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

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

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 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

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 a_n (t).

16.2  TIME DEVELOPMENT OF STATES AND TRANSITION PROBABILITY

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

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 and eigenstates are known). Now for the time-dependent Hamiltonian Ĥ, we shall have to solve time-dependent Schrodinger 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 As indicated in the Introduction above, this shall be done by expanding eigenstate ψ(r, t) in terms of unperturbed states with time-dependent coefficients: an approach known as Dirac’s method of variation of constants. So, ψ(r, t) is expanded as

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

or

Also we have

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

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

or

or

Here Ċ_n(t) denotes dC_n(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

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

one gets,

where denotes the matrix element

and

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

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 [dC_k(t)/dt] depends upon all states connected with via perturbation matrix elements [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 C_k(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 C_k(t) in powers of the parameter λ as

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

In principle, for given perturbation matrix elements 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 So, in the absence of perturbation, the coefficient C_a(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 (16.17) simply confirms that the coefficients are time-independent. Thus we may write

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

For k = a, Eq. (16.21) gives

while for k ≠ a (16.21) gives

In writing Eqs (16.22), we have used the above mentioned initial conditions, that is, and 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 (i.e., the probability that at time t, the system has made transition from its initial state to the state is

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

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 Then from Eq. (16.24) we have

So that |C_a(t)|² ≈ 1. Therefore, so far as initial state 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 state is

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

 

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

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

But the second integral above is zero, so

So (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 to state as

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

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

   around the initial energy 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 of the system by inducing transition 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 (t, ω_ka) = 0 at t = 0 for all values of ω_ka. Plot of (t, ω_ka) as function of t for a fixed value of ω_ka = ω₀
   

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

   

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

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

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).

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

Therefore, total (first-order) transition probability P⁽¹⁾(t), that the system at time t has made a transition away from its initial state , is obtained by summing the transition probabilities (16.30) over all find states 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 and defined as the number of energy levels in the unit energy interval around So denotes the number of energy levels lying in the interval and

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

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 . The probability that in time t, the system initially in state will make a transition to any of the final states is given as

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

 

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

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

Now using the standard result

We get

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

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 and state are degenerate, that is, when In this situation, we have from Eq. (16.28),

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

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 ,

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

 

 

In fact, the sum of two probabilities (the probability that the system has made a transition away from its initial state and the probability that the system is still in its initial state 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

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

 

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

or

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

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 and eigenstates :

 

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

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 as

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

Figure 16.3 Adiabatic perturbation

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

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

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

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

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 ψ_η of the total Hamiltonian Ĥ₀ + Ĥ′; in absence of the perturbation Ĥ′, the eigenstate was . 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 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 (at t = 0) to (say) (at time t).

EXERCISES

Exercise 16.1

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

 

 

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

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

 

 

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

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

 

Here V₀ is less than 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 ).

Exercise 16.5

A two-level system is represented by the Hamiltonian

Now a time-dependent perturbation

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

SOLUTIONS

Solution 16.1

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

We have

so

Solution 16.2

The given electric field produces the perturbation

 

 

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

where

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

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

so

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

Solution 16.3

Just like in the above exercise, we have

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

So, we get

And the probability of excitation to the 2 p state is

Solution 16.4

From Eq. (16.22b)

Here

So

and the transition probability at time T is

Solution 16.5

Amplitude of second state at time t is given as

From this we can easily obtain

 

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.