Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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5.3 Non-stationary perturbation theory 143

The eigenfunctions 

(0)

(x, t) and eigenvalues E

(0)

of the unperturbed Hamilto-

nian are assumed to be known, i.e., the solutions of the following equation are

known:

(0)



(0)

(x , t) = ih

∂

∂t



(0)

(x , t). (5.42)

For the nth quantum state they have the following form:



(0)

(x , t) = e

−iE

(0)

t/h

(0)

(x ). (5.43)

As follows from Eq. (5.41), the states of the perturbed system depend on time

and the energy of the system is not a constant. Since the system does not have

stationary states, the problem is not in ﬁnding these states but in the calculation of

the system’s wavefunction, which depends on time, and in ﬁnding the probability

of the system’s transition under the inﬂuence of the perturbation from the initial

state to the ﬁnal state. Therefore, the method of approximate solution of the

Schr

odinger equation, which is used for the case of a stationary perturbation, is

not applicable here and must be modiﬁed.

Let us express the wavefunction (x, t) in the form of an expansion over the

complete set of unperturbed orthonormalized wavefunctions, 

(0)

(x, t):

(x, t) =



(t)

(0)

(x , t), (5.44)

where the expansion coefﬁcients b

(t) are functions only of time, and the magni-

tude |b

(t)|

deﬁnes the probability of the system being in the state with energy

(0)

at the instant of time t. By substituting the expansion (5.44) into Eq. (5.41)

we obtain the following system of differential equations:







(0)

(x , t) + b

∂

(0)

(x , t)

∂t







(0)

V (t)





(0)

(x , t). (5.45)

Let us multiply both sides of Eq. (5.45)by

(0)∗

(x, t) and integrate over the entire

space. Taking into account the form of the unperturbed wavefunctions (5.43) and

their orthonormality, we get



iω

, (5.46)

where we introduced the frequency ω

= (E

(0)

− E

(0)

)/h

and the matrix ele-

ment of the perturbation operator,

(t) =



(0)∗

(x )

V (t)ψ

(0)

(x )dx. (5.47)

The system of equations (5.46) is exact and it is equivalent to the initial equa-

tion (5.41). In order to solve Eq. (5.46) approximately, let us assume that at

t ≤ 0 the system was in the nth quantum state with the wavefunction 

(0)

Then, for the initial instant of time all the coefﬁcients with the index k = n

144 Approximate methods of ﬁnding quantum states

in the expansion (5.44) equal zero and the coefﬁcient b

(0) = 1. Beginning

from the instant t = 0, the system is affected by the small perturbation and as

a result the wavefunction of the initial state changes a little with time. There-

fore, let us look for the coefﬁcients b

(t) for the instant t > 0intheformof

expansion:

(t) = b

(0)

(t) +b

(1)

(t) +b

(2)

(t) +···, (5.48)

where b

(0)

(t) = b

(0)

(0) = δ

. The correction b

(1)

(t) has the same order of small-

ness as the perturbation; b

(2)

(t) is quadratic with respect to the perturbation,

and so on. On substituting the expansion (5.48) into Eq. (5.46), we obtain the

equation for the correction of the ﬁrst order of smallness:

(1)



iω

(0)

= V

iω

, (5.49)

where we omitted all terms of second and higher orders of smallness with respect

to the perturbation. By integrating the differential equation (5.49) we get

(1)

(t) =



(t)e

iω

dt. (5.50)

The probability of ﬁnding the system at instant t in the state with energy E

(0)

the ﬁrst order of perturbation theory is deﬁned as

=|b

(t)|

=|b

(1)

(t)|=





(t)e

iω



. (5.51)

Thus, knowing the dependence of the perturbation on time, we can ﬁnd the

probability of transition of the system from the initial state, 

(0)

, to the ﬁnal

state, 

(0)

. Equation (5.51) is very often written as

4π

(ω

)

