Notebook

The relativistic Duffin–Kemmer–Petiau sextic oscillator¶

F Yasuk, M Karakoc and I Boztosun, Physica Scripta, Volume 78, 045010, 2008.
https://iopscience.iop.org/article/10.1088/0031-8949/78/04/045010/meta

Import AIM library¶

In [1]:

# Python program to use AIM tools
from asymptotic import *

Definitions¶

Variables¶

In [2]:

# variables, l0, and s0
En, m, hbar, c, J, r, r0 = se.symbols("En, m, hbar, c, J, r, r0")
w, q = se.symbols('w, q')
beta1, beta2, A0, A1, A2, A3, A4 = se.symbols("beta1, beta2, A0, A1, A2, A3, A4")

$\lambda_0$ and $s_0$¶

In [3]:

l0 = 2*beta1 + 4*beta2*r - 2/r
s00 = -beta1**2 + 6*beta2 + 2*beta1/r - 4*beta1*beta2*r - 4*beta1**2*r**2
s01 = -(En**2 + A0 - A1/r**2 - A2*r**2 + A3*r**4 - A4*r**6)/(hbar**2*c**2)
s0 = s00 + s01

Case: q = $\omega$ = 0.1¶

s states¶

Numerical values for variables¶¶

In [4]:

nhbar, nm, nc = o* 1, o* 1, o* 1

nJ = o* 0
nq, nw = o* 1/10, o* 1/10

nbeta1, nbeta2 = o* 2, o* 2
nr0 = -o* (nbeta1 - se.sqrt(nbeta1**2 + 8*nbeta2))/(4*nbeta2)

nA0 = nm*nc**2*(3*nhbar*nw-nm*nc**2)
nA1 = nhbar**2*nc**2*nJ*(nJ+1)
nA2 = nc**2*(nm**2*nw**2+5*nhbar*nq)
nA3 = 2*nm*nc**2*nq*nw
nA4 = nq**2*nc**2

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta1:nbeta1, beta2:nbeta2}
ps0 = {beta1:nbeta1, beta2:nbeta2, r0:nr0, A0:nA0, A1:nA1, A2:nA2, A3:nA3, A4:nA4, hbar:nhbar, c:nc}

Initialize AIM solver¶

In [5]:

%%time
# pass lambda_0, s_0 and variable values to aim class
sexticOsc_J0 = aim(l0, s0, pl0, ps0)
sexticOsc_J0.display_parameters()
sexticOsc_J0.display_l0s0(2)
sexticOsc_J0.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

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

$\displaystyle \beta_{1} = 2,~~\beta_{2} = 2$

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

$\displaystyle A_{0} = -7/10,~~A_{1} = 0,~~A_{2} = 51/100,~~A_{3} = 1/50,~~A_{4} = 1/100,~~\beta_{1} = 2,~~\beta_{2} = 2,~~\hbar = 1,~~c = 1,~~r_{0} = (-1/8)*(2 - sqrt(20))$

$\displaystyle \begin{align} \lambda_0 &= \left[8 r + 4 - \frac{2}{r} \right.\\ s_0 &= \left[- En^{2} + 0.01 r^{6} - 0.02 r^{4} - 15.49 r^{2} - 16.0 r + 8.7 + \frac{4.0}{r} \right. \end{align} $

CPU times: user 299 ms, sys: 8.52 ms, total: 308 ms
Wall time: 309 ms

Calculation of Taylor series coefficients of λ0λ0 and s0s0¶

In [6]:

%%time
# create coefficients for improved AIM
sexticOsc_J0.c0()
sexticOsc_J0.d0()
sexticOsc_J0.cndn()

CPU times: user 3min 13s, sys: 624 ms, total: 3min 13s
Wall time: 3min 13s

The solution¶

In [7]:

%%time
sexticOsc_J0.get_arb_roots(showRoots='+r', printFormat="{:25.17f}")

