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

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A.1 Classical dynamics of particles 313

where the mass of an electron is m

= 9.1 × 10

−31

kg and the charge of an electron

is e = 1.6 ×10

−19

C. This is a very small electric ﬁeld and can be neglected in most

practical situations.

For a non-relativistic particle, i.e., a particle that moves with a velocity v that is

substantially smaller than the speed of light in vacuum, c = 3 ×10

−1

, Newton’s

second law of Eq. (A.5) is often written, using Eq. (A.3), in the form

= F. (A.11)

Here the mass of a particle, m, is considered a constant that does not depend on the velocity

of the particle. In many practical cases, m depends on time and Eq. (A.11) becomes invalid.

This is one of the reasons why we will use Eq. (A.5) rather than Eq. (A.11) throughout

the remainder of the book.

One more standard expression for Newton’s second law is

= F, (A.12)

which combines Eq. (A.11) with Eq. (A.4). Equation (A.12) is a second-order differential

equation and its solution has two constants of integration. Thus, to ﬁnd the function r

and its ﬁrst derivative dr/dt as a function of time, t, it is necessary to have two initial

conditions, for example, r(t

) = r

and v(t

) = v

,orp(t

) = p

A.1.2 The dynamics of a system of particles

For n interacting particles the system of equations describing their behavior has the form

= F



k=1

, i = 1, 2,...,n, (A.13)

where F

is the resultant of all external forces acting on the i th particle and F

are the

internal forces acting between the ith and kth particles, while F

= 0. On summing up all

the equations of motion of particles in the system and taking into account that according

to Newton’s third law of motion F

=−F

(therefore



i=1



k=1

= 0), we obtain

the equation

= F, (A.14)

where

P =



i=1

. (A.15)

314 Appendix A. Classical dynamics of particles and waves

Equation (A.14) has exactly the same form as Eq. (A.5) but with two substantial differ-

ences. Here P is the sum of the momenta of all particles of the system and

F =



i=1

(A.16)

is the resultant force equal to the sum of all external forces F

acting on the individual

particle denoted by i . It is convenient to introduce other variables characterizing the

system as a whole. The mass of the system is obviously equal to the sum of the masses of

the individual particles that comprise the system:

M =



i=1

. (A.17)

By combining Eqs. (A.15)and(A.17), we can introduce the velocity V

of the system

using Eq. (A.3):

(A.18)



i=1



i=1

. (A.19)

By continuing this analogy, we can introduce the coordinate R

of a point:

= V

. (A.20)

By substituting Eq. (A.19) into Eq. (A.20)weﬁndthat



i=1



i=1

. (A.21)

The position vector R

(t) describes the trajectory of the system as a whole. Therefore, it is

called the vector of the center of mass of the system. Equations (A.14), (A.18), and (A.20)

for the system of particles are the same as Eqs. (A.5), (A.3), and (A.4) for an individual

particle. This means that we can treat any large system as a point particle with all the

mass of the system, M

, concentrated at the so-called center of mass, R

, as long as the

positions of the individual particles of the system are not the subject of our interest and

the size of a system is small relative to the coordinate space of interest.

A.1 Classical dynamics of particles 315

For a so-called closed system, the sum of all external forces, F in Eq. (A.14) is equal

to zero. Thus, the resultant momentum of the closed system, P, is constant, i.e.,

= 0, (A.22)

P =



i=1

= constant, (A.23)

and



i=1



i=1

= constant. (A.24)

Equations (A.22)and(A.23) describe one of the main principles of mechanics written in

mathematical form – the principle of conservation of momentum, which implies that the

sum of the momenta of a closed system of particles that exert forces on each other is a

constant.FromEq.(A.24) it follows that, when the sum of all external forces acting on

the system of interacting particles, F, is equal to zero, then the velocity of the system’s

center of mass, V

, is constant.

