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

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4.2 An electron in a spherically-symmetric potential well 113

the corresponding quantum levels is equal to

. (4.47)

The normalization condition of the wavefunction X

(r) gives us the value of

the constant A

√

2/a. As a result the radial function, X

(r), in the case of

electron conﬁnement being considered here has the form

(r) =



sin(kr)

. (4.48)

In the case of electron motion with orbital angular momentum L = 0, the solution

of Eq. (4.44) depends on the orbital quantum number l. This solution can be

written in the form of special functions called cylindrical Bessel functions J

l+1/2

(r) = C

√

l+1/2

(kr). (4.49)

In the general case when the index l is a non-integer number the Bessel functions

cannot be reduced to well-known elementary mathematical functions, but in the

case of integer values of l, the Bessel functions can be expressed in terms of

trigonometric functions as follows:

l+1/2

(x ) = (−1)





x dx





sin x



. (4.50)

Note that [d/(x dx)] in Eq. (4.50) is an operator, which must be applied to

(sin x/x) in this equation l times. Taking into account Eqs. (4.49) and (4.50), the

solutions of Eq. (4.44) for the ﬁrst three values of the orbital quantum number

(l = 0, 1, 2) are the following functions of coordinate r:

(r) = C



πk

sin(kr),

(r) = C



πk



sin(kr)

− cos(kr)



, (4.51)

(r) = C



πk





sin(kr)

− cos(kr)



− sin(kr)



Here C

√

πk/a and C

and C

can be found from the general normalization

condition for the total wavefunction:



|ψ(r,ϕ,θ)|

dr sin θ dθ dϕ = 1, (4.52)

where we integrate over the coordinate r from 0 to a, over the angle θ ,from0to

π, and over the angle ϕ,from0to2π .

From Eq. (4.49), using the boundary condition (4.45)forr = a we can deter-

mine the values of wavevector k that satisfy the condition J

l+1/2

(ka) = 0:

k =

, (4.53)

114 Additional examples of quantized motion

Table 4.1. Values of g

for the ﬁrst few states

nl 10 11 12 20 13 21 22 30

π 4.49 5.76 2π 6.992 7.73 9.09 3π

where the parameters g

are the roots of the corresponding Bessel functions.

From Eqs. (4.53) and (4.40) we can ﬁnd the expression for the energy of the

allowed stationary states:

. (4.54)

For states with l = 0 the corresponding roots are g

= nπ. Values of g

for the

ﬁrst few states are given in Table 4.1.

Thus, in a spherically-symmetric well described by the potential (4.41)the

stationary states are deﬁned by three quantum numbers, n, l, and m. The radial

wavefunctions (4.43) correspond to these quantum numbers and the energy of

the corresponding states is deﬁned by Eq. (4.54).

Example 4.1. Consider an electron in a spherically-symmetric potential well

with ﬁnite potential barriers (compare this with Eq. (4.41)):

U (r) =



0, 0 ≤ r ≤ a,

, r > a.

(4.55)

Solve Eq. (4.39) and ﬁnd the normalized radial function X (r). Also ﬁnd the

equation that deﬁnes the energy of stationary states with zero orbital angular

momentum for E < U

Reasoning. Zero orbital angular momentum corresponds to the orbital quantum

number l = 0. Within the potential well where U (r) = 0, Eq. (4.39) coincides

with Eq. (4.44) and its solutions for l = 0 have already been found (Eqs. (4.43)

and (4.46)). As before, we have B

= 0. Now, we need to deﬁne the normalization

constant A

that would be different from A

√

2/a that was found for the

quantum well with barriers of inﬁnite height (see Eq. (4.48)).

Outside of the potential well where U (r) = U

the square of the wavenumber

is a negative number. This is why we introduce a new parameter for E < U

− E)

, (4.56)

and Eq. (4.39) can be written in the form



− κ



X(r) = 0. (4.57)

The solution of Eq. (4.57) in the region r > a is an exponentially decreasing

function. Thus, the expression for the radial wavefunction inside and outside the

4.3 Quantum harmonic oscillators 115

potential well is

X(r) =



sin(kr)/r, 0 < r < a,

(1/r)e

−κr

, r ≥ a.

(4.58)

From the condition of continuity of the wavefunction X (r) and its derivative we

obtain the following two equations:

sin(ka) = A

−κa

, (4.59)

[ka cos(ka) − sin(ka)] =−A

(1 + κa)e

−κa

. (4.60)

By solving this system of equations and taking into account the expressions for

k and κ (Eqs. (4.40) and (4.56), respectively) we obtain the dispersion equation

which deﬁnes the energy of an electron in the stationary states with l = 0:

cot



√



=−



− 1. (4.61)

For the coefﬁcients A

and A

we obtain the relationship

= A

sin(ka)e

κa

√

+ κ

κa

. (4.62)

The normalization condition of the radial function, X (r) (see Eq. (4.58)), can be

written as follows:





sin

(kr)dr +

+ κ

2κa



∞

−2κr



= 1. (4.63)

By carrying out the integration in Eq. (4.63) we obtain the following expression

for the normalization constant A



1 + κa

. (4.64)

Note that, for U

→∞, when κ →∞ the wavefunction coincides with

Eq. (4.48).

