Matrix elements for the morse and coulomb interactions

Scientia Magna, June, 2007 by M. Enciso-Aguilar, J. Lopez-Bonilla, I. Toledo-Toledo

Abstract We show the usefulness of the hypervirial theorem to obtain the matrix elements (m|[e.sup.-[gamma]au]|n> for the Morse potential. Besides, it is known that the hydrogenlike atom can be studied as a Morse oscillator, then here we prove that these fact leads to an interesting method to calculate <n[l.sub.2|[r.sup.k]|n[l.sub.1]> for the Coulomb interaction.

Keywords Hypervirial theorem, coulomb and Morse potentials, langer transformation, matrix elements.

[section] 1. Introduction

In [1] it was applied the hypervirial theorem (HT) to determine matrix elements for the one-dimensional harmonic oscillator, here we shall employ the HT to obtain (m|[e.sup.-[gamma]au]|n> for the Morse field [2]. An important aspect of the HT is that it not need explicitly the wave function, it uses only the potential V and the corresponding energy levels En. In Sec. 2 the HT permits: a). to show that (m|[e.sup.-[gamma]au]|n> = (m|[e.sup.-2au]|n> , b). to calculate <m|u|n> and (m|[e.sup.-[gamma]au]|n>, [gamma] = 3, 4,..., if we know (m|[e.sup.-[beta]au]|n>, [beta] = 1, 2. The Sec. 3 has the general expressions of [3,4] for (m|[e.sup.-[gamma]au]|n>, with the comment that yet [5] it is not verified the total equivalence between the result of [3] and the associated relation of [4].

For the hydrogenic atom its radial wave function 1/r [g.sub.nl] depends of the principal (n) and orbital (l) quantum numbers, which are associated to eigenvalues for energy and angular momentum, respectively. Lee [6] showed that the Langer transformation [7] permits to study a non-relativistic hydrogenlike system as a vibrational Morse oscillator (MO), such that n gives the parameters of the Morse well and l determines an energy level in these well. In Sec. 4 we exhibit this result of Lee.

In according with [6] the function [g.sub.nl] is proportional to the corresponding (MO) wave function, which means that the matrix elements <n[l.sub.2]|[r.sup.[??]]|n[l.sub.1]> of the hydrogenic atom are equivalent to <[N.sub.2]|[e.sup.-[gamma]u]|[N.sub.1]>, [gamma] = [??] 2, of its MO. Thus the knowledge on Morse matrix elements can be used to determine <[r.sup.[??]]> for the Coulomb potential. In Sec. 5 we apply this approach to obtain <n[l.sub.2]|[r.sup.[??]]|n[l.sub.1]>, [??] = integer [greater than or equal to] -2, without factorization techniques [8,9] as in [10]; we reproduce as particular cases the elements <nl|[r.sup.[??]]|nl>, [??] = [ or -] 1, [ or -] 2, deduced analytically by Landau-Lifshitz [11].

[section] 2. Hypervirial theorem

One aim of our work is the calculation of matrix elements for the Morse potential:

<m|f(r)|n> = [[integral].sup.[infinity].sub.0] [psi].sub.m] f (r)[[psi].sub.n]dr, (1)

where 1/r [[psi].sub.n] is the radial wave function satisfying the Schrodinger equation (in natural units m = [??] 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2)

The analytical procedure employs the explicit formulae of [[psi].sub.n] and f(r), and it makes directly the integral (1); however, in [1] we see that the HT evaluates (1) [for the harmonic oscillator] without the explicit form of the wave function. The Schrodinger equation has all information on our quantum system, and the HT has a part of these information which permits to study (1) without the explicit use of [[psi].sub.n].

From (2) it is easy to obtain the HT [1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3)

where the prime means d/dr. We note that, in general, (3) ask us to know the energy spectrum corresponding to potential V (r). Here we consider the Morse interaction [2,12] which represents an approximation to vibrational motion of a diatomic molecule.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4)

such that D is the dissociation energy (well depth), [r.sub.0] is the nuclear separation, and a is a parameter associated with the well width, being a/2[pi][square root of 2D] the frequency of small classical vibrations around [r.sub.0].

Now we shall make examples for particular functions f(r) to illustrate how the HT (3) gives information on matrix elements.

I. f(r) = r - [r.sub.0].

Then from (3) and (4) we deduce that:

2aD<m|[e.sup.-au] - [e.sup.-2au]|n> = [([E.sub.m] - [E.sub.n]).sup.2]<m|u|n>, (5)

with two cases:

a). m = n.

Thus (5) implies an identity for diagonal elements:

<n|[e.sup.-au]|n> = <n|[e.sup.-2au]|n>, (6)

and we observe that here (6) was obtained without the explicit knowledge of [[psi].sub.m] and En: In Sec. 3 we shall employ the formulae of [3,4] to show (6) with <n|[e.sup.-au]|n> = 1/k (k-2n-1), which was demonstrated by Hu[alpha]aker-Dwivedi [9] with the factorization method.

b). m < n.

