Abstract
In this work we propose an alternative semiclassical iterative approach to obtain the Yukawa Potential, where the temporal evolution is replaced by the number of iterations. In addition, our analytical approach was able to provide an exact value very close to the adopted semi-empirically for the Yukawa magnitude scale constant.
Keywords:
Yukawa Potential; analytical model; meson
1. Introduction
Yukawa in the first half of the 20th century proposed that the nucleons were bound in the nucleus through an interaction that was mediated by mesons, [1][1] H. Yukawa, Proc. Phys. Math. Sot. Japan 17, 48 (1935). this hypothesis was later confirmed by Lattes et al. [2][2] C.M.G. Lattes, H. Muirhead, G.P.S. Occhialini and C.F. Powell, Nature 159, 694 (1947). This interaction is short-range and its potential was originally derived as being
where is a magnitude scaling constant and e is the range of the potential.
Short-range potentials of type
(where f(r) is a function that depends on the spacial dimensions) have already been studied before Yukawa [1][1] H. Yukawa, Proc. Phys. Math. Sot. Japan 17, 48 (1935). in the most varied areas, [3][3] J.S. Rowlinson, Physica A 156, 15 (1989). namely: celestial mechanics [4][4] P.S. Laplace, Oeuvres de Laplace (Imprim. Roy., Paris, 1847), v. 5, p. 480., electrostatics [5][5] C. Neumann, Allgemeine Untersuchungen über das Newton'sche Princip der Fernwirkungen mit besonderer Rücksicht auf die elektrischen Wirkungen (Teubner, Leipzig, 1896), p. 30., inter-molecular interactions [6][6] J.S. Rowlinson, J. Stat. Phys. 20, 197 (1979)., fluid theory [7][7] L.S. Ornstein and F. Zernike, Proc. Roy. Acad. Sci. Amsterdam 17, 793 (1914). and electrolysis [8[8] L.G. Gouy, J. Phys. Theor. Appl. 9, 457 (1910)., 9[9] D.L. Chapman, Phil. Mag. 25, 475 (1919).].
Due to their importance, appearance in diverse physical systems and due to still remains an object of study [10[10] C.O. Edet, P.O Okoi and S.O. Chima, Rev. Bras. Ens. Fís. 42, e20190083 (2020). [11] U.S. Okorie, C.O Edet, A.N Ikot, G.J Rampho and R. Sever, Indian J. Phys. (2020). [12] C.O. Edet and P.O. Okoi, Rev. Mex. Fís. 65, 333 (2019). [13] C.O. Edet, K.O. Okorie, H. Louis and N.A. Nzeata-Ibe, Indian J. Phys. 94, 243 (2020). [14] C.O. Edet, U.S. Okorie, A.T. Ngiangia and A.N. Ikot, Indian J. Phys. 94, 425 (2020). [15] B.I. Ita, H. Louis, O.U. Akakuru, N.A. Nzeata-Ibe, A.I. Ikeuba, T.O. Magu, P.I. Amos and C.O. Edet, Bulg. J. Phys. 45, 323 (2018).-16[16] C.N. Isonguyo, K.J. Oyewumi and O.S. Oyun, Int. J. Quantum Chem.118, e25620 (2018).], we have decided to propose an alternative approach to obtain this short-range potential.
2. Iterative Approach
Let us consider here a semiclassical system and consider the relative motion of a test particle of mass m in a two-body system. At an initial time t0 its position and momentum are r0 and p0 respectively. After the first iteration
and
where , and are the increments of their respective variables.
The system again follows a straight trajectory until its second iteration at the instant
which changes its momentum to
in position
Thus, after the nth iteration
and
In the equation (11) we use the relation
The equations (9), (10) and (11) are generic and valid for any semiclassical physical system.
3. Obtaining the Yukawa Potential
Let us consider a law of force proportional to the inverse of the square of the distance between two particles, i.e. like Newton/Coulomb's Law
where is a constant of proportionality.
Let us consider that the field-generating particle (gravitational or electric) is at the origin of the considered coordinate system and that the test particle is in the position r0 with momentum p0 and that with , this ensures that the two particles are very close to each other (such as inside a nucleus). We will also consider , this ensures that the iterative process approaches the continuous fields picture as expected, it is important to note that there are analytic approaches in which the fields are considered as discrete, [17[17] M.M. de Souza, J. Phys. A: Math. Gen. 30, 6565 (1997). [18] M.M. de Souza and R.N. Silveira, Class. Quantum Grav. 16, 619 (1999).-19[19] B.R. Segatto, J.C.S. Azevedo and M.M. de Souza, J. Phys. A: Math. Gen. 36, 5115 (2003).] is not the case in question. Defining the time increment as being constant, i.e., , equations (11) and (10) then become
and
respectively. The last term of equation (15) refers to the increase in position which is much smaller than one, consequently we can rewrite equation (15) as:
Using recursively the relation
which can be proof by induction hypothesis [20][20] B.R. Segatto, Rev. Bras. de Ens. Fís, 42, e20180346 (2020). and equality , after successive reiterations we have:
Equation (18) gives us the linear momentum of the test particle after the nth iteration, however, it is clear that the quantity here must be a continuous variable. To recover (or approximate) the continuous picture it is convenient to do . Thus, for the following approximation is valid [20][20] B.R. Segatto, Rev. Bras. de Ens. Fís, 42, e20180346 (2020).