, (5.52)

where we introduced the Fourier transform of the matrix element of the pertur-

bation operator (5.47), which corresponds to the frequency ω

(ω

) =

2π



∞

−∞

(t)e

iω

dt. (5.53)

One of the most important applied problems is the case of periodic perturbation

when

V (t) =

iωt

. (5.54)

In this case the probability of the corresponding transition during a sufﬁciently

long time interval (t  2π/ω) is given by the expression

2π

)

tδ(E

(0)

− E

(0)

− h

ω), (5.55)

5.3 Non-stationary perturbation theory 145

012

−−

Figure 5.1 Dirac’s

δ-function can be thought

of as a function

(x) = [1/ (α

√

π)]e

−x

/ α

when α tends to zero.

Then its amplitude tends

to ∞, while its width

tends to zero. The solid

line corresponds to

α = 0.1, the dashed line to

α = 0.5, and the

dash–dotted line to α = 1.

where Dirac’s δ-function, δ(E

(0)

− E

(0)

− h

ω), shows the resonant character of

the corresponding quantum transition, which is possible only at h

ω = E

(0)

−

(0)

. Therefore, the probability of the corresponding transition is not equal to

zero if the magnitude h

ω is close to the energy difference E

(0)

− E

(0)

. Let us ﬁrst

introduce Dirac’s δ-function and then continue our analysis.

Dirac’s delta-function can be deﬁned by the following two relationships:

δ(x) =



0, x = 0,

∞, x = 0,

(5.56)



∞

−∞

f (x)δ(x)dx = f (0), (5.57)

where f (x) is an arbitrary function, which is continuous at the point x = 0. For

convenience Dirac’s δ-function can be presented in the form of the limit of the

-function:

δ(x) = lim

α→0

(x ), (5.58)

where

(x ) =

√

−x

/α

. (5.59)

The function shown in Fig. 5.1 can be transformed into Dirac’s δ-function by

equating the parameter α to zero. Dirac’s δ-function belongs to the mathematical

class of so-called generalized functions, which can be considered as an extreme

case for several continuous functions that depend on some accessory parameter.

146 Approximate methods of ﬁnding quantum states

Very often the δ-function is introduced as

lim

K →∞





−K

ikx



= 2πδ(x) (5.60)

or in shorter notation as

2π



∞

−∞

ikx

dk = δ(x). (5.61)

Now let us continue our analysis of quantum transitions. For the rate of the

transition probability the following expression is used:

2π

)



(0)

− E

(0)

− h



. (5.62)

If the perturbation of an atom is caused by electromagnetic radiation, whose

wavelength is much greater than the dimensions of the atom, then the perturbation

operator,

V , can be expressed as

V =−d · E(t ). (5.63)

Here, E is the electric ﬁeld and d =−er is the dipole moment of the electron

in an atom, which is under the inﬂuence of the alternating electric ﬁeld of the

incident electromagnetic wave. Under the inﬂuence of this wave the atom can

make a transition from the nth to the mth state with the following probability:

4π

E(ω

)

, (5.64)

where

E(ω

) =

2π



∞

−∞

E(t)e

iω

dt. (5.65)

The system can make such a transition if E(ω

) = 0, i.e., if there is a spectral

line in the emission spectrum that corresponds to this transition, and if the

corresponding matrix elements of the operator of dipole moment,

(t) =−



(0)∗

(r)er(t)ψ

(0)

(r)dV, (5.66)

are not equal to zero. These conditions deﬁne the so-called selection rules for

the quantum systems to have allowed states.