4.3 Quantum harmonic oscillators

4.3.1 An electron in a parabolic potential well

Modern methods of fabrication of crystalline nanostructures using computerized

technological regimes allow us to design potential wells with the desired pro-

ﬁles. Alongside the rectangular potential wells that we considered in the previous

chapter, nowadays more complex dependences of U (x, y, z) are realized. Of spe-

cial interest are structures with parabolic potential wells, where, in contrast to

Eqs. (4.16) and (4.47), the distance between levels is constant and is determined

116 Additional examples of quantized motion

only by the parameter of the parabolic well. Moreover, in a structure with an arbi-

trary non-homogeneous potential proﬁle U (x, y, z) near the equilibrium position

that corresponds to the minimum, the potential energy, U (r), can be expanded in

a Taylor series, taking into account the ﬁrst three terms:

U (r) = U



r=r

(r − r

) +



r=r

(r −r

)

. (4.65)

Here, U

= U (r

Let us assume that the origin of coordinates, (x

, y

, z

), is at the bottom

of a potential well, i.e., let us measure the energy of an oscillating particle

from U

= U (x

, y

, z

) = 0. The term with the ﬁrst derivative for any potential

proﬁle at its minimum is equal to zero. Then, the particle’s potential energy can

be represented with the following function:

U (x , y, z) =



+ β



, (4.66)

where x, y, z are displacements from the position (x

, y

, z

) since we put the

origin of the new coordinate system onto U (x

, y

, z

). The coefﬁcients β

are

deﬁned as

∂

∂r

. (4.67)

Classical motion of a particle in such a potential takes place under the inﬂuence

of the force

F =−∇U =−



iβ

x + jβ

y + kβ



. (4.68)

Note that F vanishes at the origin. In each of three independent directions a

particle with mass m executes simple harmonic motion (see Eq. (A.73)) with

angular frequency



, (4.69)

where α = x, y, z. The region within which this particle is allowed to move

is limited by the interval of values of the corresponding coordinates −A

≤

≤ A

, where the amplitude of oscillations, A

, is related to the energy of the

particle’s motion in this direction as



mω

. (4.70)

At the turning points r

=±A

the corresponding projection of the velocity v

of the particle vanishes. Motion outside of the interval (−A

, A

)impliesthat

the kinetic energy of a particle is negative, which is meaningless in classical

mechanics. A particle that moves in such a way is called a classical harmonic

oscillator. Examples of such motion in real structures, alongside the electron

oscillations, are the oscillations of atoms and ions in a crystalline lattice.

4.3 Quantum harmonic oscillators 117

If the amplitude of the particle’s oscillations is comparable to its de Broglie

wavelength, then such a particle is called a quantum oscillator. An exact solution

of the steady-state oscillatory motion of a quantum harmonic oscillator, which

can be represented by a bound electron in a parabolic potential (Eq. (4.66)), must

be based on the Schr

odinger equation. In this case the Schr

odinger equation has

the form



−



∂

∂x

∂

∂y

∂

∂z





ψ(x, y, z) = Eψ(x, y, z).

(4.71)

Since the potential energy is a sum of three terms, each of which depends on one

coordinate only, by analogy to Eq. (4.3) the general solution of the Schr

odinger

equation (4.71) can be expressed in the form

ψ(x, y, z) = ψ

(x )ψ

(y)ψ

(z), (4.72)

where the wavefunctions ψ

)(α = x, y, z) describe the electron motion along

the α-directions. The total energy can be written as a sum of individual energies:

E = E

+ E

.AsaresultEq.(4.71) can be rewritten as a set of three

independent differential equations (compare with Eqs. (4.7)–(4.10)):

(α)



−



(α) = 0,α= x, y, z. (4.73)

Thus, the solution of the Schr

odinger equation (4.71) that describes the motion of

a three-dimensional harmonic oscillator is reduced to three equations of motion

of one-dimensional harmonic oscillators.

