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

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4.3 Quantum harmonic oscillators 123

Note that we did not consider the solution of Eq. (4.74)forthez-coordinate since

= 0 corresponds to the case of a free particle, for which the solution was

found earlier.

A three-dimensional harmonic oscillator

The harmonic coefﬁcients of the three-dimensional spherical oscillator in the x-,

y-, and z-directions are equal to each other:

= β

= β. (4.104)

In this case the solutions of the Schr

odinger equations (4.73)give

E = h



N +



, (4.105)

where N = n

+ n

, the frequency is ω =

√

β/m

, and each of the wave-

functions ψ

)inEq.(4.72) has the form (4.89). The quantum numbers n

are

positive integer numbers 0, 1, 2,...The order of degeneracy of the N th energy

level is equal to (see Example 4.4 later)

(N + 1)(N + 2)

. (4.106)

Thus, for a three-dimensional harmonic oscillator the ground state with N = 0

and E

= 3h

ω/2 is non-degenerate (g

for N = 0 is equal to unity). The energy

state with N = 1 and E

= 5h

ω/2 corresponds to three wavefunctions with the

set of quantum numbers n

, n

, and n

equal to (1, 0, 0), (0, 1, 0), and (0,

0, 1). Therefore, the order of degeneracy, g

, is equal to three. The energy level

= 7h

ω/2 has the order of degeneracy g

equal to six (the quantum numbers

are (1, 1, 0), (1, 0, 1), (0, 1, 1), (2, 0, 0), (0, 2, 0), and (0, 0, 2)), and so on.

The degeneracy of the energy levels is connected with the spherical symmetry

of the oscillator. Indeed, for β

= β

the three terms of the potential energy

in Eq. (4.71) are equal to U (r) = βr

/2. The Schr

odinger equation for U (r)in

the spherical coordinate system has the following form:

nlm

(r,θ,ϕ) = X

(r)Y

(θ,ϕ), (4.107)

where the spherical harmonic functions Y

(θ,ϕ) are deﬁned by Eq. (4.37).

The quantum state of an electron in a spherically-symmetric potential, as we

have already noted, is deﬁned by a set of three quantum numbers (n, l, m): the

principal, n, orbital, l, and orbital magnetic, m, quantum numbers. The N th

energy level (4.105) is deﬁned by a certain combination of quantum numbers n

and l: N = 2n +l, where n and l are positive integer numbers 0, 1, 2,...Since

at N = 0 the quantum numbers are n = 0 and l = 0, the ground energy state,

, of a spherical oscillator is non-degenerate. At N = 1 there can be three sets

of quantum numbers (n, l, m):(0,1,1),(0,1,0),and(0,1,−1). Therefore,

the corresponding energy level is triply degenerate. The energy level N = 2

124 Additional examples of quantized motion

corresponds to six states with quantum numbers (1, 0, 0) and (0, 2, m), where

m =−2, −1, 0, 1, 2.

Example 4.2. An electron is conﬁned in a one-dimensional potential well, U(x).

The wavefunction in the stationary state is deﬁned as

ψ(x) = Ce

−γ x

, (4.108)

where C and γ are constants and γ>0. Find the energy of the electron, E, and

the form of the function U (x), if U (0) = 0.

Reasoning. Let us ﬁnd the second derivative of the function ψ(x):

= 2Cγ e

−γ x

(2γ x

− 1). (4.109)

Let us write the one-dimensional Schr

odinger equation taking into account the

forms of the functions ψ (x) and d

ψ/dx



4γ

− 2γ +

(E − U )



−γ x

= 0. (4.110)

Let us write this equation for x = 0 and take into account U(0) = 0. As a result

we get

E =

γ h

. (4.111)

By substituting this expression for the energy, E,inEq.(4.110) we ﬁnd the

expression for the potential energy U(x):

U (x ) =

2γ

. (4.112)

On comparing the given wavefunction, ψ(x), and the corresponding potential,

U (x), with Eqs. (4.89) and (4.66) we conclude that the electron is a quantum

harmonic oscillator.

