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

Подождите немного. Документ загружается.

3.7 Problems 103

Figure 3.15 A rectangular

symmetric potential

well with respect to the

origin of coordinates

with barriers of height

= 10 eV.

Problem 3.7. An electron is in a one-dimensional rectangular potential well

with width equal to L. One of the barriers has inﬁnite height: U (x) →∞at

x < 0. The height of the second barrier at x > L is ﬁnite and is equal to U

In the region 0 ≤ x ≤ L the potential energy is equal to zero: U (x) = 0 (see

Fig. 3.3). What are the conditions on the value of U

for there to be at least

one energy level or only one level in the well? If the energy level is equal to

E = U

/2, how does this condition change?

Problem 3.8. An electron is in a symmetric rectangular potential well with

width L = 10 nm and with barriers of ﬁnite height: U

= 10 eV. The electron

wavefunction is symmetric with respect to the coordinate origin placed at the

center of the well (see Fig. 3.15), in contrast to the case shown in Fig. 3.5.

For which values of quantum numbers, n, can we apply the approximation of

barriers with inﬁnite height? (Answer: n  52.)

Problem 3.9. An electron with energy E = 32 eV, which is moving in the

positive direction along the x-direction, encounters an obstacle – an inﬁnitely

long rectangular potential step. Find the reﬂection, R, and transmission, D,

coefﬁcients of the electron de Broglie waves for a given potential step in two

cases: (a) the height of the potential step is lower than the electron energy

= 30 eV) and (b) the height of the potential step is higher than the electron

energy (U

= 34 eV). (Answer: (a) R ≈ 0.36 and D ≈ 0.64; (b) R ≈ 1.0 and

D ≈ 0.)

Problem 3.10. Estimate the transmission coefﬁcient, D, of an electron with

energy E = 18 eV for passage through a potential barrier with the following

104 Layered nanostructures

dependence of potential energy on coordinate:

U (x ) = U



1 −



(3.196)

in the region |x|≤L and

U (x ) = 0 (3.197)

in the region |x| > L. Here, U

= 20 eV and L = 2 nm. (Answer: D ≈ 0.24.)

Chapter 4

Additional examples of quantized motion

In the previous chapter we have analyzed the peculiarities of quantized elec-

tron motion in layered structures with one-dimensional potential wells. From

the mathematical point of view, the solution of the Schr

odinger equation for

one-dimensional potential proﬁles is much simpler. However, many quantum

objects, such as atoms, molecules, and quantum dots, are three-dimensional

objects. Thus, in order to analyze electron motion in such objects we need to ﬁnd

solutions of the Schr

odinger equation for three-dimensional potential proﬁles.

The electron motion in spaces with dimensionality higher than one, especially

for rectangular potential proﬁles with inﬁnite potential barriers, is not so difﬁcult

to analyze. At the same time we have to keep in mind that such potential proﬁles

frequently represent some approximation of the more complex, real potential pro-

ﬁles. Depending on the type of structure and on the form of the potential proﬁle,

the electron motion may be limited in two directions (two-dimensional quanti-

zation) or in three directions (three-dimensional quantization). In this chapter

we will show that the existence of discrete energy levels in the electron spec-

trum is an intrinsic feature of electron motion in potential wells of any form and

dimensionality.

4.1 An electron in a rectangular potential

well (quantum box)

In the previous chapter we studied the electron motion in one-dimensional poten-

tial wells. An electron’s motion along one direction was conﬁned by the potential

proﬁle and the momentum in this direction was quantized. In the two other direc-

tions the motion of the electron was free. For the speciﬁc potential proﬁles that

we studied we were able to express the solution of the Schr

odinger equation

in the form of a product of three separate wavefunctions. In the direction of

electron conﬁnement the wavefunction is a standing wave and in the direction

of free motion the wavefunctions have the form of traveling waves. Here, we

will consider the main features of the electron energy spectrum in a rectangular

potential box with impenetrable walls. The limitation of electron motion in each

105

106 Additional examples of quantized motion

of the three directions is deﬁned by the following conditions:

0 ≤ x ≤ L

, 0 ≤ y ≤ L

, 0 ≤ z ≤ L

, (4.1)

where L

, L

, and L

are the dimensions of the box in the x-, y-, and z-directions,

respectively. The above-mentioned conditions are realized if the electron potential

energy satisﬁes the following conditions:

U (r) =



0, 0 ≤ r

≤ L

∞, r

< 0, r

> L

(4.2)

where α = x, y, and z and r

= α. In this case the Schr

odinger equation inside

the potential box, where U(r) = 0, takes the form

−



∂

∂x

∂

∂y

∂

∂z



ψ(x, y, z) = Eψ(x, y, z). (4.3)

Since the electron cannot penetrate the region where the potential tends to inﬁnity,

for r

< 0 and r

> L

the electron wavefunction, ψ(x, y, z), is equal to zero

in those regions. Therefore, at the boundaries of the box the following boundary

conditions should hold:

ψ(0, y, z) = ψ(x, 0, z) = ψ(x, y, 0) = ψ(L

, y, z) = ψ (x, L

, z) = ψ(x, y, L

) = 0.

(4.4)

For the potential proﬁle of Eq. (4.2) and the boundary conditions of Eq. (4.4)the

wavefunction can be written as a product of three wavefunctions, each of which

depends on a single coordinate,

ψ(x, y, z) = ψ

(x )ψ

(y)ψ

(z), (4.5)

and the total electron energy, E, can be written as a sum of energies, E

, in each

of the three directions of conﬁnement:

E = E

+ E

. (4.6)

Here, each of the wavefunctions ψ

) describes the electron motion along one

speciﬁc coordinate r

and this motion is independent from motion in the two

other directions, r

= r

. On substituting Eqs. (4.5) and Eqs. (4.6) into Eq. (4.3)

and dividing by the wavefunction of Eqs. (4.5), we obtain

(x )



∂

(x )

∂x

(x )



(y)



∂

(y)

∂y

(y)



(z)



∂

(z)

∂z

(z)



= 0. (4.7)

Each of the three terms on the left-hand side of Eq. (4.7) depends only on one

independent variable, r

. Since the sum of these three terms is equal to zero for

any values of the variables x, y, and z, each of the terms must itself be equal

4.1 An electron in a rectangular potential well 107

to zero. Therefore, Eq. (4.7) may be presented as a system of three independent

equations (compare with Eqs. (3.6)–(3.8)):

(x )

+ k

(x ) = 0, (4.8)

(y)

+ k

(y) = 0, (4.9)

(z)

+ k

(z) = 0, (4.10)

where we introduced the parameters

√

, (4.11)

which are the projections of the wavevector k onto the coordinate axes, α = x, y,

and z. The solution of each of Eqs. (4.8)–(4.10), taking into account boundary

conditions (4.4) as in the case of a one-dimensional potential well, has the form

(x ) =



sin(k

x), (4.12)

(y) =



sin(k

y), (4.13)

(z) =



sin(k

z), (4.14)

where

=±

πn

(4.15)

and n

= 1, 2, 3,... It follows from Eqs. (4.11) and (4.15) that the eigenvalues

of energy, E

, are deﬁned by the expression

. (4.16)

As a result, the eigenfunctions of Eq. (4.3)havetheform

(x , y, z) =



sin



πn



sin



πn



sin



πn



. (4.17)

The corresponding eigenvalues are

E = E





. (4.18)

Here, the quantum numbers n

, n

, and n

are positive integers. The wavefunc-

tions (4.12)–(4.14) and (4.17) are orthonormal, which can be veriﬁed directly by

substituting them into Eq. (2.144).

The most important property of the electron quantization in a three-

dimensional potential box, as in the case of the one-dimensional conﬁnement,

108 Additional examples of quantized motion

is the quantization of the electron energy. The energy levels are deﬁned by

the set of three quantum numbers, n

, n

, and n

(Eq. (4.18)). The minimum

energy with the quantum numbers n

= n

= 1 corresponds to the ground

state:

min

= E

111





. (4.19)

This means that the state of the electron with minimum energy is not a state of

rest. This property is of purely quantum origin and is the direct consequence

of the uncertainty relations (2.118). Indeed, the state of rest must correspond to

the electron energy and momentum being equal to zero. Since these values are

deﬁned exactly, the uncertainties in the momenta p

, p

, and p

are equal

to zero too. Thus, according to the uncertainty relations (2.118) the electron state

at rest corresponds to complete uncertainty in coordinates, since in this case x,

y, and z tend to inﬁnity. But this contradicts the limited uncertainties of the

electron coordinates, because we know for sure that the electron is in the box and

the uncertainty in coordinates cannot be larger than the dimensions of the box,

i.e., x ≈ L

, y ≈ L

, and z ≈ L

As a result, the electron motion in a limited space, in contrast to the case

of the free electron in inﬁnite space, cannot have zero momentum and, as a

consequence, zero uncertainty of the momentum. From the uncertainty relations

(2.118) it follows that for the electron conﬁned in a box the uncertainty in the

momenta is the following:

p

∼

,p

∼

,p

∼

. (4.20)

Since projections p

of momentum p should be of the order of the uncertainty

p

, we may conclude that Eqs. (4.15) are direct consequences of Eq. (4.20). It

follows from Eqs. (4.15) and (4.18) that the smallest differences between discrete

values of the momentum k

and energy E

correspond to the α-axes with

the largest value of the length L

. For example, if L

> L

, L

then the set of

quantum numbers n

= n

= 1 and n

= 2 corresponds to the energy state next

to E

111

, namely E

112

. In this case the energy E

112

differs from the minimal energy

111

by E:

E = E

112

− E

111

3π

. (4.21)

By changing the set of quantum numbers (n

, n

) we can calculate the energy

of the conﬁned electron for an arbitrary electron quantum state for the given

dimensions of the box.

4.2 An electron in a spherically-symmetric potential well 109

Figure 4.1 The spherical

system of coordinates.

4.2 An electron in a spherically-symmetric potential well

4.2.1 The Schr

odinger equation in spherical coordinates

Let us consider ﬁrst the motion of an electron in a spherically-symmetric poten-

tial:

U = U (r), (4.22)

where the potential U (r) depends only on the distance r =



+ y

+ z

from

the origin of the coordinate system (r = 0).

Taking into account the spherical symmetry of the potential, the Schr

odinger

equation may be solved using spherical coordinates, in which the position of the

electron in space is given by the radial distance r and two angles – polar and

azimuthal,θ andϕ, respectively. The origin of coordinates, r = 0, coincides with

the center of the potential proﬁle U (r); the azimuthal angle takes values in the

interval 0 ≤ ϕ ≤ 2π ; and the polar angle takes values in the interval 0 ≤ θ ≤ π

(see Fig. 4.1). The wavefunction can be written as a function of the spherical

coordinates, i.e., ψ(r) = ψ (r,θ,ϕ).

The Laplace operator, ∇

, in spherical coordinates includes radial and spher-

ical parts, ∇

and ∇

ϕ,θ

, respectively:

∇

=∇

∇

ϕ,θ

. (4.23)

The radial part of the Laplace operator can be written as

∇

∂

∂r



∂

∂r



∂

∂r

∂

∂r

(4.24)

110 Additional examples of quantized motion

and its spherical part can be written as

∇

ϕ,θ

sin θ

∂

∂θ



sin θ

∂

∂θ



sin

∂

∂ϕ

. (4.25)

Thus, the Schr

odinger equation for an electron in a spherically-symmetric poten-

tial well can be written as



∇

ϕ,θ

(E − U (r))



ψ(r,ϕ,θ) = 0. (4.26)

Since the potential that acts on the electron, U (r), does not depend on the angles

ϕ and θ, the variables in Eq. (4.26) can be separated and the wavefunction can

be expressed in the form of a product of two functions – a radial function,

X(r), which depends only on the coordinate r, and an angle-dependent function

Y (ϕ, θ):

ψ(r,ϕ,θ) = X(r)Y (ϕ, θ). (4.27)

4.2.2 Eigenvalues and eigenfunctions of the

angular-momentum operators

The angular component of the wavefunction Y (ϕ, θ)ofEq.(4.27) in the case

of the spherically-symmetric potential with arbitrary U(r) is an eigenfunction of

the operator for the square of the angular momentum, i.e., it is the solution of the

equation

(

− L

)Y (ϕ, θ) = 0. (4.28)

Here L

is the eigenvalue of the operator

, which coincides up to a constant

factor with the angular part of the Laplace operator (4.25):

=−h

∇

ϕ,θ

. (4.29)

Operators of the projections of angular momentum

(α = x, y, z) are deﬁned

by the following expressions for rectangular and spherical coordinates, respec-

tively:

=−ih



∂

∂z

− z

∂

∂y



= ih



cos ϕ cot θ

∂

∂ϕ

+ sin ϕ

∂

∂θ



, (4.30)

=−ih



∂

∂x

− x

∂

∂z



= ih



sin ϕ cot θ

∂

∂ϕ

− cos ϕ

∂

∂θ



, (4.31)

=−ih



∂

∂y

− y

∂

∂x



=−ih

∂

∂ϕ

. (4.32)

To write these expressions we used the equation for the angular momentum of a

particle L = r ×p and found the projections of angular momentum by writing

L as the determinant:

= yp

− zp

, L

= zp

− xp

, L

= xp

− yp

. (4.33)

4.2 An electron in a spherically-symmetric potential well 111

Then, in Eq. (4.33) we substitute the variables by their corresponding operators.

In the coordinate representation, that we use, the operator of coordinate is the

coordinate itself, and the operators of the projections of momentum are deﬁned

by Eqs. (2.159):

=−ih

∂

∂r

where α = x, y, z are Cartesian coordinates and r

= α. In order to change

Cartesian coordinates to spherical coordinates we used the relations

x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (4.34)

The eigenfunctions in Eq. (4.28)arespherical harmonics Y

l,m

(ϕ, θ), which are

deﬁned by two integer quantum numbers, l and m. The quantum number l, which

is called the orbital quantum number, has the values l = 0, 1, 2,..., and quantum

number m, which is called the orbital magnetic quantum number, has the values

m = 0, ±1, ±2,...,±l. (4.35)

Using the normalizing conditions of Eq. (2.140) for spherical harmonics we have



2π





l,m

(ϕ, θ)



sin θ dθ dϕ = 1. (4.36)

We get the following expressions for the spherical harmonics for the ﬁrst few

values of quantum numbers l and m:

0,0

√

4π

, Y

1,0



4π

cos θ, Y

2,0



16π

(3 cos

θ −1),

1,±1

=∓



8π

sin θ e

±iϕ

, Y

2,±1

=∓



16π

sin(2θ)e

±iϕ

2,±2



32π

sin

θ e

±2iϕ

. (4.37)

The spectrum of eigenvalues of the operator

, which is deﬁned by Eq. (4.28),

is discrete:

= l(l + 1)h

. (4.38)

Each eigenvalue L

corresponds to 2l + 1 eigenfunctions Y

l,m

(ϕ, θ), where the

orbital magnetic quantum number, m, is deﬁned by Eq. (4.35). Thus, each state

with a given value of the square of the orbital angular momentum, L

,is(2l + 1)-

fold degenerate.

4.2.3 The equation for the radial function

On substituting the wavefunction (4.27) into the Schr

odinger equation (4.26),

separating the radial part of this equation, and taking into account Eq. (4.38),

112 Additional examples of quantized motion

we obtain



+ k

−

U (r)

−

l(l + 1)



X(r) = 0, (4.39)

where

. (4.40)

Let us consider the simplest case of U(r):

U (r) =



0, r ≤ a,

∞, r > a,

(4.41)

where r is the distance from the center of the sphere. It is clear that with such

a potential the region outside the sphere is not available for the electron and its

wavefunction at the boundary is equal to zero, i.e.,

X(r) = 0, r ≥ a. (4.42)

The behavior of an electron inside of the region r < a is described by the

time-independent Schr

odinger equation Eq. (3.1) with the potential U(r) = 0.

Then, Eq. (4.39) with the help of the substitution

X(r) =

R(r)

(4.43)

can be reduced to the following equation:



+ k

−

l(l + 1)



R(r) = 0. (4.44)

The function R(r) must satisfy the following two boundary conditions:

R(0) = 0andR(a) = 0. (4.45)

The ﬁrst of these conditions is written by taking into account that the wavefunction

X(r) must be ﬁnite throughout the entire region of the electron’s motion, in

particular at r = 0. The second condition comes from the continuity of the

wavefunction at the boundary of the potential well r = a and the boundary

condition (4.42).

Let us consider ﬁrst the purely radial motion of the electron when its orbital

angular momentum is equal to zero, i.e., L = 0. The orbital quantum number

l = 0 corresponds to such a motion and to the following solution of Eq. (4.44):

(r) = A

sin(kr) + B

cos(kr). (4.46)

Note that for l = 0Eq.(4.39) and its solution Eq. (4.46) coincide with the case

of the one-dimensional quantum well considered earlier. Taking into account the

boundary conditions (4.45), we ﬁnd that B

= 0 and k = nπ/a, where the radial

quantum number n is a positive integer number: n = 1, 2, 3,... The energy of