4.3.2 A one-dimensional harmonic oscillator

Let us consider now the one-dimensional motion of an electron along the x-

axis in a parabolic potential well U (x) = βx

/2. This motion is described by

Eq. (4.73), which for α = x takes the form





E −



ψ(x) = 0. (4.74)

Here, the indices of ψ

, E

, and β

are omitted for convenience. The wavefunc-

tion, ψ(x), and its derivative, dψ(x)/dx, must be ﬁnite and continuous for all

values of x.Forx →±∞the electron potential energy tends to inﬁnity. As a

result, the wavefunction ψ(x) at large distances from the equilibrium position

tends to zero. Equation (4.74) can be rewritten by introducing dimensionless

variable ξ = x/x

= x

√

ω/h

ψ(ξ )

dξ



− ξ



ψ(ξ ) = 0. (4.75)

118 Additional examples of quantized motion

Here, x

√

ω is the turning point for a classical oscillator in its ground

state E

= h

ω/2. Let us introduce into Eq. (4.75) the dimensionless parameter

λ =

. (4.76)

Then, Eq. (4.75) takes the form

ψ(ξ )

dξ

+ (λ − ξ

)ψ(ξ ) = 0. (4.77)

We will seek the solution of this equation in the form of the following product

of two functions:

ψ(ξ ) = f (ξ )e

−ξ

. (4.78)

The unknown function f (ξ ) must behave in such a way that as ξ →∞the wave-

function ψ(ξ ) is bounded. By substituting Eq. (4.78) into Eq. (4.77) we derive

the following equation for the function f (ξ):

f (ξ )

dξ

− 2ξ

d f (ξ)

dξ

+ (λ − 1) f (ξ) = 0. (4.79)

The solution of this differential equation can be found in the form of the following

power series:

f (ξ ) =



k=0

. (4.80)

Let us ﬁnd the ﬁrst and second derivatives of the function f (ξ ):

d f (ξ)

dξ



k=0

k−1

, (4.81)

f (ξ )

dξ



k=0

k(k − 1)a

k−2

. (4.82)

Let us substitute Eqs. (4.80)–(4.82) into Eq. (4.79). As a result we obtain



k=0

k(k − 1)a

k−2

− 2ξ



k=0

k−1

+ (λ − 1)



k=0

= 0. (4.83)

In order for a power series of the form



to be identically equal to zero, all

the coefﬁcients c

of the series must be equal to zero. Let us ﬁnd from Eq. (4.83)

the coefﬁcient c

before ξ

and equate it to zero:

= (k + 1)(k + 2)a

k+2

− (2k + 1 −λ)a

= 0. (4.84)

As a result we obtain the following recurrence formula, which relates coefﬁcients

k+2

and a

k+2

2k + 1 −λ

(k + 1)(k + 2)

. (4.85)

The wavefunction (4.78) will be limited as ξ →∞if the power series (4.80)

has a ﬁnite number of terms. To satisfy this condition the power series (4.80)

4.3 Quantum harmonic oscillators 119

must end at some value of k. For example, for a

= 0 the coefﬁcient a

k+2

must

be equal to zero. Then, according to Eq. (4.85) the coefﬁcients after a

k+2

must

be equal to zero too. Thus, the function f (ξ) reduces to a kth-order polynomial.

The condition a

k+2

= 0, taking into account Eq. (4.85), gives us the expression

for the nth energy level, E

λ = 2n + 1, (4.86)

= 2n + 1, (4.87)

which we can rewrite as

= h



n +



, (4.88)

where the quantum number, n, has the values n = 0, 1, 2,... The quantum

number n deﬁnes the energy of oscillatory motion of a quantum oscillator and it

is called the oscillatory quantum number.

The nth-order polynomial, f (ξ ), is one of the so-called Hermite polynomials,

which are usually denoted as H

(ξ ). Then, the normalized wavefunction of a

one-dimensional quantum oscillator in the nth energy state can be written as

(ξ) = A

(ξ)e

−ξ

= A

(ξ)e

−ξ

. (4.89)

The normalization constant A

in this case depends on the quantum number n

and is equal to

√



π h



1/4

. (4.90)

The Hermite polynomials, H

(ξ ), in a general case can be deﬁned as follows:

(ξ) = (−1)

dξ



−ξ



. (4.91)

For the ﬁrst ﬁve values of the quantum number n the Hermite polynomials have

the following values:

(ξ) = 1, H

(ξ) = 2ξ, H

(ξ) = 4ξ

− 2,

(ξ) = 8ξ

− 12ξ, H

(ξ) = 16ξ

− 48ξ

+ 12.

