In this paper we give an alternative treatment of the Schrodinger equation with the Morse potential, which based on the exact summation of the Feynman perturbation series in its original form. Using Fourier transform we establish a recurrence equation between terms of the perturbation series. Finally, by the inverse Fourier transform and some technical tools of the ordinary differential equations of the second order, we can compute the exact sum of the perturbation series which is the Green’s function of the problem.
In quantum mechanics, the class of potentials for which Schrödinger equation can be exactly solved has been extended considerably by using different methods. The popular and widely one used in quantum mechanics is the perturbation theory leading to solve the problems approximately. Furthermore, among problems that can be exactly solved, there are few whose solutions can be obtained exactly by summing up the perturbation series in the path integral formalism [
The Morse potential is one of the important potentials in physics, which raises many interests in many areas specialy in molecular physics and is used for the description of the interaction between the atoms in diatomic molecules. The Schrödinger equation for the Morse potential has been solved exactly or studied by different methods recently, for example, [
In the paper [
In this work, we will use the same technique in [
We are interested to calculate the propagator, say the Green’s function relative to the one-dimensional Morse potential:
which can be written as:
where
where
We can show that the Feynman propagator takes the form:
where:
And
Taking the Fourier transform of
we write this last formula as:
where:
and using Equations (1) and (9), then (8) becomes:
(10)
we take now the Fourier transform on the end point
i.e.
where:
and
From Equation (12), we see that all termes
where we note
Let now compute the first and the second terms of Equation (12):
and
and so on, we can see that all terms
and if we bring together all terms in power of
where the coefficients
or:
with
Now noting the series in Equation (17) by:
we can easily check that
Here we have to note that this equation is equivalent to those governing Green’s function itself but written in an other form where we have put
Return now to Equation (17) and if we take the inverse Fourier Transorm on
Then if we note by:
we see that:
and with the Fourier transform properties, we have:
Then from these last formulas and the recurrence formula of
with
where the coefficients
with
where
Finally by recurrence, we can prove that:
where the coefficients
with
Then from Equations (29) and (33) we have:
we have to note that
Knowing that if the following limits exist:
then
So in that case and from the formulas (35), (25), we are able to write the Green’s function
Knowing that the generating functions of
then if we use the recurrence formula (34) it’s easy to deduce that these generating functions are given by:
where
and they satisfy respectively the following differential equations:
hence we can conclude that the Green’s function
where
for which it’s obvious that:
from this formula and with the same way as above, if we take the Fourier transform on initial point
then from Equation (42) and this last formula (44) we have for
and
for which Equation (44) i.e.
(47)
Knowing that
which is the same as the following differential equation:
for
and
which are two linearly independent solutions of Equation (49), and since
and
Finally we get that the Green’s function for Morse potential takes the form:
and since
where
In this work, we have calculated the Green’s function for the Morse potential using the perturbation method in the path integral formalism. This contribution concerns, for the first time, the calculation of the energy Green’s function of the system by summing exactly the perturbation series with the introduction of the Fourier transform and some results concerning the Green’s function of the ordinary differential equations of the second ordre. We will consider a generalization of this method specialy for other special potentials in the exponential form.