ACCURATE ITERATIVE AND PERTURBATIVE
SOLUTIONS OF THE YUKAWA POTENTIAL

M. Karakoc and I. Boztosun, IJMPE Vol. 15, No. 6 (2006) 1253–1262.
https://www.worldscientific.com/doi/abs/10.1142/S0218301306004806

Import AIM library



In [1]:

    
# Python program to use AIM tools
from asymptotic import *

Definitions

Variables



In [2]:

    
# symengine (symbolic) variables for lambda_0 and s_0 
En, m, hbar, L, r, r0 = se.symbols("En, m, hbar, L, r, r0")
beta, alpha, A = se.symbols("beta, alpha, A")

$\lambda_0$ and $s_0$



In [3]:

    
# lambda_0 and s_0
l0 = 2*beta-2/r
s0 = -beta**2 - (2*En*m)/hbar**2 + L/r**2 + L**2/r**2 + (2*beta)/r - (2*A*se.exp(-alpha*r)*m)/(hbar**2*r)

Case: A=4

s states

Numerical values for variables



In [4]:

    
## Case: A = 4, L=0
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: 4, 
       m: o* 1/2, 
       L: 0, 
       hbar: 1, 
       r0: nr0}

Initialize AIM solver



In [5]:

    
%%time
# pass lambda_0, s_0 and variable values to aim class
yukawa_A4L0 = aim(l0, s0, pl0, ps0)
yukawa_A4L0.display_parameters()
yukawa_A4L0.display_l0s0(0)
yukawa_A4L0.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 4,~~L = 0,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- \frac{2 A m e^{- \alpha r}}{\hbar^{2} r} - \frac{2 En m}{\hbar^{2}} + \frac{L^{2}}{r^{2}} + \frac{L}{r^{2}} - \beta^{2} + \frac{2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{4.0 e^{- \frac{r}{5}}}{r} \right.
                \end{align}
                $






    



CPU times: user 285 ms, sys: 8.81 ms, total: 294 ms
Wall time: 296 ms

Calculation necessary coefficients



In [6]:

    
%%time
# create coefficients for improved AIM
yukawa_A4L0.c0()
yukawa_A4L0.d0()
yukawa_A4L0.cndn()









    



CPU times: user 2min 47s, sys: 481 ms, total: 2min 47s
Wall time: 2min 47s

The Solution



In [7]:

    
%%time
yukawa_A4L0.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001   -2.22608382037941285277
0011   -3.25646198730204924445   -0.22880426749406252625
0021   -3.25646424490443058235   -0.39503930505322610330
0031   -3.25646424490722525117   -0.39934060289808198926
0041   -3.25646424490722525404   -0.39942480823044205382   -0.00752426627166027709
0051   -3.25646424490722525404   -0.39942615062318136347   -0.01999610015762137179
0061   -3.25646424490722525404   -0.39942617037004934161   -0.02379348447266779420
0071   -3.25646424490722525404   -0.39942617065134327804   -0.02503093065297177558
0081   -3.25646424490722525404   -0.39942617065529516410   -0.02544763075010264581
0091   -3.25646424490722525404   -0.39942617065535033512   -0.02558994246491339011
0101   -3.25646424490722525404   -0.39942617065535110306   -0.02563879250512245213
0111   -3.25646424490722525404   -0.39942617065535111373   -0.02565558271377236651
0121   -3.25646424490722525404   -0.39942617065535111388   -0.02566135365381434051
0131   -3.25646424490722525404   -0.39942617065535111388   -0.02566333656258396165
0141   -3.25646424490722525404   -0.39942617065535111388   -0.02566401769819825872
0151   -3.25646424490722525404   -0.39942617065535111388   -0.02566425162394128720
0161   -3.25646424490722525404   -0.39942617065535111388   -0.02566433195250032477
0171   -3.25646424490722525404   -0.39942617065535111388   -0.02566435953491229269
0181   -3.25646424490722525404   -0.39942617065535111388   -0.02566436900553236619
0191   -3.25646424490722525404   -0.39942617065535111388   -0.02566437225727462679
0201   -3.25646424490722525404   -0.39942617065535111388   -0.02566437337375074847
CPU times: user 21.7 s, sys: 61 ms, total: 21.8 s
Wall time: 21.6 s

Case: A=8

s states



In [8]:

    
%%time

## Case: A = 8, L=0
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: 8, 
       m: o* 1/2, 
       L: 0, 
       hbar: 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A8L0 = aim(l0, s0, pl0, ps0)
yukawa_A8L0.display_parameters()
yukawa_A8L0.display_l0s0(2)
yukawa_A8L0.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A8L0.c0()
yukawa_A8L0.d0()
yukawa_A8L0.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 8,~~L = 0,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{8.0 e^{- \frac{r}{5}}}{r} \right.
                \end{align}
                $






    



CPU times: user 2min 53s, sys: 396 ms, total: 2min 53s
Wall time: 2min 53s



In [9]:

    
# the solution
yukawa_A8L0.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  -13.45216764075882570553
0011  -14.45812555697346708940   -2.61632538308431238185   -0.28824411798510504500
0021  -14.45812571278409411507   -2.61667826125750169323   -0.61447052737404830354
0031  -14.45812571278417740313   -2.61667826544806846945   -0.61924266082608469430   -0.05840432675425859014
0041  -14.45812571278417740340   -2.61667826544809997335   -0.61928152780715240586   -0.09283945581502052609
0051  -14.45812571278417740340   -2.61667826544809997355   -0.61928175291669448691   -0.09806370223569206829
0061  -14.45812571278417740340   -2.61667826544809997355   -0.61928175406646802023   -0.09881717168967634571
0071  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407204835129   -0.09891833742141017116
0081  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207482857   -0.09893127685318019373
0091  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495287   -0.09893289030164978911
0101  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893308906739952169
0111  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311341667168776
0121  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311639169670450
0131  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311675473885342
0141  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311679901463340
0151  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311680441285971
0161  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311680507093461
0171  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311680515115271
0181  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311680516116377
0191  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495345   -0.09893311680533847665
0201  -14.45812571278417740340   -2.61667826544809997355   -0.61928175407207495339   -0.09893311700471986687

p states



In [10]:

    
%%time

## Case: A = 8, L=1
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 8, 
       m: o* 1/2, 
       L: o* 1, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A8L1 = aim(l0, s0, pl0, ps0)
yukawa_A8L1.display_parameters()
yukawa_A8L1.display_l0s0(2)
yukawa_A8L1.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A8L1.c0()
yukawa_A8L1.d0()
yukawa_A8L1.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 8,~~L = 1,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{8.0 e^{- \frac{r}{5}}}{r} + \frac{2.0}{r^{2}} \right.
                \end{align}
                $






    



CPU times: user 2min 51s, sys: 460 ms, total: 2min 51s
Wall time: 2min 51s



In [11]:

    
# the solution
yukawa_A8L1.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  Nothing to show!
0011   -2.58352549387679236942   -0.34844883497962867628
0021   -2.58369238325680081047   -0.59025168732310849287
0031   -2.58369238520909205900   -0.59381799881889381975   -0.04557938861762899515
0041   -2.58369238520910751068   -0.59384787419974706197   -0.07707964001368260549
0051   -2.58369238520910751079   -0.59384805923443831063   -0.08218598649344847093
0061   -2.58369238520910751079   -0.59384806026347272262   -0.08299486802785128445
0071   -2.58369238520910751079   -0.59384806026896293800   -0.08311703264160825208
0081   -2.58369238520910751079   -0.59384806026899174050   -0.08313488556733795849
0091   -2.58369238520910751079   -0.59384806026899189057   -0.08313745430382058891
0101   -2.58369238520910751079   -0.59384806026899189135   -0.08313782192244365963
0111   -2.58369238520910751079   -0.59384806026899189136   -0.08313787449537357159
0121   -2.58369238520910751079   -0.59384806026899189136   -0.08313788202193675437
0131   -2.58369238520910751079   -0.59384806026899189136   -0.08313788310122289182
0141   -2.58369238520910751079   -0.59384806026899189136   -0.08313788325624572842
0151   -2.58369238520910751079   -0.59384806026899189136   -0.08313788327854590599
0161   -2.58369238520910751079   -0.59384806026899189136   -0.08313788328175800129
0171   -2.58369238520910751079   -0.59384806026899189136   -0.08313788328222118456
0181   -2.58369238520910751079   -0.59384806026899189136   -0.08313788328228837097
0191   -2.58369238520910751079   -0.59384806026899189136   -0.08313788328256910233
0201   -2.58369238520910751079   -0.59384806026899189130   -0.08313788361370151810