Example 4.3. For a one-dimensional quantum harmonic oscillator in its ground

state calculate the probability of ﬁnding the particle outside of the classically

allowed region, i.e., |x| > x

, where x

is the amplitude of oscillations of the

classical oscillator.

Reasoning. According to Eq. (4.88) the energy of a one-dimensional oscillator

in the ground state is equal to E

= h

ω/2. The wavefunction of this state, ψ

(x),

according to Eq. (4.89) has the form

(x ) =





1/4

−bx

, (4.113)

with b = mω/h

, where m is the mass of oscillating particle and ω is the frequency

of the oscillations. The maximum displacement of the classical oscillator from

its equilibrium position is deﬁned by the equality of its kinetic energy to zero at

the turning points (the points where the direction of the particle’s motion changes

4.3 Quantum harmonic oscillators 125

to the opposite). As we indicated when discussing Eqs. (4.95) and (4.96), the

kinetic energy at x = 0 is equal to the potential energy at x = x

, i.e.,

, (4.114)



. (4.115)

It follows from Eq. (4.115) that the parameter b = 1/x

The probability of ﬁnding the oscillating particle inside of the potential well

|x|≤x

is as follows:

inside



−x

|ψ

(x )|

dx =





1/2



−x

−bx

dx. (4.116)

Let us substitute the variable in this integral:

√

bx = u, dx =

√

. (4.117)

As a result we arrive at the integral

inside

√



−1

−u

du =

√



−u

du. (4.118)

The integral in Eq. (4.118) is a probability integral, J , which in general form is

deﬁned as

J (ξ ) =

√



−u

du. (4.119)

Its values for various magnitudes of the upper limit of integration ξ are given

in numerous handbooks. The particular value for ξ = 1isJ (1) = 0.843. Thus,

inside

= 0.843, and the probability of ﬁnding the particle in the ground state

outside of the potential well is equal to

outside

= 1 − P

inside

= 0.157. (4.120)

Example 4.4. Find the order of degeneracy of the N th electron energy level in a

spherically-symmetric parabolic potential well with the energy spectrum deﬁned

by Eq. (4.105).

Reasoning. For the given value of N the order of degeneracy is equal to the

number of possible permutations of the three quantum numbers n

, n

, and

, whose sum is equal to N . Let us ﬁnd the number of such permutations for

the given number n

. The number n

will change within the range from 0 to

N − n

+ 1 (the number n

has the same values in the opposite order), i.e.,

) = N − n

+ 1. (4.121)

126 Additional examples of quantized motion

After summation of this expression over the possible values of n

we ﬁnd the

order of degeneracy of the N th energy level, i.e.,



) =



(N − n

+ 1)

= (N + 1) + (N + 1 − 1) +··· + (N + 1 − N )

= (N + 1) + N + (N − 1) + (N − 2) +···+ 2 + 1. (4.122)

Here we can use the formula for the sum, S

, of an arithmetic sequence:

= a

+ (a

+ d) + (a

+ 2d) +···+[a

+ (k − 1)d] =

k(a

+ a

)

. (4.123)

For the arithmetic sequence (4.122) d =−1 and k = N + 1. Thus, the sum

(4.122), g

, is equal to

(N + 1)(N + 2)

. (4.124)

4.4 Phonons

4.4.1 Phonon gas

We show that in a one-dimensional crystalline lattice during the excitation of the

wave motion the total mechanical energy can be represented as a sum of energies

of independent harmonic oscillations (normal modes), which differ from each

other by the values of the wavenumber q, i.e.,

E =



. (4.125)

At the same time the energy of an individual normal mode, expressed in terms

of normal coordinates x

and corresponding momenta p

, is equal to

mω

. (4.126)

The quantum-mechanical description of each normal mode can be considered as

a set of quantum harmonic oscillators, whose energy is quantized and is equal to

= h





. (4.127)

We see that the quantum oscillator can change its energy only by an amount

equal to E

= h

n

. In accord with the selection rules for the quantum

transitions of a quantum oscillator, the quantum number n

can change only by

unity, i.e.,

n

=±1. (4.128)

If n

=−1, the crystalline lattice makes a transition to one of the allowed states,

which has a lower energy level. The energy h

is transferred to the current

4.4 Phonons 127

carriers or to the surrounding medium. For n

=+1, the crystalline lattice

makes a transition to one of the allowed states, which has a higher energy level.

