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Supplementary subject: Quantum Chemistry
Perturbation theory
6 lectures, (Tuesday and Friday, weeks 4-6 of Hilary term)
Chris-Kriton Skylaris (chris-kriton.skylaris @ chem.ox.ac.uk ) Physical & Theoretical Chemistry Laboratory South Parks Road, Oxford
February 24, 2006
Bibliography
All the material required is covered in “Molecular Quantum Mechanics” fourth edition by Peter Atkins and Ronald Friedman (OUP 2005). Specifically, Chapter 6, first half of Chapter 12 and Section 9.11.
Further reading: “Quantum Chemistry” fourth edition by Ira N. Levine (Prentice Hall 1991). “Quantum Mechanics” by F. Mandl (Wiley 1992). “Quantum Physics” third edition by Stephen Gasiorowicz (Wiley 2003). “Modern Quantum Mechanics” revised edition by J. J. Sakurai (Addison Wesley Long- man 1994). “Modern Quantum Chemistry” by A. Szabo and N. S. Ostlund (Dover 1996).
- 1 Introduction Contents
- 2 Time-independent perturbation theory
- 2.1 Non-degenerate systems
- 2.1.1 The first order correction to the energy
- 2.1.2 The first order correction to the wavefunction
- 2.1.3 The second order correction to the energy
- 2.1.4 The closure approximation
- 2.2 Perturbation theory for degenerate states
- 3 Time-dependent perturbation theory - Hamiltonian 3.1 Revision: The time-dependent Schr¨odinger equation with a time-independent
- 3.2 Time-independent Hamiltonian with a time-dependent perturbation
- 3.3 Two level time-dependent system - Rabi oscillations
- 3.4 Perturbation varying “slowly” with time
- 3.5 Perturbation oscillating with time
- 3.5.1 Transition to a single level
- 3.5.2 Transition to a continuum of levels
- 3.6 Emission and absorption of radiation by atoms
- 4 Applications of perturbation theory
- 4.1 Perturbation caused by uniform electric field
- 4.2 Dipole moment in uniform electric field
- 4.3 Calculation of the static polarizability
- 4.4 Polarizability and electronic molecular spectroscopy
- 4.5 Dispersion forces
- 4.6 Revision: Antisymmetry, Slater determinants and the Hartree-Fock method
- 4.7 Møller-Plesset many-body perturbation theory
The terms ψ(1) n and E n(1) are called the first order corrections to the wavefunction and energy respectively, the ψ(2) n and E(2) n are the second order corrections and so on. The task of perturbation theory is to approximate the energies and wavefunctions of the perturbed system by calculating corrections up to a given order.
Note 2.1 In perturbation theory we are assuming that all perturbed quantities are func- tions of the parameter λ, i.e. Hˆ(λ), En(λ) and ψn(r; λ) and that when λ → 0 we have
H^ ˆ(0) = Hˆ(0), En(0) = E n(0) and ψn(r; 0) = ψ n(0) (r). You will remember from your maths course that the Taylor series expansion of say En(λ) around λ = 0 is
En = En(0) +
dEn dλ
λ=
λ +
d^2 En dλ^2
λ=
λ^2 +
d^3 En dλ^3
λ=
λ^3 + · · · (6)
By comparing this expression with (5) we see that the perturbation theory “corrections” to
the energy level En are related to the terms of Taylor series expansion by: E n(0) = En(0),
E n(1) = dE dλn |λ=0, E n(2) = (^) 2!^1 d
(^2) En dλ^2 |λ=0,^ E
(3) n =^ 3!^1 d
(^3) En dλ^3 |λ=0, etc. Similar relations hold for the expressions (3) and (4) for the Hamiltonian and wavefunction respectively.
Note 2.2 In many textbooks the expansion of the Hamiltonian is terminated after the first order term, i.e. Hˆ = Hˆ(0)^ + λ Hˆ(1)^ as this is sufficient for many physical problems.