A.1.3 Rotational motion of a particle

Apart from the particle’s momentum, p, there is another important characteristic of a

moving particle. It is the particle’s angular momentum, L, about a given point:

L = r × p, (A.25)

where r is the position vector that deﬁnes the position of a particle with respect to a given

point. The main equation that deﬁnes the change of angular momentum, L, of a particle

can be derived from Eq. (A.25), if we take the derivatives of the left-hand and right-hand

sides of this equation:

× p + r ×

. (A.26)

Since dr/dt = v and p = mv,

× p = m(v ×v) = 0 (A.27)

because the vector product v × v is equal to zero. Using Eq. (A.5) we arrive at the equation

= τ, (A.28)

316 Appendix A. Classical dynamics of particles and waves

where

τ = r ×F (A.29)

is the so-called torque with respect to the given point. If the resultant torque, τ,ofall

forces acting on the particle is equal to zero, then

= 0andL = constant. (A.30)

The last equation reﬂects one of the main principles of mechanics – the principle of

conservation of angular momentum: the angular momentum is independent of time and it

is conserved when the net torque is equal to zero.

As an example let us consider the motion of a particle in a ﬁeld of central forces (or

central ﬁeld). The vector of force, F, in the central ﬁeld always coincides with the radius

vector, r, directed from the central point to a point where the particle is located, and its

magnitude depends only on the distance, r , between these two points, i.e.,

F(r) = F(r)e

, (A.31)

where e

= r/r is a unit vector directed along the radius vector r. Note that in all cases

of a particle’s motion in the central ﬁeld we have F (r) < 0inEq.(A.31). The torque of

this force with respect to the central point is

= r × F = F(r)

(

r × e

)

= 0, (A.32)

because the vectors r and e

are collinear. This is why in the solar system the motion of

planets around the Sun is along ﬂat elliptic orbits – the angular momentum, L, of each of

the planets is a constant vector.

A.1.4 Work and potential energy

Aforce,F, moving a particle by a certain distance, does a certain work. For the displace-

ment dr, the work done, dW , is deﬁned by the scalar product, F · dr:

dW = F ·dr = F

dx + F

dy + F

dz = F |dr|cos α = F

dr, (A.33)

where α is the angle between the direction of force, F, and the direction of displacement,

dr; F

= F cos α is the projection of the force, F, on the direction of displacement, dr;

and |dr|=dr. The total work done by the force, F, to move a particle from the point r

to the point r

is given by the following deﬁnite integral:



F · dr =



dr. (A.34)

Among all forces that can act on particles, we single out a wide class of forces whose

work does not depend on the path followed between points r

and r

(i.e., the work done

A.1 Classical dynamics of particles 317

Figure A.2 The work done

by the conservative forces

around a closed path is

equal to zero.

between points r

and r

is the same for paths 1, 2, and 3 in Fig. A.2),andforwhichthe

work, W

, of moving a particle along a closed path is equal to zero:

F · dr = 0. (A.35)

Here we introduced the symbol of integration around a closed path,

. An example of

a closed path is the transfer of a particle from point r

to r

along path 2 and then from

point r

to point r

along path 3 (see Fig. A.2). The forces that satisfy Eq. (A.35)are

called conservative forces, and they include, for example, gravitational and electrostatic

forces. Note that for the forces of friction the work done depends on the path and therefore

Eq. (A.35) is not valid for the forces of friction.

For the conservative force, F, we can introduce a function, U (r), that depends on the

coordinate, r, and relates to the force, F,as

F =−∇U(r), (A.36)

where we introduced the so-called differential operator, ∇:

∇ = i

∂

∂x

+ j

∂

∂y

+ k

∂

∂z

. (A.37)

The function U (r) is called potential energy. Taking into account Eq. (A.36) and sub-

stituting it into Eq. (A.34), we obtain the relationship between work W

and potential

energy U (r):

=−



∇U (r) ·dr =−



(2)

(1)



∂U

∂x

dx +

∂U

∂y

dy +

∂U

∂z



, (A.38)

318 Appendix A. Classical dynamics of particles and waves

where the function under the integral sign is the so-called total differential,dU (x, y, z),

of the function U(x, y, z). The expression for the work done takes the form

=−



(2)

(1)

dU(x, y, z) =−



(2)

(1)

dU(r) =−(U

−U

) =−

[

U (r

) − U (r

)

]

(A.39)

Thus, the work of conservative forces does not depend on the path, but depends on the

initial and ﬁnal coordinates, r

and r

, only. Moreover, the work of these forces is equal

to the difference of potential energies at the initial and ﬁnal positions of the particle, and

hence does not depend on the choice of the origin of the coordinate system for U (r). Let

us consider as examples of potential ﬁelds homogeneous and non-homogeneous electric

ﬁelds.

