Squeezed coherent states and the one-dimensional Morse quantum system

Abstract

The Morse potential one-dimensional quantum system is a realistic model for
studying vibrations of atoms in a diatomic molecule. This system is very close
to the harmonic oscillator one. We thus propose a construction of squeezed
coherent states similar to the one of harmonic oscillator using ladder
operators. Properties of these states are analysed with respect to the
localization in position, minimal Heisenberg uncertainty relation, the
statistical properties and illustrated with examples using the finite number of
states in a well-known diatomic molecule.

Full text

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Squeezed coherent states and the one-dimensional
Morse quantum system
M Angelova1A Hertz2and V Hussin3
1Mathematical Modelling Lab, School of Computing, Engineering and Information Sciences,
Northumbria University, Newcastle NE2 1XE, UK
2D´ epartement de Physique, Universit´ e de Montr´ eal, Montr´ eal, Qu´ ebec, H3C 3J7, Canada
3D´ epartement de Math´ ematiques et de Statistique, Universit´ e de Montr´ eal, Montr´ eal, Qu´ ebec,
H3C 3J7, Canada
E-mail: maia.angelova@northumbria.ac.uk, anaelle.hertz@umontreal.ca,
veronique.hussin@umontreal.ca
Abstract.
studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic
oscillator one. We thus propose a construction of squeezed coherent states similar to the one
of harmonic oscillator using ladder operators.
respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical
properties and illustrated with examples using the finite number of states in a well-known
diatomic molecule.
The Morse potential one-dimensional quantum system is a realistic model for
Properties of these states are analysed with
1. Introduction
Coherent and squeezed states are known to be very important in many fields of physics. Coherent
states were discovered in 1926 by Schr¨ odinger [1], while squeezed states were introduced by
Kennard in 1927 [2]. However, these works were, in the main, ignored or forgotten until the
sixties, when these states became very popular and received a lot of attention from both fields,
mathematics and physics. Among many important papers, let us mention the works of Glauber
[3], Klauder [4, 5], and Nieto [6]. In the particular field of quantum optics, the books of Walls and
Milburn [7], Gazeau [8] and Rand [9] are very good reading which also consider the applications.
The study of squeezed states for systems admitting an infinite discrete spectrum, obtained as
a generalisation of coherent states, has been recently the center of much attention (see, for
example, [10, 11, 12, 13, 14]).
In modern developments, coherent states (CS) are standardly defined in three equivalent ways:
displacement operator method, ladder (annihilation) operator method and minimum uncertainty
method (for review see for example [6]). Initially defined for the case of the harmonic oscillator,
coherent states have been generalised for other systems. We can use, for example, the definition
of Klauder [15] saying that they are obtained as the following superposition of energy eigenstates
{|ψn?,n ∈ N}
ψ(z) =
?N(|z|2)
1
?
n∈I
zn
?ρ(n)|ψn?. (1)
arXiv:1111.1974v1 [math-ph] 8 Nov 2011

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The sum is taken over all the discrete values of n and the set I is usually infinite. The parameter
z is a complex variable in general, N is a normalization factor and {ρ(n),n ∈ N} is a set of
strictly positive parameters, usually depending on the energy of the system under consideration.
These last quantities correspond to a moment problem (see [15] for details).
For a quantum system which admits an infinite discrete spectrum {|ψn?,n = 0,1,...} and
ladder operators A−and A+acting on the energy eigenstates as
?
these coherent states are defined as eigenstates of A−. We thus get [15]
A−|ψn? =k(n) |ψn−1?, A+|ψn? =
?
k(n + 1) |ψn+1?, (2)
ρ(n) =
n
?
i=1
k(i),ρ(0) = 1. (3)
Note that the quantity k(n) is not unique and can be chosen to impose additional constraints
to the ladder operators. In particular, for the harmonic oscillator, we have k(n) = n,
[A−,A+] = I,Hho= ?ω(A+A−+1
2), (4)
and the expression (1) gives the usual coherent states.
Now, for a quantum system which admits a finite discrete spectrum like the one which involves
the Morse potential, various constructions of coherent states have been adapted [16, 17, 18, 19].
In a recent paper [20], we have used ladder operators [21, 22] to construct different types of
coherent states of the Morse potential and have compared them with the so-called Gaussian
coherent states [23]. In particular, such a construction has been inspired by the approach
mentioned above (see the formula (1)) but where the set I of values of n is now finite. The
coherent states are not exactly eigenstates of the annihilation operator A−but we have shown
[20] that, in practice, the last terms on the right hand side of the sum in (1) does not contribute
significantly. In some approaches (see, for example, [24]) these states are called pseudo-coherent
states.
To our knowledge, squeezed coherent states for the Morse potential have not been constructed.
One of the aims of this paper is thus to show that such a construction can be closely related
to the one for infinite spectrum systems. In fact, these states would be almost eigenstates of a
linear combination of the ladder operators. Another aim is to investigate the behaviour of those
squeezed coherent states with respect to localization, minimal uncertainty relation, dispersion
and also to compare their statistical properties with the ones of the harmonic oscillator system.
In Section 2 we give a review of relevant results on squeezed coherent states and minimal
uncertainty relations for a quantum system with infinite spectrum. In Section 3, starting with
the definition of the Morse model and its ladder operators, we define the corresponding squeezed
coherent states. We thus get two types of the states called oscillator-like and energy-like.
In Section 4, uncertainty relation, dispersion in x, localisation and phase-space trajectories
are compared for special values of the parameters associated with coherence and squeezing.
Statistical properties of the squeezed states of diatomic molecules are also investigated. We end
the paper with some conclusions in Section 5.
2. Squeezed coherent states for a quantum system with infinite spectrum
As in the case of harmonic oscillator, general squeezed coherent states [6], for a quantum system
with an infinite discrete energy spectrum, may be constructed as the solutions of the eigenvalue
equation:
(A−+ γA+)ψ(z,γ) = z ψ(z,γ), z,γ ∈ C. (5)