Let us consider now the perturbation, which after switching on stays small

and continues to act for an inﬁnitely long time. In this case the expression for

the probability of transition (5.51) is not valid since for the upper limit (t →∞)

the integral diverges. We can show that in this case the probability of transition

is deﬁned by the expression





iω



. (5.67)

5.3 Non-stationary perturbation theory 147

If the perturbation

V (t) is small and weakly changes during the time interval

= 2π/ω

, then the magnitude of the integral in Eq. (5.51) and, correspond-

ingly, the probability of transition are small. In the limit of inﬁnitesimally slow

changes of the perturbation, the probability of any transition between the energy

levels tends to zero. In another limiting case, that of very fast (sudden) switching

on of the perturbation, the derivatives dV

/dt tend to inﬁnity at the instant of

switching on of the perturbation. Taking this into account, we can take the slowly

changing factor e

iω

out of the integral in Eq. (5.67). After this the integral can

be easily evaluated and as a result we obtain the following expression:

. (5.68)

The transition probability of a system at a sudden switching on of the pertur-

bation can also be found in cases when the perturbation is not small. Let the

system be in one of the quantum states ψ

(0)

of the unperturbed Hamiltonian

If the change of the Hamiltonian happens during a time that is small in compari-

son with the time of transition, 2π/ω

, then the wavefunction of the system

does not have enough time to change and will stay the same as it was before

the switching on of the perturbation. Thus, it will not be an eigenfunction of

the new Hamiltonian,

H =

V , and, therefore, the initial state ψ

(0)

will not

be stationary. The probability of the transition of a system to a new stationary

state is deﬁned according to Eq. (5.44) by the corresponding expansion coef-

ﬁcient of the function ψ

, which is decomposed over the set of unperturbed

eigenfunctions ψ

(0)





∗

(0)



. (5.69)

Example 5.2. Under the inﬂuence of a suddenly switched-on homogeneous

electric ﬁeld E the charged oscillator makes a transition from the ground state

to the excited state. Find the probability of such a transition if the mass of the

oscillator is equal to m and its charge is equal to e.

Reasoning. The potential energy of the charged oscillator in a homogeneous

electric ﬁeld can be written as follows:

U (x ) =

mω

− e

x =

mω

(x − x

)

−

2mω

, (5.70)

where the coordinate x

is equal to x

= e

/(mω

). Equation (5.70) has the

form of the potential energy of a linear oscillator with shifted equilibrium position

. If the wavefunction of the unperturbed oscillator has the form ψ

(0)

(x), then

for the perturbed oscillator this function must have the form ψ

(x − x

). The

wavefunctions of the unperturbed harmonic oscillator (4.89)havetheform

(0)

(ξ) = A

(ξ)e

−ξ

148 Approximate methods of ﬁnding quantum states

where ξ = x

√

mω/h

. The probability of transition from the ground state (n = 0)

to the excited state (n = k) is deﬁned according to Eq. (5.51) by the corresponding

coefﬁcient b

of the expansion of the perturbed wavefunction in the series (5.44)

over the unperturbed wavefunctions, i.e.,

=|b





∞

−∞

(0)∗

(x )ψ

(0)

(x − x

)dx



. (5.71)

Using the explicit form of the wavefunctions, let us determine the expansion

coefﬁcient b

(−1)

√

πk!

−ξ



∞

−∞

−ξξ

dξ

−ξ

+2ξξ

dξ, (5.72)

where

= x



mω

e|E|

√

By integrating Eq. (5.72) by parts k times we reduce the integral in Eq. (5.72)to

the following form:



∞

−∞

−ξξ

dξ

−ξ

+2ξξ

dξ = ξ



∞

−∞

−ξ

+ξξ

dξ = ξ

√

πe

. (5.73)

On substituting this expression into Eq. (5.72) and then into Eq. (5.71) we obtain

the following expression for the transition probability:

−ξ

. (5.74)

5.4 The quasiclassical approximation

If the de Broglie wavelength of a particle is small in comparison with the charac-

teristic dimensions of the region within which the particle’s motion takes place,

then the behavior of a particle is approximately classical. The formal passage to

the limit from the quantum description of the particle’s behavior to the classical

one can be brought about by making the reduced Planck constant, h

, tend to zero

in the expression for the particle’s wavefunction, which can be presented as

(r, t) = Ae

iS(r,t)/h

. (5.75)