Each individual quantum of energy, h

, corresponds to an elementary wave

excitation of the lattice, i.e., it corresponds to a quasiparticle with deﬁnite energy

and momentum, which is called a phonon. The energy of such a quasiparticle is

equal to

= h

, (4.129)

and this is related to its momentum by:

= h

q. (4.130)

The corresponding quantum transition of an oscillating lattice from one energy

state to another is accompanied by the emission (creation) or absorption (anni-

hilation) of a phonon. The transition from the set of normal modes to the set

of wave excitations allows us to utilize the main idea of quantum mechanics –

wave–particle duality. In this case we have a manifestation of the duality of

quasiparticles (phonons) and wave excitations in a crystal. The velocity of such

a quasiparticle is deﬁned by the group velocity of the wave motion of the lattice,

i.e.,

∂ω

∂k

∂ε

∂p

. (4.131)

During the interaction of other particles or quasiparticles with lattice oscillations

their energy can change only by ±h

and their momentum by h

q. For this

reason, for the interaction of phonons with electrons, i.e., for the description

of the processes of creation and annihilation of phonons, it is convenient to

use the laws of conservation of energy and momentum of the particles. Let us

note that the phonon’s momentum is usually called quasimomentum.Thisis

because the quantities h

q and h

(q ± 2π b

) are physically equivalent. Here b

the unit wavevector in the momentum space, which will be discussed in Chapter

7. Phonons can also interact with each other. This kind of interaction explains

many of the thermal phenomena which take place in crystals. It is necessary to

remember that in the processes of scattering of phonons by each other, during

which an exchange of energy and momentum takes place, the number of phonons

must be at least three.

Only those normal modes of a crystal which are excited higher than the

crystal’s lowest energy level



(4.132)

can interact with electrons or with each other. Namely, the excitations above

the zero oscillations of Eq. (4.132) are called phonons. Zeroth oscillations of a

normal mode establish a constant energy background, which during scattering

of phonons by each other and by electrons does not participate in scattering

128 Additional examples of quantized motion

processes. The total number of phonons in a crystal is deﬁned by their sum over

all normal modes of a crystal (i.e., over all quantum oscillators), i.e.,

N =



. (4.133)

During interactions the number of phonons changes and, in contrast to the case

of a molecular gas, phonons can be created or annihilated. The energy state of

a crystal is deﬁned by the total number of phonons, which as a whole can be

considered as a phonon gas. The energy of a phonon gas is deﬁned by the energy

excitations above the thermal energy background E

, i.e.,

ph.gas



. (4.134)

A very important feature of a phonon gas is the dependence of the number of

phonons on temperature. At thermodynamical equilibrium (at temperature T )the

average number of phonons with wavenumber q in the unit volume of a crystal

is given by the expression



(T )



/(k

T )

− 1

, (4.135)

which coincides with the analogous expression (2.41) for the number of photons

in the unit volume of a cavity ﬁlled by equilibrium thermal radiation. From

Eq. (4.135) it follows that the number of phonons with energy ε

= h

at high

temperatures (T  h

) is deﬁned as

n

≈

, (4.136)

whereas at low temperatures (T  h

) it is deﬁned as

n

≈e

−h

/(k

T )

. (4.137)

Thus, at high temperatures, such that the thermal energy k

T is sufﬁciently high

to excite a large number of modes with energy h

, the number of phonons in

a crystal is large. At low temperatures, for which the thermal energy k

T is not

sufﬁciently high to excite lattice oscillations, i.e., k

T  h

, the number of

phonons in a crystal is exponentially small, since n

≈e

−h

/(k

T )