Case: A=16

s states



In [12]:

    
%%time

## Case: A = 16, L=0
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 16, 
       m: o* 1/2, 
       L: o* 0, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A16L0 = aim(l0, s0, pl0, ps0)
yukawa_A16L0.display_parameters()
yukawa_A16L0.display_l0s0(2)
yukawa_A16L0.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A16L0.c0()
yukawa_A16L0.d0()
yukawa_A16L0.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 16,~~L = 0,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{16.0 e^{- \frac{r}{5}}}{r} \right.
                \end{align}
                $






    



CPU times: user 2min 53s, sys: 436 ms, total: 2min 54s
Wall time: 2min 54s



In [13]:

    
# the solution
yukawa_A16L0.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  -35.90433528151765141106  -17.90433528151765141106
0011  -60.83959411455164152945  -13.02741330503248532897   -4.39416089534789882041   -1.46028332602987413583
0021  -60.85900611778662673023  -13.02731559698770316509   -4.39417726867497787866   -1.60029878564889252205
0031  -60.85903279533303488128  -13.02731559641380264116   -4.39417726867760316650   -1.60041447655023284251
0041  -60.85903282299755577594  -13.02731559641378727702   -4.39417726867760325603   -1.60041450470027026594
0051  -60.85903282302548302996  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490449513
0061  -60.85903282302551367246  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515265
0071  -60.85903282302551371163  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0081  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0091  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0101  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0111  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0121  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515273
0131  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515274
0141  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515274
0151  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490515305
0161  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490517015
0171  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470490667535
0181  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450470510967902
0191  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325607   -1.60041450474616584562
0201  -60.85903282302551371170  -13.02731559641378727528   -4.39417726867760325606   -1.60041451695281073576

p states



In [14]:

    
%%time

## Case: A = 16, L=1
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 16, 
       m: o* 1/2, 
       L: o* 1, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A16L1 = aim(l0, s0, pl0, ps0)
yukawa_A16L1.display_parameters()
yukawa_A16L1.display_l0s0(2)
yukawa_A16L1.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A16L1.c0()
yukawa_A16L1.d0()
yukawa_A16L1.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 16,~~L = 1,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{16.0 e^{- \frac{r}{5}}}{r} + \frac{2.0}{r^{2}} \right.
                \end{align}
                $






    



CPU times: user 2min 51s, sys: 468 ms, total: 2min 52s
Wall time: 2min 52s



In [15]:

    
# the solution
yukawa_A16L1.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  -17.90433528151765141106
0011  -12.99107034549051500501   -4.36208955656926947220   -1.45795554685741697188
0021  -12.99103533725527656652   -4.36210053490250182720   -1.57357991881273162125   -0.42928858560796030635
0031  -12.99103533706039992987   -4.36210053490408066568   -1.57367299249004705503   -0.49334208977036406493
0041  -12.99103533706039481597   -4.36210053490408071018   -1.57367301584411823083   -0.49480334124509462367
0051  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818289336   -0.49482236117942318037
0061  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350969   -0.49482255139593078132
0071  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255311146014216
0081  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312627993389
0091  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312640559114
0101  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312640664801
0111  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312640665667
0121  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312640665133
0131  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350978   -0.49482255312640639530
0141  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818350979   -0.49482255312638543259
0151  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818351001   -0.49482255312351184698
0161  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818352292   -0.49482255248424342913
0171  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584818468534   -0.49482232810177492906
0181  -12.99103533706039481539   -4.36210053490408071020   -1.57367301584834515665   -0.49470074344085646091
0191  -12.99103533706039481539   -4.36210053490408071020   -1.57367301588157091140
0201  -12.99103533706039481539   -4.36210053490408071020   -1.57367302599636994149

Case: A=24

s states



In [16]:

    
%%time

## Case: A = 24, L=0
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 24, 
       m: o* 1/2, 
       L: o* 0, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A24L0 = aim(l0, s0, pl0, ps0)
yukawa_A24L0.display_parameters()
yukawa_A24L0.display_l0s0(2)
yukawa_A24L0.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A24L0.c0()
yukawa_A24L0.d0()
yukawa_A24L0.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 24,~~L = 0,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{24.0 e^{- \frac{r}{5}}}{r} \right.
                \end{align}
                $






    



CPU times: user 2min 51s, sys: 540 ms, total: 2min 52s
Wall time: 2min 52s



In [17]:

    
%%time
# the solution
yukawa_A24L0.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  -58.35650292227647711659  -40.35650292227647711659
0011 -139.13653387574502899132  -31.66301817646597603646  -11.68287862297764665710   -5.04495719546218526029
0021 -139.25742393258019657380  -31.43136501424337512912  -11.69980571386560753277   -5.04418470133869590089
0031 -139.25932195590774533382  -31.43127747871788354739  -11.69980641354646150149   -5.04418465378360724838
0041 -139.25934783673424127553  -31.43127745071106829863  -11.69980641363918808868   -5.04418465376794081739
0051 -139.25934813951377692548  -31.43127745069839616344  -11.69980641363923232659   -5.04418465376791612696
0061 -139.25934814283657500485  -31.43127745069838647527  -11.69980641363923239407   -5.04418465376791597917
0071 -139.25934814287233467833  -31.43127745069838646194  -11.69980641363923239436   -5.04418465376791597642
0081 -139.25934814287272258990  -31.43127745069838646191  -11.69980641363923239436   -5.04418465376791597628
0091 -139.25934814287272691639  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597627
0101 -139.25934814287272696680  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597626
0111 -139.25934814287272696743  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597626
0121 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597627
0131 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597629
0141 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597647
0151 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791597845
0161 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791601244
0171 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376791689938
0181 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376795150192
0191 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239437   -5.04418465376993522912
0201 -139.25934814287272696744  -31.43127745069838646191  -11.69980641363923239436   -5.04418465393423727654
CPU times: user 22.2 s, sys: 76.3 ms, total: 22.3 s
Wall time: 22.1 s

p states



In [18]:

    
%%time

## Case: A = 24, L=1
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 24, 
       m: o* 1/2, 
       L: o* 1, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A24L1 = aim(l0, s0, pl0, ps0)
yukawa_A24L1.display_parameters()
yukawa_A24L1.display_l0s0(2)
yukawa_A24L1.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A24L1.c0()
yukawa_A24L1.d0()
yukawa_A24L1.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 24,~~L = 1,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{24.0 e^{- \frac{r}{5}}}{r} + \frac{2.0}{r^{2}} \right.
                \end{align}
                $






    



CPU times: user 2min 52s, sys: 496 ms, total: 2min 52s
Wall time: 2min 52s



In [19]:

    
%%time
# the solution
yukawa_A24L1.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001  -40.35650292227647711659  -22.35650292227647711659
0011  -31.47595742908929494341  -11.65614536051688704547   -5.01396586679733283626   -2.05650750253059879100
0021  -31.39384360867558161412  -11.66528444450607425693   -5.01346555599166318358   -2.17706957209930807571
0031  -31.39381361058830369400  -11.66528480221943162391   -5.01346552688647371917   -2.17709498078266634406
0041  -31.39381360114239924640  -11.66528480226596544457   -5.01346552687705535167   -2.17709498198874758034
0051  -31.39381360113816328969  -11.66528480226598747457   -5.01346552687704057903   -2.17709498198884571443
0061  -31.39381360113816006838  -11.66528480226598750809   -5.01346552687704049050   -2.17709498198884708002
0071  -31.39381360113816006396  -11.66528480226598750824   -5.01346552687704048885   -2.17709498198884716842
0081  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048877   -2.17709498198884717955
0091  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048876   -2.17709498198884717628
0101  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048876   -2.17709498198884714938
0111  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048876   -2.17709498198884696868
0121  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048876   -2.17709498198884504334
0131  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048878   -2.17709498198881061666
0141  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704048889   -2.17709498198778659367
0151  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704049019   -2.17709498193825653145
0161  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704051279   -2.17709497813993860871
0171  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687704110919   -2.17709452750079619490
0181  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687706465023   -2.17701360425487743639
0191  -31.39381360113816006395  -11.66528480226598750824   -5.01346552687843032166   -2.14560858944792750436
0201  -31.39381360113816006395  -11.66528480226598750824   -5.01346552699290530785
CPU times: user 23.2 s, sys: 84.2 ms, total: 23.2 s
Wall time: 23.1 s

d states



In [20]:

    
%%time

## Case: A = 24, L=2
nbeta = 3
nr0 = o* 1/nbeta

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta: nbeta}
ps0 = {beta: nbeta, 
       alpha: o* 2/10, 
       A: o* 24, 
       m: o* 1/2, 
       L: o* 2, 
       hbar: o* 1, 
       r0: nr0}

