The Morse Oscillator’s Free Motion as the One-
Dimensional Analogue for the Kepler Problem Volume 47- Issue 1
Valery Astapenko1* and Mikhail Mutafyan2
1Moscow Institute of Physics and Technology (National Research University), 9, Institutskiy Pereulok, Russia
2Sheridan College Institute of Technology and Advanced Learning, Canada
Received: October 31, 2022; Published: November 07, 2022
*Corresponding author: Valery Astapenko, Moscow Institute of Physics and Technology (National Research University), 9,
Institutskiy Pereulok, Dolgoprudny, 141701, Russia
We obtained the expressions for Morse oscillator classical motion with explicit dependence
on initial coordinate and velocity. These formulas provide the easy comparison
with results of numerical calculations, and in addition to previous considerations,
they take into account the possible different sign of initial velocity of the oscillator.
The analogy between the free motion of a Morse oscillator and motion of a particle in
gravitation field (i.e. the Kepler problem) is drawn up.
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It is well known that the harmonic oscillator model can be employed
for weakly excited mechanical systems. When excitation
grows, anharmonicity begins to be noticeable and more sophisticated
models should be used. One of these models is the Morse
oscillator which is especially useful for the description of diatomic
molecule oscillations. For example, the experimental value of the
dimensionless the anharmonicity parameter for the CO molecule
is xe = 0.00612. The anharmonicity parameter serves as a measure
of the deviation of the real oscillator’s discrete spectrum from the
equidistant approximation. In the Morse model, xe = 0.00608, so
the relative error is only 0.65%. The Morse model is mostly used
in quantum mechanics. Particularly, the exact wave function of a
particle in the Morse potential is well-known for many decades [1].
Classical applications of the Morse model are not so numerous. To
our knowledge, the first work with the exact analytical solution for
the Morse oscillator’s classical motion is paper [2]. This problem
was also considered 30 years later [3]. In these papers, similar expressions
were obtained while using slightly different approaches.
Regardless of the abovementioned articles, we have also found
analytical expressions for the free motion of the Morse oscillator
[4]. Our formulas were obtained using the energy conservation law
without solving any differential equations. They also were written
in another form when compared with the results of [2,3]. Below we
introduce our consideration of the problem which provides some
additions to previous results [2,3]. Moreover, we demonstrate the
analogy between the free motion of a Morse oscillator and motion
of a particle in a gravitation field.
The potential energy of Morse oscillator is given [1] by the following
formula:
where D is the binding energy, k is the parameter of the
potential, x is the displacement of the oscillator’s coordinate
from its equilibrium position. In the case of a diatomic molecule,
, where r and re are the current and the equilibrium
distances between the nuclei, respectively. While the displacement
from the equilibrium position is small, when x < 1/k , the Morse
potential function transforms into the parabolic dependence. This
relation of the potential energy on the coordinate is typical for a
harmonic oscillator:
. In Figure 1, the graph of the Morse potential energy is plotted using the parameters
of a carbon monoxide (CO) molecule, together with its corresponding
harmonic approximation. Comparing the graphs of the
Morse potential and the harmonic potential energies, the Morse potential
energy’s asymmetrical shape is remarkable. This shape has
the horizontal asymptote U = 0 as x tends to infinity, x > 1/k
.This asymptote divides the energy spectrum of the oscillator into
two parts: the positive part with U > 0 and the negative part with
U < 0 . The negative part of the spectrum corresponds to the oscillator’s
finite motion, and on the contrary, the positive part of the
spectrum corresponds to the infinite motion. When the motion is
infinite, the coordinate increases indefinitely, and the oscillations
turn into expansion. If we use the Morse potential energy to describe
a diatomic molecule, the negative part of the energy spectrum
will correspond to the oscillations of the atoms forming a molecule
in a limited space. The positive part of the energy spectrum
in this case will correspond to the dissociated state of the molecule,
in which the distance between atoms grows up to infinity. Within
such model, the binding energy D receives the meaning of the molecule’s
dissociation energy. Hence, the Morse potential describes
both the atoms oscillatory motion and their expansion.
Figure 1 The Morse potential energy plotted using the parameters of a CO molecule (solid thick curve) and its corresponding harmonic approximation (dash-dotted curve).
Consider the free motion of a Morse oscillator for given total
energy ε measured from the bottom of a potential well (Figure
1). It is convenient to introduce a dimensionless displacement of
the coordinate from its equilibrium position, y = k x , and dimensionless
time,
is own frequency of the oscillator in harmonic approximation). In the future, for brevity, we will call
these variables simply the coordinate and time.
Using such definitions, the Morse oscillator’s motion equation
can be written as the following:
Note that the dimensionless Eq. (2) does not contain the
binding energy D and the parameter of the potential k , so it is
universal for the given type of potential energy. To determine the dependence y(τ), it is convenient to use the law of conservation of energy, which in dimensionless variables can be written as the
following equation:
where
is dimensionless energy. Here, the first
term on the left side of the equation corresponds to the kinetic
energy, while the second and the third terms represent the
potential energy. The solution of (3) depends on the magnitude
of the dimensionless energy
. Three cases, which correspond to three motion modes, can be differentiated: the first one, when
, the second, when
, and the third, when
. Below it will be demonstrated that the first case corresponds to the finite motion, i.e., oscillations of the Morse oscillator, while the latter two cases correspond to the infinite motion, i.e., the
expansion mode. For a finite movement which corresponds to the
oscillatory mode,
. Solving (3), the following dependence is obtained [3, 4]:
where τ0 is the integration constant (which we will later
assume is zero) and
.
The (4) yields the dependence y(τ) in an implicit form. To
obtain the explicit form of the function y(τ), the integral on the
right-hand side of (4) needs to be calculated and then y needs to
be expressed via the dimensionless time τ . The resulting formula
is
is the initial phase of the oscillations dependent on initial
displacement
.
The sign in front of the square root depends on the sign of
initial velocity of the oscillator
.
The resulting expression is valid while the dimensionless energy
. This corresponds to the negative part of the energy spectrum. The harmonic approximation is valid for low excitation
energies of the Morse oscillator, i.e., for
. The
Eq. (6) then turns into the well-known expression for free oscillations
of a harmonic oscillator. Using the dimensionless variables, it
can be written in the form:
From (5) and (7) it follows that the oscillation period for the
Morse oscillator depends on the energy and is given by the expression
Figure 2 The Morse oscillator’s motion for various values of dimensionless energy: ε~ = 0.1 (solid curve), ε~ = 0.3 (dotted curve), and ε~ = 0.6 (dashed curve) and for the negative initial velocity.
It should be noted that this is the period of anharmonic
oscillations. The motion of the Morse oscillator, defined as the
dependence of the dimensionless coordinate on the dimensionless
time, are shown in (Figure 2) for different values of the dimensionless
energy. It is noticeable that when the energy is low, the
oscillations are symmetrical in accordance with the Eq. (7) which
describes the harmonic oscillator’s motion. With energy increasing,
the amplitude of oscillations scales up, the motion becomes
asymmetric and anharmonic and oscillation period increases. This
anharmonicity is explained by the Morse potential energy’s asymmetry
with respect to the equilibrium coordinate (x = 0 in (Figure
1)) – this asymmetry does is not exist in the harmonic oscillator’s
potential energy.
The Eq. (8) suggests that with the increasing energy, the period
also increases, and with ε →D, T (Morse) →∞ Thus, upon
reaching the boundary of the negative part of the spectrum where
, the periodic movement of the Morse oscillator becomes
aperiodic. The law of motion for the energy laying on this boundary,
ε = D, following from (4) in the dimensionless variables is
given by the following formula:
where
is the value of the dimensionless coordinate at the initial moment of time. It is required that ln
so that the radicand in the right side of equality (9) is non-negative.
The sign plus in (9) relates to positive initial velocity and
sign minus to the negative one. When the time tends to infinity,
τ →∞ , the dimensionless coordinate of the Morse oscillator for
increases logarithmically: y ∝ ln τ . In this case, obviously,
the dimensionless velocity
, i.e., it decreases to zero at the infinity, so that the oscillator’s energy reduces to zero.
For the energies in the positive part of the spectrum where
, the calculation of the integral in (4) leads to the following law of
motion:
Figure 3 The Morse oscillator’s infinite motion for various dimensionless energies and different signs of initial velocity: solid curve – ε~ =1, 0 0 y < , dotted curve – ε~ =1, 0 0 y > ; dashed curve – ε~ =1.5 , 0 0 y < , dotted-dashed curve – ε~ =1.5 , 0 0 y > .
where
is the constant determined by the initial conditions, and
is the dimensionless velocity at infinity. Upper sign
in (11) corresponds to negative initial velocity and lower one to
the positive initial velocity. From the (10) it follows that when time
tends to infinity
, the oscillator’s coordinate
. This suggests that when the Morse oscillator’s total energy is
positive, i.e., when
, the expansion happens linearly in time. The time dependences of the Morse oscillator’s infinite
motion are demonstrated in Figure 3. For these graphs, the initial
coordinate is assumed to be equal to zero, y0 = 0. The solid
and dotted curves for
describes the scattering according to the logarithmic law of (9) for a sufficiently long time, τ ≥ 3. In this case,
, and the energy vanishes at the infinity. The dashed and dotted-dashed curves correspond to the
expansion for relative energies greater than one,
. It can be
seen that in these cases the dependence y(τ) becomes linear
when time tends to infinity, i.e., the movement occurs at a constant
speed. This follows from (10), as mentioned above. For the negative
initial velocity, the function y(τ) has a minimum. Further
analysis shows that with the increase of the initial coordinate, this
minimum shifts to the larger values of τ. It should be noted that
for the comparison of analytical formulas with numerical solutions
for infinite motion one can use the following initial conditions for
velocity:
, for the given energy
and coordinate y0 ; here
is normalized
potential energy as a function of dimensionless coordinate.
The three described modes of Morse oscillator’s one-dimensional
motion have an extended analogy in the three-dimensional
case. Their counterpart is the motion of a particle in a three-dimensional
Coulomb attraction field or in a gravitational field. Any
movement in such a field can occur in three modes: in an ellipse
when the energy is negative, in a parabola when the energy equals
to zero, or in a hyperbola when the energy is positive. This motion
is known as the Kepler problem. The role of the dimensionless
energy
in the case of the Kepler problem is determined by the square of the orbit’s eccentricity e2 :
Indeed, in the Kepler case, the total energy
, while in the case of the Morse oscillator,
. As it is known [5], when e2 < 1, the movement occurs on an ellipse, when e2 = 1 – on a parabola, and when e2e2 > 1 – on a hyperbola.
It is obvious that in this model, e > 0 . These movement types
are in accordance with the abovementioned Morse oscillator’s free
motion modes. For the period of motion along an ellipse, we have
, and for the period of Morse oscillations, we have
. When the eccentricity
or the dimensionless energy approaches one, the period of motion
tends to infinity for both models, although according to different
laws. We would like to also note that if the eccentricity is zero in
the Kepler problem, e = 0 , the motion happens along a circle.
However, if
, the coordinate of the Morse oscillator turns to zero in accordance with the (5). The circle in the Kepler problem
converts into a single point for the Morse oscillator. When moving
along a hyperbola when
, we have the following asymptotic dependence for the radius vector in the Kepler problem when
time tends to infinity:
(here m is reduced mass of Kepler problem). The movement hence occurs
at a constant speed, same as in the expansion case for the Morse
oscillator when
, in accordance with (10). Thus, the free motion modes of the Morse oscillator for both oscillations and
expansion can be considered as a one-dimensional analogue of the
two-dimensional motion in the Kepler problem. These motions
have the same qualitative dependencies, though are described by
different formulas. It is worth noting that for both oscillations and
expansion, relatively simple explicit expressions were obtained
to describe the motion. This explicitness and simplicity are not
always possible for other types of potential [4,5].
The analytical formulas describing the Morse oscillator’s free
motion in dimensionless variables have been derived and analyzed.
In addition to the previous results [2,3] we took into account different
signs of the initial oscillator’s velocity. Further to that, our
formulas explicitly depend on the initial conditions, which makes
it easy to compare them with the results of a numerical solution
to this problem. The study involved three motion modes: oscillatory,
infinite with zero energy, and infinite with positive energy. The
analysis involved motion characteristics of each of these cases. For
the oscillatory mode, the relation between the period of a Morse oscillator
and its energy has been established. The analogy between
the Morse oscillator’s free motion and the motion of a particle in a
gravitational potential, i.e., the Kepler problem, has been demonstrated.