of second state at time t is given as

equation

From this we can easily obtain

 

P2(t) = |C2(t)|2
REFERENCES
  1. Schiff, L.I. 1968. Quantum Mechanics. 3rd edn., New York, NY: McGraw-Hill.
  2. Merzbacher, E. 1999. Quantum Mechanics. 3rd edn., New York, NY: John Wiley & Sons.
  3. Levi, A.F.J. 2003. Applied Quantum Mechanics. Cambridge, MA: Cambridge University Press.
  4. Griffiths, D.J. 1995. Introduction to Quantum Mechanics. NJ: Prentice Hall.
  5. Cohen-Tannoudji, C., Diu, D. and Laloe, F. 1997. Quantum Mechanics, Vol. I and II. New York, NY: John Wiley & Sons.
  6. Baym, G. 1969. Lectures on Quantum Mechanics. New York, NY: W.A. Benjamin.
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