# pass lambda_0, s_0 and variable values to aim class
yukawa_A24L2 = aim(l0, s0, pl0, ps0)
yukawa_A24L2.display_parameters()
yukawa_A24L2.display_l0s0(2)
yukawa_A24L2.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
yukawa_A24L2.c0()
yukawa_A24L2.d0()
yukawa_A24L2.cndn()









    





$\displaystyle \text{Values of the parameters of } \lambda_0:$






    





$\displaystyle \beta = 3$






    





$\displaystyle \text{Values of the parameters of } s_0:$






    





$\displaystyle A = 24,~~L = 2,~~\alpha = 1/5,~~\beta = 3,~~\hbar = 1,~~m = 1/2,~~r_{0} = 1/3$






    





$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[6 - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- En - 9.0 + \frac{6.0}{r} - \frac{24.0 e^{- \frac{r}{5}}}{r} + \frac{6.0}{r^{2}} \right.
                \end{align}
                $






    



CPU times: user 2min 51s, sys: 492 ms, total: 2min 51s
Wall time: 2min 51s



In [21]:

    
%%time
# the solution
yukawa_A24L2.get_arb_roots(showRoots='-r', printFormat="{:25.20f}")









    



0001   -4.35650292227647711659
0011  -11.59387493593599586754   -4.95178952109149376833   -2.03379603736129693331
0021  -11.59594904628543983383   -4.95159937290600434998   -2.12413868519246255675   -0.78370948519434660895
0031  -11.59594912056795599076   -4.95159936286996712583   -2.12415695424100676680   -0.81399856463852673548
0041  -11.59594912057729894619   -4.95159936286682481005   -2.12415695516397097368   -0.81419430361954735366
0051  -11.59594912057730330751   -4.95159936286681992231   -2.12415695516402831014   -0.81419491493356845353
0061  -11.59594912057730331411   -4.95159936286681989294   -2.12415695516402890406   -0.81419491638933385468
0071  -11.59594912057730331414   -4.95159936286681989238   -2.12415695516402894305   -0.81419491639245126358
0081  -11.59594912057730331414   -4.95159936286681989235   -2.12415695516402894794   -0.81419491639246063373
0091  -11.59594912057730331414   -4.95159936286681989235   -2.12415695516402894617   -0.81419491639248211088
0101  -11.59594912057730331414   -4.95159936286681989235   -2.12415695516402893248   -0.81419491639272071076
0111  -11.59594912057730331414   -4.95159936286681989235   -2.12415695516402883847   -0.81419491639756221242
0121  -11.59594912057730331414   -4.95159936286681989235   -2.12415695516402780558   -0.81419491657609909042
0131  -11.59594912057730331414   -4.95159936286681989236   -2.12415695516400870410   -0.81419492817486783373
0141  -11.59594912057730331414   -4.95159936286681989240   -2.12415695516342018181   -0.81419621028986784408
0151  -11.59594912057730331414   -4.95159936286681989293   -2.12415695513390783173   -0.81442857565788224331
0161  -11.59594912057730331414   -4.95159936286681990220   -2.12415695278599274980   -0.84952539093526896149
0171  -11.59594912057730331414   -4.95159936286682015281   -2.12415666366598294616   -1.07770666919510980203
0181  -11.59594912057730331414   -4.95159936286683028223   -2.12410277306146329267   -1.42917527872850305945
0191  -11.59594912057730331414   -4.95159936286743222042   -2.10537089461045198246   -1.91920735338859440652
0201  -11.59594912057730331414   -4.95159936291913054510
CPU times: user 25.4 s, sys: 152 ms, total: 25.6 s
Wall time: 25.4 s



In [ ]:

ACCURATE ITERATIVE AND PERTURBATIVE SOLUTIONS OF THE YUKAWA POTENTIAL

Import AIM library

Definitions

Variables

$\lambda_0$ and $s_0$

Case: A=4

s states

Numerical values for variables

Initialize AIM solver

Calculation necessary coefficients

The Solution

Case: A=8

s states

p states

Case: A=16

s states

p states

Case: A=24

s states

p states

d states

ACCURATE ITERATIVE AND PERTURBATIVE
SOLUTIONS OF THE YUKAWA POTENTIAL