Note 2.3 What is the significance of the parameter λ? In some cases λ is a physical quantity: For example, if we have a single electron placed in a uniform electric field along the z-axis the total perturbed Hamiltonian is just H^ ˆ = Hˆ(0)^ + Ez(ezˆ) where Hˆ(0)^ is the Hamiltonian in the absence of the field. The effect of the field is described by the term eˆz ≡ Hˆ(1)^ and the strength of the field Ez plays the role of the parameter λ. In other cases λ is just a fictitious parameter which we introduce in order to solve a problem using the formalism of perturbation theory: For example, to describe the two electrons of a helium atom we may construct the zeroth order Hamiltonian as that of two non-interacting electrons 1 and 2 , Hˆ(0)^ = − 1 / 2 ∇^21 − 1 / 2 ∇^22 − 2 /r 1 − 2 /r 2 which is trivial to solve as it is the sum of two single-particle Hamiltonians, one for each electron. The entire Hamiltonian for this system however is Hˆ = Hˆ(0)^ + 1/|r 1 − r 2 | which is no longer separable, so we may use perturbation theory to find an approximate solution for H^ ˆ(λ) = Hˆ(0)^ + λ/|r 1 − r 2 | = Hˆ(0)^ + λ Hˆ(1)^ using the fictitious parameter λ as a “dial” which is varied continuously from 0 to its final value 1 and takes us from the model problem to the real problem.
To calculate the perturbation corrections we substitute the series expansions of equa- tions (3), (4) and (5) into the TISE (1) for the perturbed system, and rearrange and
group terms according to powers of λ in order to get
{ Hˆ(0)ψ(0) n − E n(0) ψ(0) n }
- λ{ Hˆ(0)ψ(1) n + Hˆ(1)ψ(0) n − E(0) n ψ n(1) − E n(1) ψ(0) n } (7)
- λ^2 { Hˆ(0)ψ(2) n + Hˆ(1)ψ n(1) + Hˆ(2)ψ(0) n − E n(0) ψ(2) n − E(1) n ψ n(1) − E n(2) ψ n(0) }
- · · · = 0
Notice how in each bracket terms of the same order are grouped (for example Hˆ(1)ψ(1) n
is a second order term because the sum of the orders of Hˆ(1)^ and ψ n(1) is 2). The powers of λ are linearly independent functions, so the only way that the above equation can be satisfied for all (arbitrary) values of λ is if the coefficient of each power of λ is zero. By setting each such term to zero we obtain the following sets of equations
Hˆ(0)ψ n(0) = E(0) n ψ n(0) (8) ( Hˆ(0)^ − E n(0) )ψ n(1) = (E n(1) − Hˆ(1))ψ(0) n (9) ( Hˆ(0)^ − E n(0) )ψ n(2) = (E n(2) − Hˆ(2))ψ(0) n + (E n(1) − Hˆ(1))ψ n(1) (10) · · ·
To simplify the expressions from now on we will use bra-ket notation, representing wavefunction corrections by their state number, so ψ(0) n ≡ |n(0)〉, ψ(1) n ≡ |n(1)〉, etc.