 1. Thus,

at low temperatures the phonon gas can be considered an ideal gas, i.e., a gas

of phonons that do not interact with each other. At high temperatures, on the

other hand, because of the high concentration of phonons the interaction between

phonons becomes more pronounced. In order to take into account the interaction

between phonons it is necessary to take into consideration the anharmonicity

in the potential energy, i.e., it is necessary to include the term proportional to

the third power of the particle’s displacement from its equilibrium position. The

anharmonicity is responsible for the three-phonon interaction.

4.4 Phonons 129

The temperature is considered high or low in comparison with the Debye

temperature of the given material, T

, which is deﬁned as:

max

. (4.138)

Here the limiting frequency of oscillations is related to the parameters of the

crystalline lattice by

max







6π



1/3

, (4.139)

where





is the mean velocity for longitudinal and transverse acoustic waves

in the crystal. The Debye temperature, T

, divides the entire temperature scale

into two regions. For temperatures T  T

only long-wavelength waves, whose

energy is small in comparison with k

, are excited. For T > T

all oscillations

are excited, including the oscillations with the maximum frequency, whose energy

is about k

4.4.2 Thermal properties

The concepts of thermal oscillations of a lattice as a phonon gas, which obeys

quantum-mechanical laws, allow us to explain such very important and easily

measured thermal phenomena as heat capacity and thermal conductivity, which

we will brieﬂy consider here.

Heat capacity

It is well established experimentally that at T → 0 the heat capacity of all solids

tends to zero. From the classical point of view this experimental fact cannot be

explained. Moreover, the dependence of thermal conductivity on temperature

cannot be explained either. The Dulong–Petit law according to which the molar

heat capacity of simple solids does not depend on temperature and is equal

to 3R is based on the classical concepts. In the framework of the phonon-gas

model the heat capacity of a solid practically does not depend on temperature

for T > T

and substantially decreases with decreasing temperature for T < T

The observed decrease of the heat capacity at low temperatures is due to a simple

reason: the energy which corresponds to one oscillatory degree of freedom is

too low to excite high-frequency phonons. Thus, the temperature for which the

Dulong–Petit law is valid must be higher than the characteristic temperature T

for a given material. The values of the Debye temperature for several crystalline

materials are given in Table 4.2.

Figure 4.5 shows the dependence of the molar heat capacity, c,ofsimple

crystalline materials on the dimensionless temperature T/T

. This type of the

dependence c = f (T/T

) is characteristic for most crystals.

130 Additional examples of quantized motion

Table 4.2. The Debye temperature, T

, for several materials

Material T

(K) Material T

(K)

Beryllium 1160 Zinc 308

Magnesium 406 Aluminum 418

Iron 467 Diamond 2000

Platinum 229 Silicon 658

Silver 225 Germanium 366

Gold 165 Lead 94

0.3 0.6 0.9 1.2 1.5

0.2

0.4

0.6

0.8

1.0

Figure 4.5 The

dependence of the molar

heat capacity, c,onthe

dimensionless

temperature T/ T

By considering the problem of internal energy of crystals from the quantum-

mechanical point of view, Debye proved that at temperatures close to absolute

zero the internal energy is proportional to the fourth power of temperature:

E =

3π

, (4.140)

where R is the universal gas constant. From this relation we can obtain the

expression for the molar heat capacity of a simple (i.e., consisting of one type of

atom) crystal at low temperatures:

c =

12π

. (4.141)

Therefore, near absolute zero the heat capacity of a crystal is proportional to the

third power of temperature. The region of such dependence covers the entire tem-

perature interval from 0 to temperatures close to 0.1T

. At higher temperatures

(up to T = T

) there is an intermediate region. In this region the dependence

changes from the law (4.141) to the Dulong–Petit law, which becomes valid at

4.4 Phonons 131

T > T

. Note that a small increase of heat capacity at T > T

and a deviation

from the Dulong–Petit law are due to the high concentration of phonons and the

fact that a phonon gas is not ideal. This means that at high temperatures it is

necessary to take into account phonon–phonon interactions, which are absent in

an ideal phonon gas.