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The mixing of A−and A+is said to be controlled by a squeezing parameter γ and z is called
the coherent parameter. The coherent states are special solutions when γ = 0. Conditions on γ
must be imposed for the states to be normalisable.
Squeezed coherent states (SCS) based on su(2) or su(1,1) algebras [12, 13] and also direct
sums of these algebras with the algebra h(2) [14], have been constructed using group theoretical
methods. This involves, in particular, the operators displacement D and squeezing S similar to
the ones of the harmonic oscillator. In fact, for su(2) or su(1,1) algebras, k(n) is a quadratic
function of n.
More generally, equation (5) may be solved by using a direct expansion of ψ(z,γ) in the form
ψ(z,γ) =
1
?Ng(z,γ)
∞
?
n=0
Z(z,γ,n)
?ρ(n)
|Z(z,γ,n)|2
ρ(n)
|ψn?, (6)
with
Ng(z,γ) =
∞
?
n=0
, (7)
where ρ(n) = (3). Indeed, for the case γ ?= 0, inserting (6) into (5), we get a 3-term recurrence
relation
Z(z,γ,n + 1) − z Z(z,γ,n) + γ k(n) Z(z,γ,n − 1) = 0, n = 1,2,...
and without restriction, we take Z(z,γ,0) = 1 and thus Z(z,γ,1) = z.
For the harmonic oscillator, ρ(n) = n! and we get explicitly [14, 25]:
(8)
Zho(z,γ,n) =
[n
2]
?
i=0
n!
i!(n − 2i)!(−γ
2)iz(n−2i)= (γ
2)
n
2Hermite[n,
z
√2γ]. (9)
It is well-known that |γ| < 1 for the states to be normalizable in this case. In (9), we see that the
definition of the Hermite polynomials has been extended on the field C. These polynomials have
been shown to have interesting properties in terms of orthogonality, measure and resolution of
the identity [26].
Let us recall that for the harmonic oscillator, these states minimize the Schr¨ odinger-Robertson
uncertainty relation [27] which becomes the usual Heisenberg uncertainty relation (HUR) for γ
real. Indeed, we get
(∆xop)2=
2, (∆pop)2=
1
1 + γ−11
1 − γ−1
2
(10)
and
HUR = (∆xop)2(∆pop)2=1
4.(11)
we see that this implies the reduction of the ”quantum noise” on one of the observables while
increasing it on the other. In the following we will treat the case when the quantum noise
is reduced on the observable x because we will be concerned with a good localisation in the
position.
3. The Morse potential and different types of squeezed coherent states
The Morse potential quantum system is a realistic model for studying vibrations of atoms in
a diatomic molecule. Since this system is very close to the harmonic oscillator, the squeezed
coherent states will be constructed following the procedure given for the harmonic oscillator,
but we will deal with a finite number of eigenstates. We will show that the states constructed
in this way will thus be well localized for some values of the coherent and squeezing parameters.