2.1.1 The first order correction to the energy
To derive an expression for calculating the first order correction to the energy E(1), take equation (9) in ket notation
( Hˆ(0)^ − E n(0) )|n(1)〉 = (E n(1) − Hˆ(1))|n(0)〉 (11)
and multiply from the left by 〈n(0)| to obtain
〈n(0)|( Hˆ(0)^ − E(0) n )|n(1)〉 = 〈n(0)|(E n(1) − Hˆ(1))|n(0)〉 (12) 〈n(0)| Hˆ(0)|n(1)〉 − E n(0) 〈n(0)|n(1)〉 = E n(1) 〈n(0)|n(0)〉 − 〈n(0)| Hˆ(1)|n(0)〉 (13) E n(0) 〈n(0)|n(1)〉 − E n(0) 〈n(0)|n(1)〉 = E n(1) − 〈n(0)| Hˆ(1)|n(0)〉 (14) 0 = E n(1) − 〈n(0)| Hˆ(1)|n(0)〉 (15)
where in order to go from (13) to (14) we have used the fact that the eigenfunctions of the unperturbed Hamiltonian Hˆ(0)^ are normalised and the Hermiticity property of Hˆ(0) which allows it to operate to its eigenket on its left
〈n(0)| Hˆ(0)|n(1)〉 = 〈( Hˆ(0)n(0))|n(1)〉 = 〈(E n(0) n(0))|n(1)〉 = E n(0) 〈n(0)|n(1)〉 (16)
because we have assumed non-degeneracy of the zeroth-order problem (i.e. E n(0) −E k(0) = 0 ). To proceed in our derivation for an expression for |n(1)〉 we will employ the iden- tity operator expressed in the eigenfunctions of the unperturbed system (zeroth order eigenfunctions):
|n(1)〉 = ˆ 1 |n(1)〉 =
k
|k(0)〉〈k(0)|n(1)〉 (26)
Before substituting (25) into the above equation we must resolve a conflict: k must be different from n in (25) but not necessarily so in (26). This restriction implies that the first order correction to |n〉 will contain no contribution from |n(0)〉. To impose this restriction we require that that 〈n(0)|n〉 = 1 (this leads to 〈n(0)|n(j)〉 = 0 for j ≥ 1. Prove it! ) instead of 〈n|n〉 = 1. This choice of normalisation for |n〉 is called intermediate normalisation and of course it does not affect any physical property calculated with |n〉 since observables are independent of the normalisation of wavefunctions. So now we can substitute (25) into (26) and get
|n(1)〉 =
k =n
|k(0)〉
〈k(0)| Hˆ(1)|n(0)〉 E(0) n − E k(0)
k =n
|k(0)〉
H kn(1) E n(0) − E k(0)
where the matrix element H kn(1) is defined by the above equation.
2.1.3 The second order correction to the energy
To derive an expression for the second order correction to the energy multiply (10) from the left with 〈n(0)| to obtain
〈n(0)| Hˆ(0)^ − E n(0) |n(2)〉 = 〈n(0)|E(2) n − Hˆ(2)|n(0)〉 + 〈n(0)|E(1) n − Hˆ(1)|n(1)〉 0 = E n(2) − 〈n(0)| Hˆ(2)|n(0)〉 − 〈n(0)| Hˆ(1)|n(1)〉 (28)
where we have used the fact that 〈n(0)|n(1)〉 = 0 (section 2.1.2). We now solve (28) for
E n(2) E(2) n = 〈n(0)| Hˆ(2)|n(0)〉 + 〈n(0)| Hˆ(1)|n(1)〉 = H nn(2) + 〈n(0)| Hˆ(1)|n(1)〉 (29)
which upon substitution of |n(1)〉 by the expression (27) becomes
E n(2) = H(2) nn +
k =n
H nk(1) H kn(1) E (0) n −^ E
(0) k
Example 2 Let us apply what we have learned so far to the “toy” model of a system which has only two (non-degenerate) levels (states) | 1 (0)〉 and | 2 (0)〉. Let E 1 (0) < E(0) 2 and assume that there is only a first order term in the perturbed Hamiltonian and that the
diagonal matrix elements of the perturbation are zero, i.e. 〈m(0)| Hˆ(1)|m(0)〉 = H mm(1) = 0. For this simple system we can solve exactly for its perturbed energies up to infinite order (see Atkins):
E 1 =
(E
(0) 1 +^ E
(0) 2 )^ −^
[(E
(0) 1 −^ E
(0) 2 )
2 + 4|H(1)
2 ]^12 (31)
E 2 =
(E 1 (0) + E 2 (0) ) +
[(E 1 (0) − E 2 (0) )^2 + 4|H 12 (1) |^2 ]
(^12) (32)
According to equation 30 the total perturbed energies up to second order are
E 1 E(0) 1 −
|H 12 (1) |^2
E 2 (0) − E 1 (0)
E 2 E 2 (0) +
|H 12 (1) |^2
E 2 (0) − E 1 (0)
These sets of equations show that the effect of the perturbation is to lower the energy of the lower level and raise the energy of the upper level. The effect increases with the strength of the perturbation (size of |H 12 (1) |^2 term) and decreasing separation between the
unperturbed energies ( E(0) 2 − E 1 (0) term).