Here, S(r, t) is the action of the particle. As we noted in Section 2.3.3, the

reduced Planck constant h

is the quantum of action. For a free particle, the action

is deﬁned as

S(r, t) = p ·r − Et. (5.76)

The wavefunction of a free particle (2.145) can be written in the form (5.75). The

expression for the wavefunction stays the same in the case of a particle’s motion

5.4 The quasiclassical approximation 149

in a potential ﬁeld U(r). For the stationary state of the particle in the ﬁeld, U (r),

the action can be written as

S(r, t) =



p ·dr − Et = S

(r) − Et, (5.77)

where the integration is carried out along the trajectory of the particle, L.The

action, S

, depends only on the particle’s coordinates and thus S

is called the

reduced action.

In the quasiclassical case the reduced Planck constant, h

, can be considered

a small parameter and the physical magnitudes A and S can be considered

independent of h

. On substituting Eq. (5.75) into the Schr

odinger equation (2.182)

and leaving only the terms proportional to h

and h

we arrive at the following

two equations for the phase and amplitude of the wavefunction, (r, t):

∂ S

∂t

(∇ S)

+U = 0, (5.78)

∂ A

∂t

∇

S + ∇ S ∇ A = 0. (5.79)

The ﬁrst of these equations represents a well-known equation of classical mechan-

ics – the Hamilton–Jacobi equation for the particle’s action. The second equation,

after its multiplication by 2A, can be rewritten as

∂ A

∂t

+ div



∇ S



= 0. (5.80)

Here, A

=||

is the probability density of the particle’s location, and ∇ S/m =

p/m is the classical velocity of the particle. Thus, Eq. (5.80) is the continuity

equation, which shows that the probability density moves in each point of space

according to the laws of classical mechanics with the velocity p/m.

For stationary states, taking into account Eq. (5.77), the reduced action, S

and amplitude, A, satisfy the following equations:

(

∇ S

)

+U = E, (5.81)

div



∇ S



= 0. (5.82)

Let us write the wavefunction for stationary states in the case of a particle’s

one-dimensional motion in the ﬁeld U (x). Taking into account that

(

∇ S

)

(

/dx

)

,Eq.(5.81) reduces to

=±



2m[E −U (x)]. (5.83)

The solution of this equation can be written as

=±





2m[E −U (x)]dx =±



p(x)dx, (5.84)

150 Approximate methods of ﬁnding quantum states

where p(x) =

√

2m[E − U (x)] is the momentum of the particle. Then, Eq. (5.82)

and its solution can be written as

p) = 0, (5.85)

p = constant, (5.86)

where for the amplitude of the wavefunction we get

A = constant ×

√

. (5.87)

Thus, in the quasiclassical approximation the wavefunction can be considered as

the superposition of the wavefunctions of two states, which describe the motion

of the particle along the x-axis in two possible directions:

ψ(x) =

√

2m[E−U(x)]dx/h

√

−i

√

2m[E−U(x)]dx/h

. (5.88)

As we have already mentioned, the quasiclassical approximation is valid only

for those regions of the particle’s motion where its de Broglie wavelength, λ

slightly changes at distances x of the order of λ

itself. This condition is not

satisﬁed near the turning points where the direction of motion is reversed and

the velocity of the particle tends to zero. Thus, at the turning points p → 0 and

therefore the de Broglie wavelength cannot be considered small.

Example 5.3. Estimate the probability of electron transmission through the

potential barrier (3.182):

U (x ) =











0, x < −a,

(1 + x /a), −a ≤ x ≤ 0,

(1 − x /b), 0 < x ≤ b,

0, x > b,

with a linear dependence of the potential in the regions of its increase and

decrease. The electron is incident on the barrier from the negative part of the

x-axis and has the energy E < U

Reasoning. This probability can be estimated on the basis of the quasiclassical

approximation if we assume that the electron potential energy, U (x), and the

momentum, p(x), change slowly. The interval of electron motion under the

barrier is limited by the coordinates x

and x

, where

= a



− 1



, (5.89)

= b



1 −



(5.90)