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3.1. The model
The one-dimensional Morse model is given by the energy eigenvalue equation (see, for example,
[21])
?pop2
where mr is the reduced mass of the oscillating system composed of two atoms of masses m1
and m2, i.e.
mr=
variable x represents the displacement of the two atoms from their equilibrium positions, V0is
a scaling energy constant representing the depth of the potential well at equilibrium x = 0 and
β is the parameter of the model (related to the characteristics of the well, such as its depth and
width).
The finite discrete spectrum is known as
H ψ(x) =
2mr
+ VM(x)
?
ψ(x) = Eψ(x), (12)
1
1
m1+
1
m2. The potential is VM(x) = V0(e−2βx− 2e−βx), where the space
En= −?2
2mrβ2?n2, (13)
where
?n=ν − 1
2
− n = p − n,ν =
?
8mrV0
?2β2, (14)
and {n = 0,1,2,...,[p]}, with [p] the integer part of p =ν−1
the consecutive energies are not equally spaced. The following shifted energies
2. We see that for the Morse oscillator
e(n) =2mr
?2β2(En− E0) = ?2
0− ?2
n= n(2p − n) (15)
are useful for the construction of squeezed coherent states. Using the change of variable
y = νe−βx,(16)
we get the energy eigenfunctions, for the discrete spectrum in terms of associated Laguerre
polynomials, denoted by L2?n
n, as
ψν
n(x) = Nne−y
2y?nL2?n
n(y), (17)
where Nnis a normalization factor given by
Nn=
?
β(ν − 2n − 1)Γ(n + 1)
Γ(ν − n)
=
?
2β(p − n)Γ(n + 1)
Γ(2p − n + 1)
.(18)
Let us mention that p =ν−1
means that it is not an integer in practice and N is never zero as expected. But mathematically
speaking, if p is an integer, the last normalised state is ψν
usual Laguerre polynomial.
For many applications, it is convenient to introduce the number operator N such that
2
is related to the physical parameters of the Morse system. This
p(x) = e−y
2Lp(y) where Lp(y) is the
Nψν
n(x) = n ψν
n(x)(19)
and we see from (13) that the Hamiltonian operator can be in fact written as
H = −?2
2mrβ2(p − N)2.(20)

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3.2. Ladder operators
We use the definition (2) for the ladder operators of the Morse system where the set of
eigenfunctions {|ψn?} is finite and given by {ψν
the quantity k(n) is not unique and some choices have been considered in our study of coherent
states for such system [20].
Here, we will consider two different types of ladder operators.
”oscillator-like” with k(n) = n since it is associated to the h(2) algebra. The second type is
called ”energy-like” where k(n) = e(n) as given in (15). It is associated with a su(1,1) algebra.
In what follows, we will use subscripts o and e to denote an ”oscillator-like” or ”energy-like”
type, respectively.
Though our future calculations do not need the explicit form of the ladder operators, we give
them for completeness. Ladder operators for the Morse potential have been obtained in different
papers [19, 21, 28, 29]. For example, we get [21]:
n(x)} as in (17). As mentioned in the introduction,
The first type is called
A−
=
−[d
?
dy(ν − 2N) −(ν − 2N − 1)(ν − 2N)
K(N))−1[d
2y
+ν
2]
?
K(N), (21)
A+
=(
dy(ν − 2N − 2) +(ν − 2N − 1)(ν − 2N − 2)
2y
−ν
2], (22)
where K(n) is related to k(n) by
k(n) =n(ν − n)(ν − 2n − 1)
ν − 2n + 1
K(n). (23)
These relations are valid for any integer n in the interval [0,[p]−1]. Note that, for n = [p], we get a
permitted energy eigenstate ψν
[p](x) of the Morse potential but the action of the creation operators
on this state does not give zero in general. It gives a state which may not be normalisable with
respect to our scalar product. This problem has been already mentioned in some contributions
(see, for example, [21, 28]). For arbitrary p, the special choice
k(n) = n([p] + 1 − n),
[p]−1(x) =?[p]ψν
(24)
leads to A+ψν
in what follows but it is similar to the energy-like case.
The ”oscillator-like” ladder operators are now acting on the eigenfunctions (17) of the Morse
potential as
a−ψν
n−1(x),
since k(n) = n. We have identified A−with a−and A+with a+by taking
[p](x) = 0 and A+ψν
[p](x). It is not the case that we are considering
n(x) =√n ψν
a+ψν
n(x) =√n + 1 ψν
n+1(x),(25)
Ko(n) =
ν − 2n + 1
(ν − n)(ν − 2n − 1).(26)
The ”energy-like” ladder operators are obtained by identifying A±with the operators denoted
by J±taking
Ke(n) =(ν − 1 − n)(ν − 2n + 1)
(ν − n)(ν − 2n − 1)
Let us mention that we can in fact relate a±and J±. Indeed, we have:
J−= a−?2p − N, J+=
= Ko(n)(ν − 1 − n).(27)
?2p − Na+. (28)