Energy
O
E 1
E 2
E 3
E 4
E 5
E 6
Figure 1: Effect of a perturbation on energy levels. In this example the perturbation removes all the degeneracies of the unperturbed levels.
where now we use two indices to represent each state: the first index n runs over the different energy eigenvalues while the second index i runs over the d degenerate states for a particular energy eigenvalue. Since we have d degenerate states of energy E n(0) , any linear combination of these states is also a valid state of energy E n(0). However, as the perturbation parameter λ is varied continuously from 0 to some finite value, it is likely that the degeneracy of the states will be lifted (either completely or partially). The ques-
tion that arises then is whether the states ψ n,i(0) of equation (39) are the “correct” ones, i.e. whether they can be continuously transformed to the (in general) non-degenerate perturbed states. It turns out that this is usually not the case and one has to first find the “correct” zeroth order states
φ(0) n,j =
∑^ d
i=
|(n, i)(0)〉cij j = 1,... , d (40)
where the coefficients cij that mix the ψ(0) n,i are specific to the perturbation Hˆ(1)^ and are determined by its symmetry. Here we will find a way to determine the “correct” zeroth order states φ(0) n,j and the
first order correction to the energy. To do this we start from equation 9 with φ(0) n,i in
place of ψ n,i(0)
( Hˆ(0)^ − E n(0) )ψ n,i(1) = (E n,i(1) − Hˆ(1))φ(0) n,i (41)
Notice that we include in the notation for the first order energy E (1) n,i the index^ i^ since the
perturbation may split the degenerate energy level E n(0). Figure 1 shows an example for a hypothetical system with six states and a three-fold degenerate unperturbed level. Note that the perturbation splits the degenerate energy level. In some cases the perturbation may have no effect on the degeneracy or may only partly remove the degeneracy. The next step involves multiplication from the left by 〈(n, j)(0)|
〈(n, j)(0)| Hˆ(0)^ − E(0) n |(n, i)(1)〉 = 〈(n, j)(0)|E n,i(1) − Hˆ(1)|φ(0) n,i 〉 (42) 0 = 〈(n, j)(0)|E (1) n,i −^ Hˆ
(1)|φ(0) n,i 〉^ (43) 0 =
k
〈(n, j)(0)|E n,i(1) − Hˆ(1)|(n, k)(0)〉cki (44)
where we have made use of the Hermiticity of Hˆ(0)^ to set the left side to zero and we have substituted the expansion (40) for φ(0) n,i. Some further manipulation of (44) gives:
∑
k
[〈(n, j)(0)| Hˆ(1)|(n, k)(0)〉 − E n,i(1) 〈(n, j)(0)|(n, k)(0)〉]cki = 0 (45) ∑
k
(H jk(1) − E n,i(1) Sjk)cki = 0 (46)
We thus arrive to equation 46 which describes a system of d simultaneous linear equations for the d unknowns cki, (k = 1,... , d) for the “correct” zeroth order state φ(0) n,i. Actually, this is a homogeneous system of linear equations as all constant coefficients (i.e. the righthand side here) are zero. The trivial solution is obviously cki = 0 but we reject it because it has no physical meaning. As you know from your maths course, in order to obtain a non-trivial solution for such a system we must demand that the determinant of the matrix of the coefficients is zero:
|H jk(1) − E n,i(1) Sjk| = 0 (47)
We now observe that as En,i occurs in every row, this determinant is actually a dth degree polynomial in En,i and the solution of the above equation for its d roots will give
us all the E n,i(1) (i = 1,... d) first order corrections to the energies of the d degenerate
levels with energy E n(0). We can then substitute each E n,i(1) value into (46) to find the corresponding non-trivial solution of cki (k = 1,... d) coefficients, or in other words
the function φ(0) n,i. Finally, you should be able to verify that E n,i(1) = 〈φ(0) n,i| Hˆ(1)|φ(0) n,i 〉, i.e. that the expression (17) we have derived which gives the first order energy as the expectation value of the first order Hamiltonian in the zeroth order wavefunctions still holds, provided the “correct” degenerate zeroth order wavefunctions are used.