A.1.5 Kinetic and total mechanical energy

The total mechanical energy, E, of a particle is deﬁned as the sum of its potential energy,

U (r), and kinetic energy, K :

E = K + U. (A.40)

The expression for a particle’s kinetic energy, K , can be derived from the equation of

motion, Eq. (A.11), after multiplying its right-hand and left-hand sides by the displacement

vector, dr = v dt:

mv ·dv = F ·dr. (A.41)

This expression can be transformed into





= F · dr, (A.42)

because d(v

) = 2v · dv. The quantity

K =

(A.43)

is called the kinetic energy of a particle. If we integrate Eq. (A.42) from r

to r

, we will

obtain

− K

−



F ·dr. (A.44)

Here, K

= (mv

)/2andv

= v(r

) is the velocity of the particle at point i,wherei = 1, 2.

The force F under the integral sign is the sum of the conservative, F

, and non-

conservative (for example, friction), F

,forces:

F = F

+ F

. (A.45)

A.1 Classical dynamics of particles 319

The work of the conservative forces can be deﬁned as the difference of potential energies

of the particle. In accordance with Eq. (A.39),



· dr = U (r

) −U (r

), (A.46)

where U (r) is the potential energy of the conservative force. Therefore, Eq. (A.44)canbe

rewritten as

− K

−U



· dr. (A.47)

Equation (A.47) is called the work–energy theorem. The change in total mechanical

energy of a particle, E, is equal to the work of friction forces along the entire path of a

particle’s motion. In the absence of non-conservative forces, the total mechanical energy,

E = K + U , is conserved, i.e.,

= K

. (A.48)

During the process of a particle’s motion in a potential ﬁeld the kinetic energy of a particle

is transformed into its potential energy or vice versa:

− K

=−(U

−U

). (A.49)

A.1.6 Equilibrium conditions for a particle

The conditions of equilibrium play a very important role in the analysis of the motion of

a particle in a potential ﬁeld. The mechanical equilibrium of a body assumes satisfaction

of two vector conditions:



i=1

= 0, (A.50)



i=1

= 0. (A.51)

The sum of all forces and the sum of all torques acting on a particle must be equal to

zero. The ﬁrst of the above-mentioned conditions is required for physical bodies that can

be considered as particles (or material points). With Eq. (A.36) taken into account, the

equilibrium condition for a particle in the potential ﬁeld can be rewritten in the form

∇U (x, y, z) = 0. (A.52)

From the last expression, we can write the following expressions for the potential energy:

∂U

∂x

= 0,

∂U

∂y

= 0,

∂U

∂z

= 0, (A.53)

320 Appendix A. Classical dynamics of particles and waves

2min

2max

1max

1min

First potential well

Second potential well

Figure A.3 A potential

proﬁle with two stable

and x

) equilibrium

positions and one

unstable (x

) equilibrium

position.

which have to be satisﬁed at the position of equilibrium. For convenience the position of

equilibrium is often denoted by the coordinates x

, y

,andz

The position of equilibrium can be either stable or unstable. In the case of a stable

equilibrium position, when the particle is displaced out of the equilibrium, the force,

F =−∇U(r), that appears due to the dependence of U (r)onr will return the particle to

the state of equilibrium. In the case of an unstable equilibrium, when the body is displaced

out of the equilibrium, the force F =−∇U(r) will take the particle away from the state

of equilibrium. When the force depends only on a single coordinate (let us choose it to be

the x -coordinate) it is deﬁned as

F =−i

dU(x)

. (A.54)

From Eq. (A.54) it follows that for dU (x)/dx > 0 the force F is directed against the vector

i, and, in the case when dU(x)/dx < 0, F is parallel to the vector i.