0001       3.36445725728443535
0011       2.06224866748193186       3.71453601866614535
0021       1.79773849991486153       2.95853552211924131       4.21129759560483331       5.54649535945588575
0031       1.73438805878535504       2.68502761908710836       3.66764104203528586       4.70180770713959515
0041       1.72427025965769588       2.60822476751384039       3.44645622846246563       4.29618250218738717
0051       1.72357609738502431       2.59874453628846712       3.39721350565403320       4.15416305172843344
0061       1.72356698735998023       2.59852833595200994       3.39495720783361346       4.14052146658052801
0071       1.72356712662823824       2.59853205012559990       3.39500303047643314       4.14086311173335481
0081       1.72356712428290494       2.59853197531069727       3.39500187424648369       4.14085177697641391
0091       1.72356712431358431       2.59853197665893558       3.39500190340645597       4.14085218844201362
0101       1.72356712431425220       2.59853197667054612       3.39500190331404967       4.14085218204871021
0111       1.72356712431419405       2.59853197666834400       3.39500190327169256       4.14085218152151186
0121       1.72356712431419431       2.59853197666837365       3.39500190327286154       4.14085218154933829
0131       1.72356712431419441       2.59853197666837798       3.39500190327295278       4.14085218155059375
0141       1.72356712431419441       2.59853197666837795       3.39500190327295062       4.14085218155052230
0151       1.72356712431419441       2.59853197666837794       3.39500190327295025       4.14085218155051528
0161       1.72356712431419441       2.59853197666837794       3.39500190327295024       4.14085218155051524
0171       1.72356712431419441       2.59853197666837794       3.39500190327295024       4.14085218155051527
0181       1.72356712431419441       2.59853197666837794       3.39500190327295024       4.14085218155051527
0191       1.72356712431419441       2.59853197666837794       3.39500190327295024       4.14085218155051527
0201       1.72356712431419441       2.59853197666837794       3.39500190327295024       4.14085218155051527
CPU times: user 24.8 s, sys: 72.6 ms, total: 24.9 s
Wall time: 24.7 s

p states¶

In [8]:

%%time
nJ = o* 1
nA1 = nhbar**2*nc**2*nJ*(nJ+1)

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta1:nbeta1, beta2:nbeta2}
ps0 = {beta1:nbeta1, beta2:nbeta2, r0:nr0, A0:nA0, A1:nA1, A2:nA2, A3:nA3, A4:nA4, hbar:nhbar, c:nc}

# pass lambda_0, s_0 and variable values to aim class
sexticOsc_J1 = aim(l0, s0, pl0, ps0)
sexticOsc_J1.display_parameters()
sexticOsc_J1.display_l0s0(2)
sexticOsc_J1.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
sexticOsc_J1.c0()
sexticOsc_J1.d0()
sexticOsc_J1.cndn()

# the solution
sexticOsc_J1.get_arb_roots(showRoots='+r', printFormat="{:25.17f}")

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

$\displaystyle \beta_{1} = 2,~~\beta_{2} = 2$

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

$\displaystyle A_{0} = -7/10,~~A_{1} = 2,~~A_{2} = 51/100,~~A_{3} = 1/50,~~A_{4} = 1/100,~~\beta_{1} = 2,~~\beta_{2} = 2,~~\hbar = 1,~~c = 1,~~r_{0} = (-1/8)*(2 - sqrt(20))$

$\displaystyle \begin{align} \lambda_0 &= \left[8 r + 4 - \frac{2}{r} \right.\\ s_0 &= \left[- En^{2} + 0.01 r^{6} - 0.02 r^{4} - 15.49 r^{2} - 16.0 r + 8.7 + \frac{4.0}{r} + \frac{2.0}{r^{2}} \right. \end{align} $