Example 3 A typical example of degenerate perturbation theory is provided by the study of the n = 2 states of a hydrogen atom inside an electric field. In a hydrogen atom all
4 degenerate n=2 states
l=1, m=-1, 0, 1 l=0, m=0 E
(1)=
E(1)=3e E z a 0
E(1)=-3e E z a (^0)
E z
Figure 2: Pattern of Stark spliting of hydrogen atom in n = 2 state. The fourfold degeneracy is partly lifted by the perturbation. The m = ±1 states remain degenerate and are not shifted in the Stark effect.
3 Time-dependent perturbation theory
3.1 Revision: The time-dependent Schr¨odinger equation with
a time-independent Hamiltonian
We want to find the (time-dependent) solutions of the time-dependent Schr¨odinger equa- tion (TDSE)
i¯h
∂t
= Hˆ(0)Ψ(0)^ (53)
where we assume that Hˆ(0)^ does not depend on time. Even though in this section we are not involved with perturbation theory, we will still follow the notation Hˆ(0), Ψ(0)^ of representing the exactly soluble problem as the zeroth order problem as this will prove useful in the derivation of time-dependent perturbation theory that follows in the next section. According to the mathematics you have learned, the solution to the above equation can be written as a product
Ψ(0)(r, t) = ψ(0)(r)T (0)(t) (54)
where the ψ(0)(r), which depends only on position coordinates r, is the solution of the energy eigenvalue equation (TISE)
Hˆ(0)ψ(0) n (r) = E n(0) ψ(0) n (r) (55)
and the expression for T (t) is derived by substituting the right hand side of the above to the time-dependent equation 53. Finally we obtain
Ψ(0)(r, t) = ψ(0) n (r)e−iE n(0) t/¯h^
. (56)
Now let us consider the following linear combination of ψ n(0)
Ψ(0)(r, t) =
k
akψ k(0) (r)e−iE
(0) k t/¯h^ (57)
where the ak are constants. This is also a solution of the TDSE (prove it!) because the TDSE consists of linear operators. This more general “superposition of states” solution of course contains (56) (by setting ak = δnk) but unlike (56) it is not, in general, an
eigenfunction of Hˆ(0). Assuming that the ψ(0) n have been chosen to be orthonormal, which is always possible, we find that the expectation value of the Hamiltonian is
〈Ψ(0)| Hˆ(0)|Ψ(0)〉 =
k
|ak|^2 E(0) k (58)
We see thus that in the case of equation 56 the system is in a state with definite energy E n(0) while in the general case (57) the system can be in any of the states with an average energy given by (58) where the probability Pk = |ak|^2 of being in the state k is equal to the square modulus of the coefficient ak. Both (56) and (57 ) are time-dependent because
of the “phase factors” e−iE
(0) k t/¯h^ but the probabilities Pk and also the expectation values
for operators that do not contain time (such as the Hˆ(0)^ above) are time-independent.