The relation (5) leads to:

<m|u|n> = 2aD[([E.sub.m] - [E.sub.n]).sup.-2]<m|[e.sup.-au] - [e.sup.-2au]|n>, (7)

which means that all elements <m|u|n> are determined if we know <m|[e.sup.-[beta]au]|n>, [beta] = 1, 2. From the expressions of [3,4] we have the values for m [less than or equal to] n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (8)

where [GAMMA] denotes the gamma function, then (7) and (8) imply the result:

<m|u|n> = k/a[[(m - n)(k - n - m)].sup.-1], m < n (9)

deduced analytically by Gallas [13] without the HT, he uses explicitly [[psi].sub.n] and the following non-trivial identity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (10)

in this work we not need (10). The equation (6) also is consequence from (8) when m = n. In [8] it is proved (9) via ladder operators.

II. f(r) = [e.sup.-[gamma][alpha](r-[r.sup.0])]], [gamma] = 1, 2,...

Therefore from (3) and (4) we obtain that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (11)

thus, by example, if [gamma] = 1 the (11) gives us:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (12)

and if we put (4) and (8) into (12) it results the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (13)

besides it also can be deduced from expressions of [3,4]; (13) shows that <m|[e.sup.-3au]|n> is determined if we have the elements (8), which too give us <m|[e.sup.-4au]|n> when [gamma] = 2 into (11), etc., then it is evident the usefulness of the HT.

[section] 3. Matrix elements <e.sup.-[gamma]au]>

Here we exhibit general formulae for <m|[e.sup.-[gamma][alpha](r-[r.sub.0])|n>, [gamma] = 1, 2,... In fact, Vasan-Cross [3] use analytical techniques to determine these elements, obtaining thus the following expression for m [less than or equal to] n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (14)

which permits to verify (6) and (8). On the other hand, Berrondo et al [4] employ the relationship between Morse potential and the two-dimensional harmonic oscillator to deduce the corresponding relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (15)

which also reproduces (6) and (8), that is, we have the equality of (14) and (15) for [gamma] = 1, 2, however, yet it is an open problem [5] to show that both expressions are totally equivalent for any [gamma].

[section] 4. Hydrogenlike atom as a Morse oscillator

Here we exhibit the result of Lee [6]: The motion of an electron into the Coulomb field generated by a nucleus with charge Ze, is equivalent to the vibrational dynamics of a MO.

It is very well known [11] that the radial wave function 1/r[g.sub.nl] satisfies the Schrodinger equation (in natural units [??] = m = 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (16)

where n = 1, 2,..., and l = 0, 1,..., n - l. Now the quantities r, [g.sub.nl] are changed to u, [[psi].sub.N] via the Langer transformation [6,7]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (17)

with b = 4[pi][[epsilon].sub.0]/Z[e.sup.2], then (16) adopts the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (18)

which is the Schrodinger equation for a MO [3,9] with parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (19)

thus each n generates one MO with width a = 1, depth D = [n.sup.2]/2 and vibrational frequency a/2[pi][square root of 2D] = n/2[pi]. Finally, the value of l determines the eigenstate [[psi].sub.N], N = n-l-1, with energy E = - 1/2 [(l 1/2).sup.2].

[section] 5. Matrix elements for the Coulomb potential

Other aim of our work is the calculation of the matrix elements:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (20)

The factorization method [8-10] calculates (20) using ladder operators for the proper states [g.sub.n]l; the analytical approach [11] employs the explicit expression of [g.sub.nl] and determines directly the integral (20). Here we apply the Langer transformation [6,7] to obtain (20) via the relationship between the Coulomb and Morse interactions.

In fact, if we put (17) into (20):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (21)

with [N.sub.j] = n - [l.sub.j] - 1, j = 1, 2 and [gamma] = [??] 2 = 0, 1, 2,..., which means that any [r.sup.[??]] for the Coulomb potential is proportional to a matrix element of the corresponding MO. The elements <e.sup.-[gamma]u> are determined in (14):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (22)

where [b.sub.c] = k - 2[N.sub.c] - 1, c = 1, 2, and without loss of generality we have accepted [N.sub.1] [greater than or equal to] [N.sub.2] (that is [l.sub.2] [greater than or equal to] [l.sub.1]). Then (21) and (22) with k = 2n imply the exact expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (23)

which is not explicity in the literature, and it is more simple than the corresponding relation deduced in [10] using factorization techniques. Special applications of (21) and (23) are:

a). [??] = -2.

In this case [gamma] = 0, then from (21) it is immediate the proportionality:

<n[l.sub.2]|[r.sup.[??]]|n[l.sub.1]> [varies] <[N.sub.2]|[N.sub.1]> = [sigma][N.sub.1][N.sub.2], (24)

therefore only if [l.sub.1] = [l.sub.2] we have <r.sup.-2> [not equal to] 0, which is the result of Pasternack-Sternheimer mentioned in [10].

b). [l.sub.1] = [l.sub.2] = l, [??] = [ or -]1, [ or -]2.

The general expression (23) reproduces easily the following particular examples of Landau-Lifshitz [11]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (25)

Received May 14, 2007

References

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[12] Lopez-Bonilla, J., Lucas-Bravo, A. and Vidal-Beltr..n, S., Integral relationship between Hermite and Laguerre polynomials, Its application in quantum mechanics, Proc. Pakistan Acad. Sci., 42(2005), No.1, 63-65.

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M. Enciso-Aguilar, J. Lopez-Bonilla and I. Toledo-Toledo

SEPI-ESIME-Zacatenco, Instituto Politecnico Nacional

Edif. Z-4, 3er Piso, Col. Lindavista, C.P. 07738 Mexico DF

Email: jlopezb@ipn.mx