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

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A.2 Oscillatory motion of a particle 323

x(t)

Figure A.5 The

oscillations of a particle

connected to a spring

along a smooth surface.

is the angular frequency of the pendulum’s oscillations. Equation (A.64) is valid at any

displacement of the pendulum from the position of equilibrium. However, its solution at

arbitrary values of the angle α cannot be described by elementary functions. Therefore,

let us consider oscillations with small amplitude when the angle α is much less than 1

(α  1) and sin α ≈ α. In this case, Eq. (A.64) becomes a standard form of equation for

small displacements of a simple pendulum:

+ ω

α = 0. (A.66)

The following function is the solution of Eq. (A.66):

α(t) = α

cos(ωt + φ), (A.67)

where α

and φ are the amplitude and phase of oscillations, respectively. This function

deﬁnes the angular position of the pendulum at any given time. Oscillatory motion deﬁned

by a function of this type is called simple harmonic motion. These oscillations have the

most simple form, from the mathematical point of view, because they can be described by

an elementary function of Eq. (A.67).

A.2.2 One-dimensional harmonic oscillator

Another example of harmonic motion is that of a particle connected to a spring (see

Fig. A.5). Three forces act on the particle: the gravitational force, F

, the normal force,

, and the spring force, F

. The net force is equal to the spring force because the forces

and F

exactly compensate for each other. Here we neglect the force of friction (see

the next section). According to Hooke’s law the spring force F

depends linearly on the

displacement x(t) from the equilibrium position x = 0 (we chose the x-axistobeinthe

direction of the force F

) as shown in Fig. A.5:

=−kx(t), (A.68)

where k is a spring constant. Then the equation of motion can be written as

= F

, (A.69)

324 Appendix A. Classical dynamics of particles and waves

=−kx. (A.70)

Introducing the angular frequency of oscillations,

ω =



, (A.71)

we arrive at the following differential equation:

+ ω

x = 0, (A.72)

which is the same as Eq. (A.66) for small-amplitude oscillations of a simple pendulum,

which are harmonic oscillations.

The general solution of Eq. (A.72) can be found in the form of any of the following

expressions:

x(t) =







iωt

+ A

−iωt

cos(ωt) + B

sin(ωt),

C cos(ωt + φ),

(A.73)

where the constants A

1,2

, B

1,2

, C,andφ are constants of integration. Since all three

solutions satisfy the same equation, these constants are functionally related and these

relations can easily be established using well-known trigonometric relations.

Let us ﬁnd ﬁrst the relationship between the coefﬁcients A

1,2

and B

1,2

. By substituting

the formulae

cos(ωt) =



iωt

+ e

−iωt



and sin(ωt) =



iωt

− e

−iωt



(A.74)

into the second expression of Eqs. (A.73) and regrouping terms with the same exponents,

we obtain

x(t) =





iωt



−



−iωt

. (A.75)

On comparing the last equation with the ﬁrst expression of Eq. (A.73), we get

(

− iB

)

and A

(

+ iB

)

. (A.76)

Inversely,

= A

+ A

and B

= i( A

− A

). (A.77)

A.2 Oscillatory motion of a particle 325

Let us deﬁne the relation of coefﬁcients C and φ with A

1,2

by rewriting the third expression

of Eq. (A.73) in terms of the exponents:

C cos(ωt + φ) =



i(ωt+φ)

+ e

−i(ωt+φ)



iφ

iωt

−iφ

−iωt

. (A.78)

On comparing this equation with the ﬁrst expression of Eq. (A.73), we get

iφ

and A

−iφ

. (A.79)

Inversely,

C = 2



and φ =





. (A.80)

By substituting Eq. (A.76) into Eq. (A.80) we can ﬁnd the relation of the coefﬁcients C

and φ with B

1,2

. The constants themselves are deﬁned from the initial conditions.

A.2.3 The energy of harmonic oscillations

For a particle that follows simple harmonic oscillatory motion, its velocity and accel-

eration are also given by harmonic functions. Choosing the dependence of a particle’s

displacement, x, from its equilibrium position as

x(t) = C cos(ωt + φ), (A.81)

we obtain for the velocity, v,

v(t) =

=−ωC sin(ωt + φ), (A.82)

and for the acceleration, a,

a(t) =

=−ω

C cos(ωt + φ) =−ω

x(t). (A.83)

From the last equation it follows that, for harmonic oscillations, the acceleration, a,is

directly proportional to the displacement, x, and its proportionality constant is always

equal to the square of the angular frequency with a negative sign, −ω

Let us deﬁne now the total mechanical energy, E, of the oscillating particle. A particle’s

kinetic energy is deﬁned by the expression

K (t ) =

(t)

mω

sin

(ωt + φ). (A.84)

326 Appendix A. Classical dynamics of particles and waves

The potential energy is related to the force by Eq. (A.36), which for the one-dimensional

case has the form

=−

=−kx. (A.85)

From Eq. (A.85) we obtain the quadratic dependence of the potential energy, U ,on

the displacement, x:

U (x ) =

. (A.86)

By substituting the displacement x(t), deﬁned by Eq. (A.81), into Eq. (A.86) we obtain

for the potential energy of an oscillating particle the relation

U (x ) =

(t)

mω

cos

(ωt + φ). (A.87)

We see that the kinetic as well as the potential energy varies in time with doubled frequency

(since 2 sin

β = 1 − cos(2β) and 2 cos

β = 1 + cos(2β)), whereas the total mechanical

energy, E, does not depend on time, t, and stays constant:

E = K (t ) +U (t) =

mω

. (A.88)

A.2.4 Damped oscillatory motion

The oscillations of a particle become damped if in the process of the oscillatory motion a

particle loses its energy. For a number of problems, e.g., the motion of a particle in viscous

media, the occurrence of damping in an oscillatory system can be taken into account by

introducing into Eq. (A.69)thedamping force, which is proportional to the velocity of a

particle, v = dx/dt:

=−kx − b

. (A.89)

Let us introduce new parameters, b/m = 2β and k/m = ω

,whereβ is the damping

factor and ω is the frequency of the particle’s oscillations at β = 0. As a result, Eq. (A.89)

is reduced to a standard form of the equation with damped oscillations

+ 2β

+ ω

x = 0. (A.90)

Let us seek the solution of this equation in the form

x(t) = Ae

λt

. (A.91)

A.2 Oscillatory motion of a particle 327

After substitution of x(t) into Eq. (A.90), we obtain the characteristic equation for ﬁnding

the parameter λ:

+ 2βλ + ω

= 0. (A.92)

This equation has two roots:

1,2

=−β ±



− ω

. (A.93)

Depending on the magnitudes of ω and β, we can distinguish two different types of

damped oscillatory motion: aperiodic,

ω ≤ β, (A.94)

and oscillatory,

ω>β. (A.95)

In the ﬁrst case both roots λ

and λ

are real numbers and both are negative. Such

a case corresponds to the strong damping when x(t ) exponentially decreases in time

(Eq. (A.91)). The second case, when ω>β, is more interesting for us. Moreover, for a

sufﬁciently distinctive oscillatory behavior an even stronger inequality has to be fulﬁlled:

ω  β. In this case both roots are complex numbers:

1,2

=−β ±i



− β

. (A.96)

For these two roots of the characteristic equation, we can write the general solution of

Eq. (A.90)intheform

x(t) = A

+ A

= e

−βt



it

+ A

−it



= Ce

−βt

cos(t + φ)

= C(t)cos(t + φ), (A.97)

where  =



− β

is the angular frequency of damped oscillations and C(t) = Ce

−βt

Since <ω, the period of damped oscillations, T = 2π/, is always greater than the

period of free oscillations, T

= 2π/ω.

In Fig. A.6 the graph of the time-dependent displacement, x(t), of a particle from its

position of equilibrium, x = x(0) = x

, is shown for the case of damped oscillations. At

t = 0, the displacement of a particle from the position of equilibrium is equal to

x(0) = x

= C cos φ. (A.98)

To characterize the rate of damping, it is convenient to introduce a number of quantities

apart from the coefﬁcient of damping, β, such as the time of relaxation, τ , and the

logarithmic decrement of damping, δ. The time of relaxation is equal to

τ =

, (A.99)

328 Appendix A. Classical dynamics of particles and waves

(t)

− βt

−

Figure A.6 Damped

oscillations.

and it deﬁnes the time interval during which the amplitude of oscillations decreases by

afactorofe(e≈ 2.718). The logarithmic decrement of damping, δ, is introduced in the

following way:

δ = ln



C(t)

C(t + T )



= ln



−βt

−β(t+T )



= ln e

βT

= β T. (A.100)

Taking into account that β = 1/τ , we obtain

δ =

, (A.101)

where N

= τ/T is the number of full periods for which the amplitude of oscillations

decreases by a factor of e. The smaller the logarithmic decrement of damping, δ,thelarger

is the number of oscillations that a particle makes during time τ , and the more evident the

oscillatory character of a particle’s motion.

The velocity of a particle that is oscillating with a damping can be found by taking the

ﬁrst derivative with respect to time of x(t ) as deﬁned by Eq. (A.97):

v(t) =

=−Ce

−βt

[

 sin(t + φ) +β cos(t + φ)

]

. (A.102)

If at t = 0 the particle’s velocity is equal to zero, then it follows from the last equation

that the initial phase of oscillations, φ, is equal to

φ =−arctan







. (A.103)

A.2 Oscillatory motion of a particle 329

The total energy of an oscillating particle, E = K +U , decreases after each period of

oscillations by the value of the work done by friction forces, A

=−b



x(t+T )

x(t)

v dx =−2βm



t+T

(t)dt. (A.104)

This energy is transformed into the internal energy of the medium, i.e., into the thermal

energy of the medium – heat.

If the damping factor is large, β ≥ ω, the roots of the characteristic equation Eq. (A.92)

are real, and the solution of Eq. (A.90) can be written in the form

x(t) = e

−βt

√

−ω

+ A

−

√

−ω

. (A.105)

The motion of a particle described by this solution is not oscillatory and there is only an

exponential approach of the particle to the equilibrium position. The velocity and energy

of a particle exponentially drop with time, t.

A.2.5 Small-amplitude oscillations

If a particle executes oscillatory motion in an arbitrary potential ﬁeld, then, near the

position of equilibrium, x

, the potential energy of a particle, U(x), can be expanded in a

Taylor series with respect to the displacement from the position of equilibrium:

U (x ) = U





x=x

(x − x

) +





x=x

(x − x

)





x=x

(x − x

)

+···. (A.106)

Here, U

= U (x

) is the value of the potential energy at the position of equilibrium. Since,

at the position of equilibrium, the sum of all forces acting on a particle must be equal

to zero, the coefﬁcient (dU/dx)

x=x

, which deﬁnes the force acting on the particle at the

equilibrium position, is equal to zero. The coefﬁcient (d

U/dx

)

x=x

in the expression for

the potential energy, U (x), is a coefﬁcient of proportionality for the quadratic displacement

of a particle from the equilibrium position. Therefore, this coefﬁcient is actually the spring

constant k, i.e.,





x=x

= k. (A.107)

In the case of small-amplitude oscillations we can neglect in Eq. (A.106) the terms of

expansion with powers greater than two. Then, the expression for the potential energy

will take the form

U (x ) = U (x

) +

k(x − x

)

. (A.108)

330 Appendix A. Classical dynamics of particles and waves

min

U(r)

E < 0

U(r) = A/r

– B/r

(r) = A/r

(r B/r

U(x)

< 0

−x

> 0

)b()a(

max

Figure A.7 Small-amplitude oscillations in a parabolic potential (a) and in the

potential deﬁned by Eq. (A.113)(b).

Such a representation of a potential energy is called the harmonic approximation.If

we choose the position of equilibrium at the origin of coordinates, x

= 0, and put the

potential energy at this point equal to zero (U (x

) = U (0) = 0), the expression for the

potential energy, Eq. (A.108), becomes equivalent to Eq. (A.86). We can conclude that, in

the case of an arbitrary dependence of U on x for small displacements of a particle from

a stable equilibrium position, the motion of the particle is always harmonic. The terms of

the expansion proportional to (x − x

)

and higher powers of the particle’s displacement

deﬁne the anharmonic effects in the oscillatory motion of a particle.

Since the potential energy of a particle in a ﬁeld of elastic forces is quadratic with

respect to the displacement of the particle from the equilibrium position, we can say that

its motion takes place in a parabolic potential (see Fig. A.7(a)). From Fig. A.7(a) you can

see that the motion of a particle at given total energy, E, takes place within the following

range of displacements:

−x

≤ x (t) ≤ x

, (A.109)

i.e., inside of the potential well where the kinetic energy is equal to

K = E −U ≥ 0. (A.110)

The particle cannot move outside of this potential well because at x > x

or at x < −x

the kinetic energy of the particle would have to be negative, which is impossible in classical

physics. Since at the point x

the inequality U(x ) < E holds, we have K (x

) > 0. At the

point x

the value of U (x

) is greater than E (U(x

) > E), so K (x

) < 0. At the points

of maximum displacement, x =±x

, the particle’s velocity becomes equal to zero and

K (±x

) = 0. At these points the particle’s velocity changes its sign to the opposite. For

this reason these points are called turning points. The turning points of the maximum

displacement are found from the condition

U (x ) = E. (A.111)

A.2 Oscillatory motion of a particle 331

Then, according to Eq. (A.86), the turning points are deﬁned as



. (A.112)

The point x = 0 is the particle’s position of stable equilibrium.

In real situations we very often have to solve the problem of ﬁnding the period of small-

amplitude oscillations in a given potential ﬁeld. Since the force acting on the particle in

such a ﬁeld is deﬁned by Eq. (A.36), by using the corresponding expansion in series of

the given function U (x) we can write the particle’s equation of motion. By limiting the

expansion of U (x) to the quadratic term we can solve the particle’s equation of motion in

the limit of small-amplitude oscillations and ﬁnd the period of such oscillations.

Example A.1. The potential energy of a particle with mass m is given by the function

U (r) =

−

, (A.113)

see Fig. A.7(b). Find the position of equilibrium, r

, the energy of the particle, U

,atthe

position of equilibrium, the angular frequency, ω, and the period, T , of the small-amplitude

oscillations of the particle.

Reasoning. The potential energy, U (r), given above deﬁnes the oscillatory motion of ions

in metals. The coordinate r = r

as shown in Fig. A.7 is the position of equilibrium of a

particle. The ﬁrst term in the expression for U (r)(Eq.(A.113)) is related to the forces

of repulsion that deﬁne the particle’s behavior at small distances from r = 0, i.e., to the

left of r = r

. The second term is related to the forces of attraction that are signiﬁcant at

large distances, i.e., at r  r

. At the position of equilibrium, r = r

, the potential energy,

U (r), reaches its minimum, U

(at the position of equilibrium the sum of repulsion and

attraction forces is equal to zero). Thus,





r=r

=−

= 0. (A.114)

From the last equation we get

and U(0) = U (r)

r=r

−

=−

. (A.115)

To deﬁne the frequency of oscillations, we ﬁnd the coefﬁcient k,usingEq.(A.107):

k =





r=r



− B



r=r

. (A.116)

The potential energy, U (r), near the equilibrium position r

can be written in the form

U (r) = U (r)

r=r





r=r

(

r −r

)

=−

16A

(

r −r

)

. (A.117)

332 Appendix A. Classical dynamics of particles and waves

The equation of the particle’s oscillatory motion in such a ﬁeld can be written in the

following form, if we take into account the relation F =−dU/dr:

=−

(

r −r

)

. (A.118)

The right-hand side of Eq. (A.118) was obtained by differentiating Eq. (A.117) with

respect to r. To ﬁnd the solution of Eq. (A.118), let us make the change of variable

r −r

= R. Then, the last equation can be rewritten as

=−kR (A.119)

+ ω

R = 0, (A.120)

where the angular frequency of oscillations

ω =



√

and the period of oscillations

T =

4π

√

A.2.6 Probability and probability density

of oscillatory motion

Let us consider a particle oscillating harmonically as x(t) = x

sin(ωt) in the given region

of one-dimensional space and analyze the probability of ﬁnding the particle at a speciﬁc

location, x . Since the period of oscillations is

T =

2π

and the time the particle remains within the given interval dx is equal to 2 dt (one dt for

each direction of motion during the period T ), the corresponding probability is the ratio

of the time which the particle spends in the given time interval to the period:

dP(x) =

2dt(x)



− x

. (A.121)

To obtain this expression, we carried out the following transformations:

= ωx

cos(ωt) = ω



− x

sin

(ωt) = ω



− x

(A.122)