3.2 Time-independent Hamiltonian with a time-dependent per-
turbation
We will now develop a perturbation theory for the case where the zeroth order Hamil- tonian is time-independent but the perturbation terms are time-dependent. Thus our perturbed Hamiltonian has the following form
Hˆ(t) = Hˆ(0)^ + λ Hˆ(1)(t) + λ^2 Hˆ(2)(t) +... (59)
To simplify our discussion, in what follows we will only consider up to first order per- turbations in the Hamiltonian
Hˆ(t) = Hˆ(0)^ + λ Hˆ(1)(t). (60)
The purpose now of the perturbation theory we will develop is to determine the time-dependent coefficients ak(t). We begin by writing a perturbation expansion for the coefficient ak(t) in terms of the parameter λ
ak(t) = a(0) k (t) + λa(1) k (t) + λ^2 a(2) k (t) +... (68)
where you should keep in mind that while λ and t are not related in any way, we take t = 0 as the “beginning of time” for which we know exactly the composition of the system so that ak(0) = a(0) k (0) (69)
which means that a( kl )(0) = 0 for l > 0. Furthermore we will assume that
a(0) g (0) = δgj (70)
which means that at t = 0 the system is exclusively in a particular state |j(0)〉 and all other states |g(0)〉 with g = j are unoccupied. Now substitute expansion (68) into (67) and collect equal powers of λ to obtain the following expressions
a(0) k (t) − a(0) k (0) = 0 (71)
a(1) k (t) − a(1) k (0) =
i¯h
n
∫ (^) t
0
a(0) n (t′)H kn(1) (t′)eiωknt
′ dt′^ (72)
a(2) k (t) − a(2) k (0) =
i¯h
n
∫ (^) t
0
a(1) n (t′)H kn(1) (t′)eiωknt
′ dt′^ (73)
... (74)
We can observe that these equations are recursive: each of them provides an expression for a( fm )(t) in terms of a( fm −1)(t). Let us now obtain an explicit expression for a(1) f (t) by first substituting (71) into (72), and then making use of (70):
a(1) f (t) =
i¯h
n
∫ (^) t
0
a(0) n (0)H f n(1) (t′)eiωf nt
′ dt′^ =
i¯h
∫ (^) t
0
H f j(1) (t′)eiωf j^ t
′ dt′^. (75)
The probability that the system is in state |f (0)〉 is obtained in a similar manner to equation 58 and is given by the squared modulus of the af (t) coefficient
Pf (t) = |af (t)|^2 (76)
but of course a significant difference from (58) is that Pf = Pf (t) now changes with time. Using the perturbation expansion (68) for af (t) we have
Pf (t) = |a(0) f (t) + λa(1) f (t) + λ^2 a(2) f (t) +... |^2. (77)
Note that in most of the examples that we will study in these lectures we will confine ourselves to the first order approximation which means that we will also approximate the above expression for Pf (t) by neglecting from it the second and higher order terms.
Note 3.1 The previous derivation of time-dependent perturbation theory is rather rig- orous and is also very much in line with the approach we used to derive time-independent perturbation theory. However, if we are only interested in obtaining only up to first order corrections, we can follow a less strict but more physically motivated approach (see also Atkins). We begin with (67) and set λ equal to 1 to obtain
ak(t) − ak(0) =
i¯h
n
∫ (^) t
0
an(t′)H kn(1) (t′)eiωknt
′ dt′^ (78)
This equation is exact but it is not useful in practice because the unknown coefficient ak(t) is given in terms of all other unknown coefficients an(t) including itself! To proceed we make the following approximations:
- Assume that at t = 0 the system is entirely in an initial state j, so aj (0) = 1 and an(0) = 0 if n = j.
- Assume that the time t for which the perturbation is applied is so small that the change in the values of the coefficients is negligible, or in other words that aj (t) 1 and an(t) 0 if n = j.
Using these assumptions we can reduce the sum on the righthand side of equation 78 to a single term (the one with n = j for which aj (t) 1 ). We will also rename the lefthand side index from k to f to denote some “final” state with f = j to obtain
af (t) =
i¯h
∫ (^) t
0
H f j(1) (t′)eiωf j^ t ′ dt′^ (79)
This approximate expression for the coefficients af (t) is correct to first order as we can see by comparing it with equation 75.
Example 4 Show that with a time-dependent Hamiltonian Hˆ(t) the energy is not con- served.