and the equation (18) then becomes:
The two summations of the equation (20) are the partial series of the hyperbolic cosine and hyperbolic sine functions, respectively. At the asymptotic limit, i.e., these summations then become:
The equation (21) represents the momentum of the particle after time interval (equation (9)). If we consider that the particle moves with velocity
with (where c is the speed of the light), then the equation (21) becomes
with
We are considering that the system is conservative, so we can correlate the potential energy with the linear momentum through the relation
Consequently
Rewriting the equation (26) we have
Since and , then the second term of equation (27), should be nil to ensure convergence of potential. Thus
and
with
Defining and and by the equations (24) and (28) and using Heisenberg uncertainty principle we obtain
We can conclude that , a value very close to the semiempirical adopted by Gauthier et al.[21][21] N. Gauthier and S. Sherrit, Amer. J. Phys. 59, 1144 (1991). of .
4. Discussion and Conclusions
In this work we propose an alternative iterative approach to obtain the potential of Yukawa, based in the works M. M. de Souza [17][17] M.M. de Souza, J. Phys. A: Math. Gen. 30, 6565 (1997). where the classical electromagnetic field of a spinless point electron was described in a formalism with extended causality by discrete finite point-vector fields with discrete and localized point interactions (the same formalism were used to described the general relativity homogeneous field equations [18][18] M.M. de Souza and R.N. Silveira, Class. Quantum Grav. 16, 619 (1999).). The intention was not to propose a better or simpler approach to the usual ones. However we were able to obtain the Yukawa Potential without directly solving the Schrödinger equation using only the Heisenberg Uncertainty principle. In summary therefore, even if the results presented here are not new from a physical point of view, the analytical method proposed here can serve as a basis for students to create their own models.
5. acknowledgments
The author would like to thank the CAPES, CAPES and CNPq funding agencies and the anonymous referee for their suggestions.
References
- [1] H. Yukawa, Proc. Phys. Math. Sot. Japan 17, 48 (1935).
- [2] C.M.G. Lattes, H. Muirhead, G.P.S. Occhialini and C.F. Powell, Nature 159, 694 (1947).
- [3] J.S. Rowlinson, Physica A 156, 15 (1989).
- [4] P.S. Laplace, Oeuvres de Laplace (Imprim. Roy., Paris, 1847), v. 5, p. 480.
- [5] C. Neumann, Allgemeine Untersuchungen über das Newton'sche Princip der Fernwirkungen mit besonderer Rücksicht auf die elektrischen Wirkungen (Teubner, Leipzig, 1896), p. 30.
- [6] J.S. Rowlinson, J. Stat. Phys. 20, 197 (1979).
- [7] L.S. Ornstein and F. Zernike, Proc. Roy. Acad. Sci. Amsterdam 17, 793 (1914).
- [8] L.G. Gouy, J. Phys. Theor. Appl. 9, 457 (1910).
- [9] D.L. Chapman, Phil. Mag. 25, 475 (1919).
- [10] C.O. Edet, P.O Okoi and S.O. Chima, Rev. Bras. Ens. Fís. 42, e20190083 (2020).
- [11] U.S. Okorie, C.O Edet, A.N Ikot, G.J Rampho and R. Sever, Indian J. Phys. (2020).
- [12] C.O. Edet and P.O. Okoi, Rev. Mex. Fís. 65, 333 (2019).
- [13] C.O. Edet, K.O. Okorie, H. Louis and N.A. Nzeata-Ibe, Indian J. Phys. 94, 243 (2020).
- [14] C.O. Edet, U.S. Okorie, A.T. Ngiangia and A.N. Ikot, Indian J. Phys. 94, 425 (2020).
- [15] B.I. Ita, H. Louis, O.U. Akakuru, N.A. Nzeata-Ibe, A.I. Ikeuba, T.O. Magu, P.I. Amos and C.O. Edet, Bulg. J. Phys. 45, 323 (2018).
- [16] C.N. Isonguyo, K.J. Oyewumi and O.S. Oyun, Int. J. Quantum Chem.118, e25620 (2018).
- [17] M.M. de Souza, J. Phys. A: Math. Gen. 30, 6565 (1997).
- [18] M.M. de Souza and R.N. Silveira, Class. Quantum Grav. 16, 619 (1999).
- [19] B.R. Segatto, J.C.S. Azevedo and M.M. de Souza, J. Phys. A: Math. Gen. 36, 5115 (2003).
- [20] B.R. Segatto, Rev. Bras. de Ens. Fís, 42, e20180346 (2020).
- [21] N. Gauthier and S. Sherrit, Amer. J. Phys. 59, 1144 (1991).
Publication Dates
-
Publication in this collection
15 June 2020 -
Date of issue
2020
History
-
Received
07 Jan 2020 -
Reviewed
08 Apr 2020 -
Accepted
27 Apr 2020