Thermal conductivity

The phenomenon of thermal conductivity in crystals is related to phonon–phonon

interactions, which lead to a scattering of phonons on phonons. If the two opposite

ends of a crystalline specimen are kept at different temperatures, a continuous

ﬂux of heat occurs in this specimen. Quantitatively the heat ﬂux, dQ/dt, through

the cross-section of the rod S during unit time at a temperature gradient of dT /dx

is deﬁned by the expression

=−λ

S, (4.142)

where λ is the coefﬁcient of thermal conductivity. In deﬁning the coefﬁcient of

thermal conductivity it is necessary to take into account the quantum nature of

the excitation of heat waves in a crystalline lattice. Let us imagine the excited

state of a lattice as a gas of phonons, which can transfer energy in the direction

opposite to the temperature gradient. Let us choose as a characteristic length in

the x-direction the distance equal to the phonon mean-free-path length, l.The

temperature difference in the given interval (0, l) is deﬁned as

T =

l. (4.143)

For the resultant ﬂux of energy created by phonons we obtain

=−





S, (4.144)

where c is the heat capacity per unit volume of the phonon gas. Thus, the

coefﬁcient of thermal conductivity, λ, of a crystalline lattice is deﬁned by the

expression

λ =





l. (4.145)

Deeper analysis shows that at sufﬁciently high temperatures the mean-free-path

length of the phonon is inversely proportional to temperature. Therefore, the

coefﬁcient of thermal conductivity of the most non-conducting crystals at T > T

is inversely proportional to temperature since at T > T

the heat capacity of the

phonon gas does not depend on temperature.

In pure crystals at temperatures close to absolute zero the dependence of the

coefﬁcient of thermal conductivity on the dimensions of the specimen becomes

more pronounced. At low temperatures the concentration of phonons is negli-

gible. Therefore, the probability of phonons scattering on phonons is small and

132 Additional examples of quantized motion

their mean-free-path length, l, is limited by the dimensions of the specimen, L.

Thus, in the interval of temperatures being considered here the coefﬁcient of

thermal conductivity is deﬁned as

λ =





L. (4.146)

4.4.3 The analogy between phonons and photons

Note that phonons and photons, which we discussed in Chapter 2, share a number

of physical properties. If the electron can be considered as a harmonic oscilla-

tor, then during quantum transitions between adjacent energy levels a photon

(quantum of electromagnetic ﬁeld) is emitted or absorbed. Photon energy and

momentum are deﬁned by the expressions (2.35) and (2.39), which coincide with

the corresponding expressions for phonons (4.129) and (4.130).

Photons and phonons belong to the same class of particles, called bosons –

the particles which possess integer spin (we will talk about the particle property

called spin in Section 6.3). Therefore, the average number of phonons in any

given energy state, E, at thermodynamical equilibrium is given by the same

expression:



n(T )



E/(k

T )

− 1

. (4.147)

The main difference between a phonon and a photon is that the phonon is a

quasiparticle and its introduction as a notion is convenient for the mathematical

description of physical processes. In contrast to photons, which are real particles

and can exist in media as well as in vacuum, phonons can exist only in crystals.

Example 4.5. Estimate the number of phonons in a linear chain of atoms which

corresponds to a single normal oscillator with wavenumber q, if the average

energy of the oscillator is deﬁned as

ε

=γ k

T, (4.148)

where the parameter γ>1. Assume that the temperature, T , is high.

Reasoning. According to Eq. (4.135) the average number of phonons which

corresponds to a single normal oscillator is deﬁned by the average quantum

number,

n

=

/(k

T )

− 1

, (4.149)

and by the average energy of a single oscillator,

ε

=h



n

+



. (4.150)