We obviously need to use the time-dependent Schr¨odinger equation
i¯h
∂t
= Hˆ(t)Ψ ⇔
∂t
i ¯h
Hˆ(t)Ψ (80)
where the system is described by a time-dependent state Ψ. We now look for an ex- pression for the derivative of the energy 〈H〉 = 〈Ψ| Hˆ(t)|Ψ〉 (expectation value of the Hamiltonian) with respect to time. We have
∂〈H〉 ∂t
∂t
| Hˆ(t)|Ψ〉 + 〈Ψ|
∂ Hˆ(t) ∂t
|Ψ〉 + 〈Ψ| Hˆ(t)|
∂t
Note 3.2 In this section we are not really applying perturbation theory: The two level system allows us to obtain the exact solutions for the coefficients a 1 (t) and a 2 (t) (up to infinite order in the language of perturbation theory).
The probability of the system being in the state ψ 2 (0) is
P 2 (t) = |a 2 (t)|^2 =
4 |V |^2
ω^221 + 4|V |^2
sin^2
(ω^221 + 4|V |^2 )^1 /^2 t (87)
and of course since we only have two states here we will also have P 1 (t) = 1 − P 2 (t). Let us examine these probabilities in some detail. First consider the case where the two states are degenerate (ω 21 = 0). We then have
P 1 (t) = cos^2 |V |t , P 2 (t) = sin^2 |V |t (88)
which means that the system oscillates freely between the two states | 1 (0)〉 and | 2 (0)〉 and the only role of the perturbation is to determine the frequency |V | of the oscillation. The other extreme is the case where the levels are widely separated in comparison with the strength of the perturbation in the sense that ω^221 >> |V |^2. In this case we obtain
P 2 (t)
2 |V |
ω 21
sin^2
ω 21 t (89)
which shows that the probability of the system occupying state | 2 (0)〉 can not get any larger than (2|V |/ω 21 )^2 which is a number much smaller than 1. Thus the system remains almost exclusively in state | 1 (0)〉. We should also observe here that the frequency of oscillation is independent of the strength of the perturbation and is determined only by the separation of the states ω 21.
3.4 Perturbation varying “slowly” with time
Here we will study the example of a very slow time-dependent perturbation in order to see how time-dependent theory reduces to the time-independent theory in the limit of very slow change. We define the perturbation as follows
Hˆ(1)(t) =
0 , t < 0 Hˆ(1)(1 − e−kt), t ≥ 0. (90)
where Hˆ(1)^ is a time-independent operator, which however may not be a constant as for example it may depend on ˆx, and so on. The entire perturbation Hˆ(1)(t) is time- dependent as Hˆ(1)^ is multiplied by the term (1 − e−kt) which varies from 0 to 1 as t increases from 0 to infinity. Substituting the perturbation into equation (75) we obtain
a(1) f (t) =
i¯h
H f j(1)
∫ (^) t
0
(1 − e−kt
′ )eiωf j^ t
′ dt′^ =
i¯h
H f j(1)
[
eiωf j^ t^ − 1 iωf j
e−(k−iωf j^ )t^ − 1 k − iωf j
]
If we assume that we will only examine times very long after the perturbation has reached its final value, or in other words kt >> 1, we obtain
a(1) f (t) =
i¯h
H f j(1)
[
eiωf j^ t^ − 1 iωf j
k − iωf j
]
and finally that the rate in which the perturbation is switched is slow in the sense that k^2 << ω^2 f j , we are left with
a(1) f (t) = −
H f j(1) ¯hωf j
eiωf j^ t^ (93)
The square of this, which is the probability of being in state |f (0)〉 to first order is
Pf (t) = |a(1) f (t)|^2 =
|H f j(1) |^2 ¯h^2 ω^2 f j
We observe that the resulting expression for Pf (t) is no longer time-dependent. In fact, it is equal to the square modulus |〈f (0)|j(1)〉|^2 of the expansion coefficient in |f (0)〉 of the first order state |j(1)〉 as given in equation 25 of time-independent perturbation theory. Thus in the framework of time-independent theory (94) is interpreted as being the fraction of the state |f (0)〉 in the expansion of |j(1)〉 while in the time-dependent theory it represents the probability of the system being in state |f (0)〉 at a given time.