In the one-dimensional case we can easily depict the function U (x) near its equilibrium

positions. In Fig. A.3 two positions of stable equilibrium are deﬁned by the coordinates x

and x

, and near these positions the function U(x) has the form of a “potential well.” At

the position x

the state of equilibrium is unstable. The directions of the forces that appear

upon displacement of a particle from the respective equilibrium positions are depicted by

arrows. It is clearly seen that the minimum of potential energy corresponds to the position

of stable equilibrium. The conditions of a stable equilibrium state at an arbitrary point

x = x

in the one-dimensional case have the form



x=x

= 0and



x=x

> 0. (A.55)

An unstable equilibrium state is deﬁned by the conditions



x=x

= 0and



x=x

< 0. (A.56)

A.2 Oscillatory motion of a particle 321

Figure A.4 Asimple

pendulum.

If the particle’s total energy is equal to its minimal possible energy, E

= U (x

), the

particle can only be at the position of equilibrium x

. For a given proﬁle of the potential

energy, the total energy of a particle, E, cannot be less than E

; otherwise its kinetic

energy, K , would be negative. With an increase of energy, a particle can oscillate near

its equilibrium position. For example, for the energy E =

E,whereE

E < E

,the

particle will move in the region of positive kinetic energy between points x

1min

and x

1max

At E = E

the particle is in the position of unstable equilibrium, x

. After leaving

the position of equilibrium x

, the particle may travel between the ﬁrst and second

potential wells in the region from x

2min

to x

2max

. A further increase of energy above E

leads to an increase of the interval of allowed coordinates, x. At energies E = E



,where

< E



< E

, the “classical” particle can be located only in one of two potential wells

near their equilibrium positions, x

or x

. In order to transfer the particle from the ﬁrst

potential well to the second well, it is necessary to supply the energy deﬁcit, i.e., the

energy difference between E

and E



. As will be shown later, for “quantum” particles this

ban is lifted.

A.2 Oscillatory motion of a particle

A.2.1 Simple harmonic motion

If a particle follows a motion that is periodic, i.e., repetitive in time, then

r(t + nT) = r(t), (A.57)

where T is a period and n is an integer number. Along each of the Cartesian coordinates

the particle undergoes a periodic motion, which is called an oscillatory motion.

The most famous example of oscillatory motion is the oscillation of a simple pendulum,

i.e., an oscillation of a point mass on a long massless inextensible string with length l

(Fig. A.4). Let the mass of the oscillating particle be m and let us derive the equation that

describes oscillations of such a pendulum. At an arbitrary displacement of a pendulum at

an angle α, two forces act on a particle – the weight, mg, and the tension of the string, F

322 Appendix A. Classical dynamics of particles and waves

Since the pendulum undergoes rotational motion with respect to the point of suspension

O, we can write the main equation of rotational motion (Eq. (A.28)) in the form of a

projection onto the axis of rotation:

= τ, (A.58)

where L and τ are projections of the angular momentum and torque of an oscillating

particle onto the axis perpendicular to the plane of the pendulum’s oscillations. Since the

line along which the tension force of the string, F

, is directed goes through the point of

suspension, this force does not create the torque. The torque that returns the particle to

the equilibrium position is created only by the particle’s weight, mg. The magnitude of

this force’s torque, τ , is equal to

τ =−mgl sin α, (A.59)

where the negative sign occurs because at any angle of displacement, α, the torque τ

is restoring, i.e., torque pushes the particle back to the equilibrium position. For the

pendulum shown in Fig. A.4, the position angle α is positive. Note that the displacement

angle, α, around a ﬁxed axis is considered as a vector directed along the axis of rotation

according to the right-hand rule (see Fig. A.4). The angular momentum according to

Eq. (A.25)is

L = mvl = ml

 = ml

dα

, (A.60)

where the linear velocity of a particle, v, along the arc of a circle of radius l is equal to

v = l (A.61)

and the angular velocity is deﬁned as

 =

dα

. (A.62)

After substitution of L and τ into Eq. (A.58), the last equation becomes

=−mgl sin α. (A.63)

Let us rewrite Eq. (A.63) in the standard form of the equation for a simple pendulum’s

oscillatory motion,

+ ω

sin α = 0, (A.64)

where

ω =



(A.65)