1. Spherical Linear Potential: Eingenvalues and Wave Functions Swee-Ping Chia Physics Department, University of Malaya 50603 Kuala Lumpur, Malaysia E-Mail: spchia@um.edu.my S.P. Chia University of Malaya

2. = orbital quantum number, k = constant  Reparametrization to the following dimensionless form: S.P. Chia University of Malaya Spherical Linear Potential Radial equation for linear potential

3.  Boundary conditions:  For = 0, Eq. becomes Airy equation, and the solution is Airy function:  B. C. at ρ = 0 yields the eigenvalues: ε =0 =  x n x n (n = 1, 2, 3 …) are zeroes of Airy function S.P. Chia University of Malaya Spherical Linear Potential

4. Airy function S.P. Chia University of Malaya Spherical Linear Potential

5.  Asymptotic behavior of wave function U ℓ () at large values of ρ: independent of orbital quantum number.  The behaviour is the same as for = 0, but the graph just get pushed out for larger and larger values of because of the centrifugal potential. S.P. Chia University of Malaya Spherical Linear Potential Asymptotic behavior

6.  Performed using Runge-Kutta method  Integrate from infinity inwards to the origin  Perform first the integration for = 0  For = 0, Airy function is used as a guide. U ℓ=0 () is small for  is about 8 and above.  Set initial point of integration ρ initial =10. This is sufficiently far away from origin.  The value of U =0 (ρ initial ) is taken as 110 -11. With this, B.C. at large value of ρ is satisfid.  Stepsize for integration is chosen as h = 0.01. S.P. Chia University of Malaya Spherical Linear Potential Numerical integration

7.  B.C. at origin is implemented by demanding that the numerical integration of the radial equation yields U ℓ=0 () = 0 at  = 0 by varying the eigenvalue ε =0.  Start with ε = 2.1.  Vary ε slowly until U ℓ=0 () changes sign between two adjacent values of ε.  Go to the next decimal place, and repeat the process until the value of the eigenvalue is obtained to 10 decimal places. S.P. Chia University of Malaya Spherical Linear Potential Numerical integration

8.  B.C. at origin is implemented by demanding that the numerical integration of the radial equation yields U ℓ=0 () = 0 at  = 0 by varying the eigenvalue ε =0.  Start with ε = 2.1.  Vary ε slowly until U ℓ=0 () changes sign between two adjacent values of ε.  Go to the next decimal place, and repeat the process until the value of the eigenvalue is obtained to 10 decimal places.  For = 0, n = 1, the required eigenvalue obtained is ε 01 = 2.3381074105. S.P. Chia University of Malaya Spherical Linear Potential Numerical integration

9.  To obtain the next eigenvalue for = 0, namely ε 01, with n = 2.  Again we are guided by the Airy function.  As the asymptotic behaviour is shifted further by about 2.0 for the second zero, the initial value ρ initial = 12.34 is used.  The initial value of ε is set at 3.8, to be far away from the first eigenvalue.  The procedure above is repeated until the eigenvalue ε 02 = 4.0879494441 is obtained to the required precision.  Repeat for n = 3, 4, 5, until 10. S.P. Chia University of Malaya Spherical Linear Potential Numerical integration

10.  The entire procedure is repeated for = 1.  We are no longer guided by exact solution.  With an intelligent choice of initial values for integration, we are able to obtain the series of eigenvalues for = 1.  Similar procedure allows us to obtain the eigenvalues for = 2, 3,4, up to 10.  With each of the eigenvalue obtained, we integrate the radial equation again. But this time from the origin outwards, to obtain the wavefunction corresponding to this eigenvalue. S.P. Chia University of Malaya Spherical Linear Potential Numerical integration

11. Eingenvalues n = 1, = 0 E = 2.3381074105 n = 1, = 1 E = 3.3612545230 n = 1, = 2 E = 4.2481822572 n = 1, = 3 E = 5.0509256349 n = 1, = 4 E = 5.7944228023 n = 1, = 5 E = 6.4930175536 n = 1, = 6 E = 7.1559044418 n = 1, = 7 E = 7.7894193209 n = 1, = 8 E = 8.3981605247 n = 1, = 9 E = 8.9855983136 n = 1, = 10 E = 9.5544327714

13. Eingenvalues n = 2, = 0 E = 4.0879494441 n = 2, = 1 E = 4.8844518441 n = 2, = 2 E = 5.6297083770 n = 2, = 3 E = 6.3321153747 n = 2, = 4 E = 6.9992603338 n = 2, = 5 E = 7.6369667691 n = 2, = 6 E = 8.2496656905 n = 2, = 7 E = 8.8407827947 n = 2, = 8 E = 9.4130196129 n = 2, = 9 E = 9.9685473870 n = 2, = 10 E = 10.5091411019