0001       4.82187291925769935
0011       2.77436290277683605       4.51757239251514998
0021       2.31776026114239723       3.53387962016352688       4.82832238391099067       6.24051437733093774
0031       2.18609163892967229       3.14346458552742025       4.15525665322138078       5.21421277084299846
0041       2.15929167601387908       3.01161948774625525       3.85209730709065412       4.71827906112208084
0051       2.15696391699341434       2.99030773430463900       3.76752637998941861       4.51621778785434398
0061       2.15692574507381821       2.98965068332447030       3.76206600925062292       4.48938346167429866
0071       2.15692635440678831       2.98966274907362220       3.76218736811237571       4.49012058102627214
0081       2.15692634335014822       2.98966248118758365       3.76218391425178413       4.49009220361544042
0091       2.15692634351752849       2.98966248682243458       3.76218401715307969       4.49009345238929151
0101       2.15692634352009137       2.98966248683912324       3.76218401624244177       4.49009342433839500
0111       2.15692634351979751       2.98966248683045813       3.76218401610122617       4.49009342288485385
0121       2.15692634351979992       2.98966248683062559       3.76218401610652583       4.49009342299305611
0131       2.15692634351980047       2.98966248683064331       3.76218401610684465       4.49009342299665589
0141       2.15692634351980047       2.98966248683064307       3.76218401610683300       4.49009342299633993
0151       2.15692634351980047       2.98966248683064301       3.76218401610683151       4.49009342299631502
0161       2.15692634351980047       2.98966248683064301       3.76218401610683149       4.49009342299631515
0171       2.15692634351980047       2.98966248683064301       3.76218401610683149       4.49009342299631529
0181       2.15692634351980047       2.98966248683064301       3.76218401610683149       4.49009342299631531
0191       2.15692634351980047       2.98966248683064301       3.76218401610683149       4.49009342299631531
0201       2.15692634351980047       2.98966248683064301       3.76218401610683149       4.49009342299631531
CPU times: user 3min 34s, sys: 616 ms, total: 3min 35s
Wall time: 3min 35s

d states¶

In [9]:

%%time
nJ = o* 2
nA1 = nhbar**2*nc**2*nJ*(nJ+1)

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta1:nbeta1, beta2:nbeta2}
ps0 = {beta1:nbeta1, beta2:nbeta2, r0:nr0, A0:nA0, A1:nA1, A2:nA2, A3:nA3, A4:nA4, hbar:nhbar, c:nc}

# pass lambda_0, s_0 and variable values to aim class
sexticOsc_J2 = aim(l0, s0, pl0, ps0)
sexticOsc_J2.display_parameters()
sexticOsc_J2.display_l0s0(2)
sexticOsc_J2.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
sexticOsc_J2.c0()
sexticOsc_J2.d0()
sexticOsc_J2.cndn()

# the solution
sexticOsc_J2.get_arb_roots(showRoots='+r', printFormat="{:25.17f}")

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

$\displaystyle \beta_{1} = 2,~~\beta_{2} = 2$

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

$\displaystyle A_{0} = -7/10,~~A_{1} = 6,~~A_{2} = 51/100,~~A_{3} = 1/50,~~A_{4} = 1/100,~~\beta_{1} = 2,~~\beta_{2} = 2,~~\hbar = 1,~~c = 1,~~r_{0} = (-1/8)*(2 - sqrt(20))$

$\displaystyle \begin{align} \lambda_0 &= \left[8 r + 4 - \frac{2}{r} \right.\\ s_0 &= \left[- En^{2} + 0.01 r^{6} - 0.02 r^{4} - 15.49 r^{2} - 16.0 r + 8.7 + \frac{4.0}{r} + \frac{6.0}{r^{2}} \right. \end{align} $

0001       6.71796084610507275
0011       3.46876391454353238       5.29952520154266871
0021       2.81569356954559334       4.09612757090051198       5.43289869872423591       7.06102335434582362
0031       2.60425013001630686       3.59134722913684855       4.63454617543588826       5.71857232137262456
0041       2.55238230915766284       3.39727882131377914       4.25001824758429294       5.13562634414079308
0051       2.54677288983622212       3.35778858029083239       4.12196893224062712       4.86998485495169561
0061       2.54665755663931973       3.35615070252050024       4.11062667466230666       4.82326398291710868
0071       2.54665948021553734       3.35618288535783258       4.11090129123394419       4.82462842675151302
0081       2.54665944256855839       3.35618209675222476       4.11089245340522475       4.82456698944013368
0091       2.54665944322262613       3.35618211600644651       4.11089276443508312       4.82457030718275413
0101       2.54665944322897081       3.35618211596221919       4.11089275987701934       4.82457020670488945
0111       2.54665944322789604       3.35618211593420368       4.11089275947750660       4.82457020339416883
0121       2.54665944322790947       3.35618211593494049       4.11089275949773220       4.82457020376092585
0131       2.54665944322791156       3.35618211593500017       4.11089275949867385       4.82457020376923346
0141       2.54665944322791155       3.35618211593499885       4.11089275949862284       4.82457020376802745
0151       2.54665944322791154       3.35618211593499861       4.11089275949861770       4.82457020376795044
0161       2.54665944322791154       3.35618211593499860       4.11089275949861766       4.82457020376795204
0171       2.54665944322791154       3.35618211593499860       4.11089275949861769       4.82457020376795258
0181       2.54665944322791154       3.35618211593499860       4.11089275949861769       4.82457020376795262
0191       2.54665944322791154       3.35618211593499860       4.11089275949861769       4.82457020376795262
0201       2.54665944322791154       3.35618211593499860       4.11089275949861769       4.82457020376795262
CPU times: user 3min 36s, sys: 600 ms, total: 3min 36s
Wall time: 3min 36s

f states¶

In [10]:

%%time
nJ = o* 3
nA1 = nhbar**2*nc**2*nJ*(nJ+1)

# parameters of lambda_0 (pl0) and s_0 (ps0)
pl0 = {beta1:nbeta1, beta2:nbeta2}
ps0 = {beta1:nbeta1, beta2:nbeta2, r0:nr0, A0:nA0, A1:nA1, A2:nA2, A3:nA3, A4:nA4, hbar:nhbar, c:nc}

# pass lambda_0, s_0 and variable values to aim class
sexticOsc_J3 = aim(l0, s0, pl0, ps0)
sexticOsc_J3.display_parameters()
sexticOsc_J3.display_l0s0(2)
sexticOsc_J3.parameters(En, r, nr0, nmax=201, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
sexticOsc_J3.c0()
sexticOsc_J3.d0()
sexticOsc_J3.cndn()

# the solution
sexticOsc_J3.get_arb_roots(showRoots='+r', printFormat="{:25.17f}")

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

$\displaystyle \beta_{1} = 2,~~\beta_{2} = 2$

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

$\displaystyle A_{0} = -7/10,~~A_{1} = 12,~~A_{2} = 51/100,~~A_{3} = 1/50,~~A_{4} = 1/100,~~\beta_{1} = 2,~~\beta_{2} = 2,~~\hbar = 1,~~c = 1,~~r_{0} = (-1/8)*(2 - sqrt(20))$

$\displaystyle \begin{align} \lambda_0 &= \left[8 r + 4 - \frac{2}{r} \right.\\ s_0 &= \left[- En^{2} + 0.01 r^{6} - 0.02 r^{4} - 15.49 r^{2} - 16.0 r + 8.7 + \frac{4.0}{r} + \frac{12.0}{r^{2}} \right. \end{align} $

0001       8.77525795921239121
0011       4.15929959993047792       6.07848575023158345
0021       3.30463604821916714       4.65002182978383676       6.02988975486865881
0031       3.00635414470137319       4.03218572313430491       5.10759026270516701       6.21709867002266860
0041       2.92172472116760770       3.77170763260858999       4.64288107816498526       5.54929888680002931
0051       2.91054412552470313       3.70740177946627438       4.46519632461924794       5.21873246962450026
0061       2.91025538772731920       3.70385381768827230       4.44424799488364677       5.14475518696186483
0071       2.91026042038319194       3.70392834021144135       4.44479401778934940       5.14693047601440175
0081       2.91026031403008912       3.70392632226785393       4.44477407754451303       5.14681450023362899
0091       2.91026031613854039       3.70392637928431049       4.44477490974680271       5.14682236718064835
0101       2.91026031614830746       3.70392637886211645       4.44477489239945666       5.14682205633173198
0111       2.91026031614504222       3.70392637878365096       4.44477489142071161       5.14682205043574196
0121       2.91026031614509916       3.70392637878638569       4.44477489148839600       5.14682205154605937
0131       2.91026031614510571       3.70392637878655962       4.44477489149078389       5.14682205155980257
0141       2.91026031614510564       3.70392637878655383       4.44477489149059150       5.14682205155571029
0151       2.91026031614510562       3.70392637878655305       4.44477489149057576       5.14682205155550072
0161       2.91026031614510562       3.70392637878655303       4.44477489149057580       5.14682205155550964
0171       2.91026031614510562       3.70392637878655304       4.44477489149057589       5.14682205155551152
0181       2.91026031614510562       3.70392637878655304       4.44477489149057590       5.14682205155551164
0191       2.91026031614510562       3.70392637878655304       4.44477489149057590       5.14682205155551163
0201       2.91026031614510562       3.70392637878655304       4.44477489149057590       5.14682205155551163
CPU times: user 3min 33s, sys: 660 ms, total: 3min 34s
Wall time: 3min 34s

In [ ]: