Download Morse potential derived from first principles

Morse potential derived from first principles
Raimundo N. Costa Filho,1, ∗ Geová Alencar,2, † Bo-Sture Skagerstam,3, ‡ and José S. Andrade Jr.1, §
arXiv:1204.5391v1 [cond-mat.mtrl-sci] 24 Apr 2012
1
Departamento de Fı́sica, Universidade Federal do Ceará,
Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil
2
Universidade Estadual do Ceará, Faculdade de Educação, Ciências e Letras do Sertão Central,
R. Epitácio Pessoa, 2554, 63.900-000 Quixadá, Ceará, Brazil.
3
Department of Physics, The Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Centre for Advanced Study (CAS), Drammensveien 78, N-0271 Oslo, Norway
We show that a direct connection can be drawn, based on fundamental quantum principles,
between the Morse potential, extensively used as an empirical description for the atomic interaction
in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a
non-additive translation operator, whose action leads to a perfect equivalence between the quantum
harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way,
our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore
revealing a possible quantum origin for several related properties as, for example, the dissociation
energy of diatomic molecules and the deformation of cubic metals.
PACS numbers: 03.65.Ca, 03.65.Ge, 73.40.Gk
The quantum harmonic oscillator (QHO) is certainly
one of the most celebrated paradigms in quantum mechanics. Among its several important attributes, the
QHO can be solved exactly and has been consistently
used to approximate any potential function when Taylor
expanded around their minima till second order. Moreover, the fact that bosons can be conceptually modeled
in terms of a QHO readily explains its broad application, ranging from fundamental physics, in the description of a quantized electromagnetic field, to condensed
matter, for vibrational properties of molecules as well as
phonons in solids. However, anharmonic potentials are
very often required to mathematically represent physical phenomena. For instance, an adequate description of
the vibrational modes in diatomic molecules must necessarily allow for dissociation (i.e., bond breaking) of its
two bounded atomic nuclei. By increasing the energy of
the molecule, through heating for example, the system
starts to vibrate till eventually break down into two nonbounded atoms. This is an essential feature of diatomic
molecules that is not compatible with the QHO model.
As originally proposed by Phillip M. Morse in 1929 [1],
the so-called Morse potential provides a much better description for the potential energy of a diatomic molecule
than the QHO, being usually written as,
VM (x) = D(1 − e−αx )2 ,
(1)
where x is the distance between atoms, D is the well
depth related to the molecule dissociation energy, and α
is an inverse length parameter related to the curvature
of the potential at the origin. As such, this potential
has been frequently used as an empirical model for anharmonic interactions in the study of a large variety of
physical systems and conditions, including the rotating
vibrational states of diatomic molecules [2], the adsorption of atoms and molecules by solid surfaces [3], and the
deformation of cubic metals [4]. Figure 1 shows that, according to this potential, the energy difference between
levels gradually decreases as the level number n increases,
till it reaches zero, where no more bounded states are allowed.
Variants of the Morse potential have also been utilized to investigate the physical behavior of more complex
materials. As a typical example, the state-of-the-art for
atomistic modeling of semiconductor surfaces and interfaces is the Abel-Tersoff potential [5, 6] that has the form
of a Morse pair potential, but with the bond-strength parameter depending on its local environment. Also in the
investigation of thermal denaturation of double-stranded
DNA chains, the Morse potential has been successfully
applied to model hydrogen bonds connecting two bases
in a pair [7–10].
It is the purpose of this Letter to show that the Morse
potential emerges naturally as an effective interaction
when a particle is subjected to a harmonic potential in
a contracted space. This physical situation is substantiated here in terms of the following quantum operator for
non-additive translation [11, 12]:
hx|Uγ (ε)|ψi = ψ(x + ε(1 + g(γx))),
(2)
where there is no restriction on g(γx). The action of
Uγ (ε) on the bra vector hx| can therefore be expressed
as hx|Uγ (ε) = (Uγ (ε)† |xi)† = hx + ε(1 + g(γx))|. For
infinitesimal transformations we obtain that,
Uγ (δx) = 1 +
i
pγ δx,
~
(3)
and
d
i
hx|pγ |ψi = (1 + g(γx)) hx|ψi,
(4)
~
dx
where pγ is the momentum operator. Considering the
particular case where the function g(γx) = γx and a
2
finite displacement a, we can rewrite Eq. (2) as,
eγa − 1
,
hx|Uγ (a)|ψi = ψ xeγa +
γ
(5)
from which one can immediately recognize the action of
the dilation/contraction operator xd/dx. Moreover, as
defined in Eq. (2), the non-additive operator Uγ (a) corresponds to the infinitesimal generator of the q-exponential
function [13],
expq (a) ≡ (1 + (1 − q)a)1/(1−q)
(6)
with q = 1 − γ. Equation (6) represents a fundamental
mathematical definition for the generalized thermostatistics of Tsallis and its applications [14–21].
At this point, it is important to state that the momentum operator pγ is Hermitian with regard to the following
scalar product:
Z
dx
(ψ, φ) =
ψ ∗ (x)φ(x),
(7)
1 + γx
where the range of integration shall depend on the specific boundary conditions of the system under investigation. Equation (2) implies that the action of Uγ (a) is
unitary. Indeed, the measure of integration dx/(1 + γx)
is invariant under the action of the transformation, x →
y = xeγa + (eγa − 1)/γ, and the decomposition of the
unit operator takes the form,
Z
dx
1=
|xihx|.
(8)
1 + γx
The equation of motion for a particle in the xrepresentation of this dilated/contracted space corresponds to a time-dependent Schrödinger-like equation in
the form,
i~
∂
ψ(x, t) = Hψ(x, t),
∂t
η=
(9)
or, more explicitly, as
~2 ∂ 2
∂
ψ(x, t)−
ψ(x, t) = −(1 + γx)2
∂t
2m ∂x2
~2 ∂
ψ(x, t) + V (x)ψ(x, t). (11)
γ(1 + γx)
2m ∂x
However, it is more convenient to express Eq. (11) in
terms of a simple change of variables, as suggested by
ln(1 + γx)
.
γ
(12)
From this transformation, it is important to notice
that the “canonical coordinate” can be written as x =
(exp(γη) − 1)/γ. In this way, the finite interval [0, L] for
x corresponds to [0, L̄], with L̄ = ln(1 + γL)/γ, for the
variable η. Equation (11) rewritten in terms of the new
variable η becomes,
i~
∂
~2 ∂ 2
φ(η, t) + Veff (η)φ(η, t),
φ(η, t) = −
∂t
2m ∂η 2
(13)
where φ(η, t) = ψ(x(η), t) and Veff (η) = V (x(η)). Assuming that,
φ(η, t) = Φ(η) exp (−iEt/~),
(14)
the transformation (12) leads us back to a more familiar
version of the time-independent Schrödinger equation,
EΦ(η) = −
~2 d2
Φ(η) + Veff (η)Φ(η),
2m dη 2
(15)
If we now consider the problem of a standing wave in
a null potential, Veff (η) = 0, it follows that the standard
form of the plane wave solution is recovered in terms
of the transformed variable η, namely Φ(η) = e±ikη .
In the archetypal case of a harmonic oscillator, V (x) =
1
2 2
2 mω x , the transformed effective potential becomes,
Veff (η) =
where the Hamiltonian operator is H = p̂2γ /2m + V (x),
and the modified momentum operator can be written for
short as p̂γ = −i~Dγ , with Dγ ≡ (1 + γx)d/dx being a
deformed derivative in space. Equation (9) can then be
rewritten as,
2
∂
~
i~ ψ(x, t) = −
Dγ2 ψ(x, t) + V (x)ψ(x, t), (10)
∂t
2m
i~
the decomposition of the unit operator Eq. (8). More precisely, we can define a variable η through the differential
equation dη/dx = 1/(1 + γx) with boundary condition
η(0) = 0, whose solution is
mω 2 γη
(e − 1)2 ,
2γ 2
(16)
where ω is the frequency of the oscillator. Strikingly, by
identifying D ≡ mω 2 /2γ 2 and α ≡ −γ, we conclude that
Eq. (16) corresponds exactly to the expression (1) for
the Morse potential. To the best of our knowledge, this
is the first time that a connection based on fundamental
quantum principles is provided between this potential,
which has been widely utilized as a consistent description for the vibrational structure of diatomic molecules,
and the harmonic potential. A physical interpretation for
this correspondence can be made in terms of a positiondependent (effective) mass induced due to the presence
of a material body in the system [11] as, for example, the
case of electrons propagating through abrupt interfaces
in semiconductor heterostructures [23–27].
The wave function solution for the quantum Morse oscillator (QMO) has been previously determined as [1],
1
Φn (z) = An z s e− 2 z L2s
n (z),
(17)
3
20
and p must be calculated. Considering that the following
correspondence holds between expected values of a given
operator Ô in the x−space and η−space:
Z
D E
D E
(21)
Ôx = Ôη = dηψn (η)∗ Oη ψn (η),
γ=0.0
γ=0.1
γ=0.2
Veff (η)
15
10
x
and using a procedure similar to the one adopted in
Ref. [28], we obtain,
1
γ~
n+
,
(22)
hxin,x = −
mω
2
9
E
6
5
3
0
-5 0
0
-25
hpin,x = 0,
1
~
n+
,
x2 n,x =
mω
2
2
1
~γ 2
1
p n,x = m~ω n +
1−
n+
,
2
mω
2
D
γ=-0.25
5 10 15 20 25
η
-15
η
η
-5
5
FIG. 1. The effective potential given by Veff (η) = D(1 − eγη )2
for γ = 0 (solid line), γ = 0.1 (dashed line), and γ = 0.2
(dotted line). Here we use ω = 1 and adopt atomic units,
namely ~ = m = 1. As shown, the anharmonicity of the
potential depends on γ. The energy levels for γ = −0.25
are shown in the inset. The parameter D corresponds to the
dissociation energy of the diatomic molecule.
