The work objective is to describe the numerical solution method for the stationary Schrödinger equation based on the application of the integral equation identical to the Schrödinger equation. The structure of this integral equation is close to the structure of the Fredholm equation of second kind and allows obtaining the problem numerical solution. The method under study allows finding the energy eigenvalues and eigensolutions to the quantum-mechanical problems of various dimensions. The test results of the solving problems method for one-dimensional and two-dimensional quantum oscillators are obtained. The found numerical values of eigenenergy and eigenfunctions of the oscillator are compared to the known analytical solutions, and then the error of result is evaluated. The highest accuracy of the solution is obtained for the first energy levels. The numerical solution error increases with the number of the energy eigenvalue. For the subsequent energy level, the error increases almost by an order of magnitude. The solution error for the fourth energy level is less than 2% if the integration domain contains 500 elements. If the energy level is degenerate, it is possible to obtain all eigenfunctions corresponding to the given level. For this purpose, various auxiliary functions the symmetry of which is coherent with the eigenfunction symmetry are used.
Schrödinger equation, eigenvalues, eigenfunctions, numerical solution, the fundamental solutions.
Фундаментальное значение при решении квантовых механических задач описывается уравнением Шредингера [1]. Аналитические решения этого уравнения могут быть получены лишь для весьма ограниченного круга задач, преимущественно одномерных. Поэтому разработано множество приближенных методов решения уравнения Шредингера, как аналитических, с использованием теории возмущения [2–4], так и прямых численных методов. Несмотря на широкий спектр имеющихся численных методов решения уравнения Шредингера, таких, например, как метод Нумерова [5], метод диагонализации [6–7], спектральный метод [7], и других численных методов [8–12], проблема эффективных способов нахождения собственных энергий и собственных функций для основного уравнения квантовой механики, особенно при решении многомерных задач, продолжает оставаться актуальной. Ниже предложена методика решения уравнения Шредингера, основанная на приведении его к интегральному уравнению с последующим численным решением.
1. Landau, L.D., Lifshitz, E.M. Kvantovaya mekhanika. Nerelyativistskaya teoriya. [Quantum mechanics. Nonrelativistic theory.] Moscow: Nauka, 1963, 703 p. (in Russian).
2. Kesarwani, R.N., Varshni, Y.P. Eigenvalues of an anharmonic oscillators. J. Math. Phys., 1981, vol. 22, no.9, pp.1983–1989.
3. Ulyanov, V.V. Integral´nye metody v kvantovoy mekhanike. [Integral methods in quantum mechanics.] Kharkov: Vishcha shkola, 1982, 160 p. (in Russian).
4. Bender, С.М., Wu, Т.Т. Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev., 1973, vol. 7, no.6, pp. 1620–1636.
5. Killingbeck, J.P. Microcomputer Quantum Mechanics. Adam Hiller, 1983, 380 p.
6. Banerjee, К., Bhatnagar, S.P., Choudhry, V., Kanwal, S.S. The anharmonic oscillator. Proc. R. Soc. A, 1978, vol. 360, pp.575–586.
7. Cherkassky, V.A. Chislennoe reshenie uravneniya Shredingera: metod diagonalizatsii i spektral´nyy metod. [The numerical solution of the Schrödinger equation: diagonalization method and spectral method.] Вісник Харківського національного університету, 2010, no. 926, pp. 204–211 (in Russian).
8. Dineykhan, M., Efimov, G.V. The Schroedinger equation for bound state systems in the oscillator representation. Repots of Math. Phys., 1995, vol. 6, no.2/3, pp. 287–308.
9. Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., Puzynin, I.V. FESSDE, a program for the finite-element solution of the coupled-channel Schroedinger equation using high-order accuracy approximations. Соmр. Phys. Commun., 1995, vol. 85, pp. 65–74.
10. Jafarpour, М., Afshar, D. Calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential. J. Phys. A: Math. Gen., 2002, vol. 35, pp. 87–92.
11. Kvitko, G.V., Kuzin, E.L., Shott, D.V. Chislennoe reshenie uravneniya Shredingera s polinomial´nymi potentsialami. [Numerical solution of the Schrödinger equation with polynomial potentials.] IKBFU’s Vestnik, 2011, vol. 5, pp. 115–119 (in Russian).
12. Lukyanenko, A.N., Chekanov, N.A. Simvol´no-chislennoe reshenie dvumernogo uravneniya Shredingera s dvukh´´yamnym potentsialom. [Symbolic-numerical solution of two-dimensional Schrödinger equation with a double-well potential.] Universitet im. V.I. Vernadskogo, 2008, no. 3(13), vol. 2, pp.43–49 (in Russian).
13. Polyanin, A.D. Spravochnik po lineynym uravneniyam matematicheskoy fiziki. [Handbook on linear equations of mathematical physics.] Moscow: Fizmatlit, 2001, 576 p. (in Russian).
14. Vladimirov, V.S., Zharinov, V.V. Uravneniya matematicheskoy fiziki. [Equations of Mathematical Physics.] Moscow: Fizmatlit, 2004, 400 p. (in Russian).
15. Fairweather, G., Johnston, R.L. The method of fundamental solutions for problems in potential theory. C.T.H. Baker, G.F. Miller, eds. Treatment of Integral Equations by Numerical Methods. London: Academic Press, 1982, pp.349–359.
16. Knyazev, S.Yu. Chislennoe reshenie uravneniy Puassona i Gel´mgol´tsa s pomoshch´yu metoda tochechnykh istochnikov. [Numerical solution of Poisson and Helmholtz equations using the point-source method] Russian Electromechanics, 2007, no. 2, pp. 77–78 (in Russian).
17. Knyazev, S.Yu., Shcherbakova, E.E., Zaichenko, A.N. Chislennoe reshenie kraevykh zadach dlya neodnorodnykh uravneniy Gel´mgol´tsa metodom tochechnykh istochnikov polya. [Numerical Solution of the Boundary Problems with Non-Homogeneous Helmholtz Equation by Field Point-Source Method.] Russian Electromechanics, 2014, no. 4, pp. 14–19 (in Russian).
18. Knyazev, S.Yu., Shcherbakova, E.E. Reshenie zadach teplo- i massoperenosa s pomoshch´yu metoda tochechnykh istochnikov polya. [The solution of heat and mass transfer problems by the point source method.] University News. North-Caucasian region. Technical Sciences Series, 2006, no. 4, pp. 43—47 (in Russian).
19. Knyazev, S.Yu., Pustovoyt, V.N., Shcherbakova, E.E. Modelirovanie poley uprugikh deformatsiy s primeneniem metoda tochechnykh istochnikov. [Modeling the elastic strain fields by point-source method.] Vestnik of DSTU, 2015, vol. 15, no. 1(80), pp. 29–38 (in Russian).
20. Knyazev, S.Yu., Pustovoyt, V.N., Shcherbakova, E.E., Shcherbakov, A.A. Modelirovanie trekhmernykh poley uprugikh deformatsiy s pomoshch´yu metoda tochechnykh istochnikov. [Modeling of three-dimensional elastic strain fields by point-source method.] Vestnik of DSTU, 2015, vol. 15, no. 4(83), pp. 13–23 (in Russian).