5.5 Summary 151

(see Example 3.9 and Fig. 3.11). In the classically unavailable region the electron

wavefunction has the form of a damping exponent:



|p(x)|ψ(x) = e

−



|p(x)|dx/h

, (5.91)

where |p(x)|=

√

[U(x) − E]dx. Thus, the amplitude of the electron

wavefunction at the second boundary (x

= b) decreases by a factor of

exp[



|p(x)|dx/h

]. The probability of electron transmission through the barrier

is equal to the square of its magnitude, i.e.,

D = e

−2



√

[U(x)−E]dx /h

. (5.92)

This formula can be used until the exponent’s argument has become large in

comparison with unity (compare it with Eq. (3.183)). On integrating Eq. (5.92)

we arrive at the following expression:

D = e

−4

√

(

a+b

)(

−E

)

3/2

/(3h

)

, (5.93)

which coincides with the exact solution (see Eq. (3.189)) that was obtained earlier

to within a factor of order one.

5.5 Summary

1. The approximate methods are especially important for the solution of many problems

concerning electron quantum states. Perturbation theory uses weak perturbation, which

negligibly changes the state of the unperturbed system. If the Hamiltonian does not

depend directly on time, stationary perturbation theory is used; for the opposite case

non-stationary perturbation theory is used.

2. In the ﬁrst order of approximation of stationary perturbation theory the correction to

the unperturbed energy, E

(0)

, is deﬁned by the diagonal matrix element of perturbation,

which is an average perturbation, i.e., E

(1)

= W

=W

.

3. If all unperturbed quantum states are non-degenerate, then for the wavefunction and

energy the following expressions are used:

= ψ

(0)



i=n

(0)

− E

(0)

= E

(0)







i=n

(0)

− E

(0)

These relationships are valid if the matrix elements of the perturbation operator are

signiﬁcantly smaller than the distance between the unperturbed levels, i.e.,

|



(0)

− E

(0)



152 Approximate methods of ﬁnding quantum states

4. If the unperturbed energy level is s-fold degenerate, then it corresponds to s self-

orthogonal wavefunctions ψ

(0)

,...,ψ

(0)

. The perturbation splits a degenerate energy

level into s close sublevels. Each of them corresponds to its own wavefunction, which

is a linear combination of unperturbed wavefunctions ψ

(0)

5. The probability of transition of the system from the initial state to the ﬁnal state

is proportional to the square of the Fourier transform of the matrix element of the

perturbation operator, which corresponds to the transition frequency, ω

= (E

(0)

−

(0)

)/h

, i.e.,

4π

(ω

6. If the de Broglie wavelength of the particle is small compared with the characteristic

size of the region within which the particle motion takes place, then the particle

behavior can be described in the framework of the quasiclassical approximation. In

this case the wavefunction can be presented as a superposition of two states, which

describe the particle motion along and against the x-direction:

ψ = ψ

+ ψ

−

√

∓



√

2m(E−U(x))dx/h

7. The quasiclassical probability of electron transmission through the barrier is deﬁned

by the approximate expression

D = e

−2



√

(U(x)−E)dx /h

At the turning points, x

and x

, the momentum of a particle tends to zero ( p → 0)

and the de Broglie wavelength is not small. Therefore, these points must be excluded

from the integration.

5.6 Problems

Problem 5.1. The unperturbed system has two close energy levels, E

(0)

and

(0)

, which are separated by an energy comparable to the energy of perturbation.

Find the ﬁrst-order corrections to the energy of these states.

Problem 5.2. A harmonic oscillator with charge e and mass m is placed in a

homogeneous electric ﬁeld E directed along the axis of oscillations. Considering

the electric ﬁeld as a perturbation, ﬁnd the shift in the energy levels of the

oscillator.

Problem 5.3. An electron in a one-dimensional rectangular potential well with

the barriers of inﬁnite height,

U (x ) =



0, 0 ≤ x ≤ L ,

∞, x < 0, x > L ,