2
where z = νeγη , ν ≡
~, s = ν/2 − n − 1/2,
n2mω/γ
2s
−2s z
−z n+2s
Ln (z) = z e /n! d e z
/dz n is the generalized Laguerre polynomial [22], and the normalization
constant is given by,
s
n!
.
(18)
An =
Γ(ν − n)
The energies for the QMO can then be calculated as,
2
1
~2 γ 2
1
En = ~ω n +
−
n+
.
(19)
2
2m
2
This expression clearly indicates that, in the limit of
small values of γ, the QHO spectrum is recovered. One
should also note that the energy levels,
∆E = En+1 − En = ~ω −
~2 γ 2
(n + 1),
m
(20)
valid for 0 ≤ n ≤ ⌊mω/γ 2 − 1⌋, are no longer equally
spaced, and approach zero depending on γ and the quantum number n. In the main plot of Fig. 1 we show the
form of the potential (16) for different values of γ, while
in the inset the energy levels are depicted for γ = −0.25.
The potential is symmetric in γ, i.e., V (γ) = V (−γ). The
energy difference between levels decreases as the quantum number increases.
Finally, we show that an uncertainty relation for the
QMO can be disclosed in the framework of our theoretical approach. In order to do this, expected values for x
(23)
(24)
(25)
so that the uncertainty relation for the QMO can be written as,
γ2~
1
1
1−
n+
.
(26)
∆x∆p = ~ n +
2
mω
2
For the ground state, this expression is identical to the
one previously obtained in Ref. [11], namely ∆x∆p ≥
(~/2)(1 + γhxi). The uncertainty in the QMO is therefore lower than in the QHO due to the anharmonicity of
the Morse potential. Note that in both expressions for
energy (19) and uncertainty (26) the parameter γ, which
defines whether the space is contracting (γ < 0) or dilating (γ > 0), appears squared. Consequently, for the
harmonic potential in a deformed space, the energy is
symmetric under contraction or dilation.
It is interesting to note that the Morse potential can
be associated to a two-dimensional harmonic oscillator
in polar coordinates for ω = 1, and considering atomic
units, ~ = m = 1. In fact, it is this relationship that
allows one to solve the QMO using ladder operators [29],
and study this problem through supersymmetry [30]. As
a consequence, our approach also provides a conceptual
basis for the map between a 1D-QHO and a 2D-QHO in
polar coordinates. This mapping is the result of the new
commutation relation for x̂ and p̂ generated through the
translation operator introduced in [11] and used here.
Previous studies have focused on plausible modifications on the position momentum [31–33], so that a minimum length and momentum could be defined for quantum theory. In particular, Quesne et al. [34] have shown
that, if some special generalized deformed commutation
relations are employed (e.g., [−e−x , p] = i[e−x +βp2 ]), the
Morse potential can be obtained as an effective potential
of the theory. In all these studies, however, modified commutation relations are introduced in an ad hoc. manner,
i.e., they are not obtained from a first principle mechanism, like the non-additive translation operator employed
in this work. Another important point here is that the
4
parameter γ, responsible for the dilation/contraction in
the translation, corresponds exactly to the minus value
of the α parameter for the Morse potential. In the particular case of the Hydrogen molecule, for example, the
numerical value of this parameter is γ = −α = −1.4 a.u.
In summary, we have shown that a one-dimensional
harmonic potential in a space deformed by the action of
the operator defined as in Eq. (2) can be equivalent to
the Morse potential in a regular space. As the particle
travels in a different way in the deformed space, it feels
the harmonic potential as a Morse potential in regular
space. This equivalence is achieved when g(γx) = γx,
namely for the case in which the translation occurs as a
contraction in a deformed space, γ < 0. Such a physical framework has perfect analogy with the behavior
of a position-dependent (effective) mass particle on a
non-homogeneous substrate, typified by electrons moving
through abrupt interfaces in semiconductor heterostructures. In this particular situation, the anharmonic feature of the Morse potential emerges naturally from the
standard quantum harmonic oscillator. We thus conclude
that our study, based on a non-additive translation operator, provides a first principle explanation for anharmonic properties.
We thank the Brazilian Agencies CNPq, CAPES,
FUNCAP and FINEP, the FUNCAP/CNPq Pronex
grant, the National Institute of Science and Technology
for Complex Systems in Brazil, the Centre for Advanced
Study (CAS) in Norway, and the Norwegian Research
Council for financial support.
∗
†
‡
§
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[email protected]
[email protected]
[email protected]
[email protected]
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