(4.92)

According to Eq. (4.88) the electron spectrum in a one-dimensional parabolic

potential well consists of equidistant energy levels. The distance between adjacent

energy levels is

n+1

− E

= h

ω. (4.93)

The energy of the electron ground state, which corresponds to the quantum

number n = 0, is the lowest and equal to

. (4.94)

120 Additional examples of quantized motion

Since E

> 0, then the lowest energy state is not the state at rest. This distin-

guishes the quantum oscillator from its classical counterpart and it is a direct

consequence of Heisenberg’s uncertainty principle. Indeed, the uncertainty of

the electron coordinate in the ground energy state is of the order of the maxi-

mum value of the electron coordinate, i.e., x ≈ x

max

= A

. At the turning point

where x = A

the kinetic energy equals zero and the total energy coincides with

the potential energy, which, in accordance with Eqs. (4.66) and (4.70), is

E =

. (4.95)

The potential energy becomes zero at x = 0 and the total energy is equal to the

kinetic energy E = p

max

/(2m

). Thus

max

= m

ω A

. (4.96)

Thus, the uncertainty of the momentum is of the order of the maximum value of

the momentum, i.e., p ≈ p

max

= m

ω A

. If we write the uncertainty relation

as x p ≥ h

, then from the expression

x p ≥ m

ω A

≥ h

, (4.97)

we obtain the condition for the amplitude A

≥

. (4.98)

Taking into account the inequality (4.98) and (4.95), we come to the expression

E ≥

, (4.99)

which gives us the minimum energy of the quantum oscillator in the parabolic

potential well:

min

= E

. (4.100)

The oscillations of a quantum oscillator for the lowest energy state are called

zeroth oscillations. They have purely quantum origin and are not connected with

the thermal energy of an electron. Such oscillations in real systems exist at

temperatures even close to absolute zero.

The forms of the electron wavefunctions for the ﬁrst three lowest energy

levels are shown in Fig. 4.2. For the classical oscillator, which has a ﬁxed

amplitude of oscillation, the increase of energy of oscillations is connected

according to Eq. (4.95) with the increase in amplitude, A

. The interval within

which the particle is allowed to move is (−A

, A

). For the quantum oscillator

the wavefunctions are not equal to zero outside of this interval. This fact

demonstrates that there is a certain probability of ﬁnding the electron outside of

the interval (−A

, A

), where motion is forbidden classically. The distribution

of the probability density of the particle’s location in the ground state (n = 0),

which is characterized by |ψ

, is shown in Fig. 4.3.

4.3 Quantum harmonic oscillators 121

−

Figure 4.2 A quantum

harmonic oscillator: the

three lowest levels and

corresponding

wavefunctions are shown

schematically.

A−−

Figure 4.3 The probability

density distribution, |ψ

for the ground energy

state, E

. Classical turning

points are located at

x =−A

and x = A

. The

dash–dotted line shows

the classical probability

distribution of a harmonic

oscillator with the same

energy.

At high values of the quantum number, n, the behavior of a quantum oscillator

increasingly resembles the behavior of the classical oscillator, for which the

probability density changes smoothly from a minimum at x = 0 to inﬁnity at

the turning points, −A

and A

(Fig. 4.4). Note that for a parabolic potential

> A

for n > 0. The behavior of the quantum oscillator becomes classical

if its oscillation amplitude A

becomes much greater than the amplitude of the

zeroth oscillation, A

√

/(m

ω):





. (4.101)

Taking into account that the de Broglie wavelength for the maximum momentum

on the nth level, p

max

= m

ω A

, is equal to λ

= h

/ p = h

/(m

ω A

), we see

122 Additional examples of quantized motion

xA−

Figure 4.4 The probability

density distribution,

|ψ

, for the energy state

with the principal

quantum number n = 10.

The dash–dotted line

shows the classical

probability distribution of

a harmonic oscillator with

thesameenergy

(compare it with the one

in Fig. A.8).

that Eq. (4.101) satisﬁes the condition that A

is greater than λ

, i.e., an oscillator

with large amplitude (4.101)isalwaysclassical.

4.3.3 Two-dimensional and three-dimensional

harmonic oscillators

A two-dimensional harmonic oscillator

In the case of an isotropic two-dimensional quantum oscillator whose motion

takes place in the xy-plane the harmonic coefﬁcients are as follows: β

= β

and β

= 0. The Schr

odinger equations (4.73)forthex- and y-directions are

exactly the same as Eq. (4.74). The eigenvalue, E , is a sum of two energies,

and E

. The wavefunctions which are the solutions of these two differential

equations have the same form as in Eq. (4.89) and the energies E

and E

are

given by Eq. (4.88). As a result, the energy eigenvalues are

E = h

ω(N + 1), (4.102)

where the angular frequency of the isotropic two-dimensional harmonic oscillator

is ω =

√

β/m

, and the quantum number N is the sum of two quantum numbers,

and n

: N = n

+ n

. The quantum numbers n

and n

are positive integer

numbers 0, 1, 2,...The energy level E = E

with a given number N = n

+ n

corresponds to N + 1 degenerate wavefunctions. It follows that at a given value

of N the corresponding quantum number takes the values n

= 0, 1,...,N

and n

= N − n

. Because the energy level E

corresponds to N + 1 different

wavefunctions ψ(x, y), the order of degeneracy is

= N + 1. (4.103)