15. Eingenvalues n = 3, = 0 E = 5.5205598281 n = 3, = 1 E = 6.2076232936 n = 3, = 2 E = 6.8688826894 n = 3, = 3 E = 7.5046456421 n = 3, = 4 E = 8.1172594294 n = 3, = 5 E = 8.7092252157 n = 3, = 6 E = 9.2827906878 n = 3, = 7 E = 9.8398885788 n = 3, = 8 E = 10.3821625715 n = 3, = 9 E = 10.9110111351 n = 3, = 10 E = 11.4276298478

17. Eingenvalues n = 4, = 0 E = 6.7867080901 n = 4, = 0 E = 7.4056654355 n = 4, = 0 E = 8.0097029227 n = 4, = 0 E = 8.5971170119 n = 4, = 0 E = 9.1683261517 n = 4, = 0 E = 9.7243486275 n = 4, = 0 E = 10.2663285804 n = 4, = 0 E = 10.7953738715 n = 4, = 0 E = 11.3125028674 n = 4, = 0 E = 11.8186322372 n = 4, = 0 E = 12.3145803625

19. Eingenvalues n = 5, = 0 E = 7.9441335871 n = 5, = 1 E = 8.5152343025 n = 5, = 2 E = 9.0770030508 n = 5, = 3 E = 9.6272670132 n = 5, = 4 E = 10.1656631638 n = 5, = 5 E = 10.6924900563 n = 5, = 6 E = 11.2082951647 n = 5, = 7 E = 11.7137051346 n = 5, = 8 E = 12.2093518609 n = 5, = 9 E = 12.6958402842 n = 5, = 10 E = 13.1737354255

21. Eingenvalues n = 6, = 0 E = 9.0226508533 n = 6, = 1 E = 9.5576159127 n = 6, = 2 E = 10.0864598018 n = 6, = 3 E = 10.6070036903 n = 6, = 4 E = 11.1185615981 n = 6, = 5 E = 11.6210650896 n = 6, = 6 E = 12.1147236775 n = 6, = 7 E = 12.5998717256 n = 6, = 8 E = 13.0768938864 n = 6, = 9 E = 13.5461873471 n = 6, = 10 E = 14.0081425370

23. Eingenvalues n = 7, = 0 E = 10.0401743416 n = 7, = 1 E = 10.5465221484 n = 7, = 2 E = 11.0487407461 n = 7, = 3 E = 11.5447885641 n = 7, = 4 E = 12.0338571155 n = 7, = 5 E = 12.5156870352 n = 7, = 6 E = 12.9902905132 n = 7, = 7 E = 13.4578190614 n = 7, = 8 E = 13.9184948716 n = 7, = 9 E = 14.3725733416 n = 7, = 10 E = 14.8203220296

25. Eingenvalues n = 8, = 0 E = 11.0085243037 n = 8, = 1 E = 11.4914273039 n = 8, = 2 E = 11.9715073798 n = 8, = 3 E = 12.4468964073 n = 8, = 4 E = 12.9167557565 n = 8, = 5 E = 13.3807280141 n = 8, = 6 E = 13.8387083052 n = 8, = 7 E = 14.2907313649 n = 8, = 8 E = 14.7369104298 n = 8, = 9 E = 15.1774022710 n = 8, = 10 E = 15.6123864626

27. Eingenvalues n = 9, = 0 E = 11.9360155632 n = 9, = 1 E = 12.3992180547 n = 9, = 2 E = 12.8604908139 n = 9, = 3 E = 13.3181377904 n = 9, = 4 E = 13.7713336032 n = 9, = 5 E = 14.2196737520 n = 9, = 6 E = 14.6629838551 n = 9, = 7 E = 15.1012231320 n = 9, = 8 E = 15.5344306457 n = 9, = 9 E = 15.9626935296 n = 9, = 10 E = 16.3861274558

29. Eingenvalues n = 10, = 0 E = 12.8287767529 n = 10, = 1 E = 13.2750961668 n = 10, = 2 E = 13.7201166919 n = 10, = 3 E = 14.1623003710 n = 10, = 4 E = 14.6008550126 n = 10, = 5 E = 15.0353577163 n = 10, = 6 E = 15.4655936051 n = 10, = 7 E = 15.8914728614 n = 10, = 8 E = 16.3129835509 n = 10, = 9 E = 16.7301630752 n = 10, = 10 E = 17.1430801618

31.  We solved the radial Schrödinger equation for the spherical potential numerically.  Backward Runge-Kutta integration is utilized.  The asymptotic behavior of the wave function is exploited.  Eigenvalues accurate to 10 decimal places for = 0 to 10, and for n = 1 to 10. are obtained.  The corresponding wave functions are obtained. S.P. Chia University of Malaya Spherical Linear Potential Conclusion