Academic Articles & Supplements

Bound states of the Yukawa Potential from Hidden Supersymmetry

Mauro Napsuciale, Simón Rodríguez

Author

Mauro Napsuciale, Simón Rodríguez

Title

Bound states of the Yukawa Potential from Hidden Supersymmetry

Description

In this notebook we provide a code for the calculation of eigenvalues and eigenstates for the bound states of the Yukawa Potential following the algorithm described in the following papers i) Phys.Lett.B 816 (2021) 136218 (arXiv: 2012.12969), ii) arXiv:2102.07160, submitted for publication to Progress of Theoretical and Experimental Physics (PTEP); you can freely use it but we kindly ask you to cite these papers as the original source.

Solution to the Yukawa potential from hidden supersymmetry

In this notebook we provide a code for the calculation of eigenvalues and eigenstates for the bound states of the Yukawa Potential following the algorithm described in the following papers i) Phys.Lett.B 816 (2021) 136218 (arXiv: 2012.12969),ii) arXiv:2102.07160, submitted for publication to Progress of Theoretical and Experimental Physics (PTEP), you can freely use it but we kindly ask you to cite these papers as the original source.The solution uses the hidden supersymmetry of the Yukawa potential and an expansion of the potential V(r)=-

in powers of δ=

where

denotes the Bohr radius of the system. Energy levels are given by Rydberg units, i.e. E=

, and this code yield results for ϵ. The eigestates are given by

nlm

(r,θ,ϕ)=

(r,δ)

(θ,ϕ)

with

(r,δ)

(x,δ)

, where x=

.Our solution uses an expansion of the potential to order “MaxOrder”, and the code yields the solution as a Taylor series in δ to any order k between k=0 and k=”MaxOrder”. Also, the code yields the solution

(x,δ)

for n between n=1 and n=”DualOrder”. The value of both “MaxOrder” and “DualOrder” can be set in the next section (Input Parameters) and are the only parameters you should change in this notebook. The core of the code is in the section named “Supersymmetric Hamiltonians”. You must just run the cells in this section. We do not recomend you to modify them.The last section, “Eigenvalues, eigenstates and phenomenology”, yields the solutions of the Yukawa potential. We give some examples of how to ask for the eigenstates and eigenvalues and to make some consistency checks. This section includes some phenomenology, the calculation of critical screenning lengths and probabilities.

Input Parameters

MaxOrder=31;DualOrder=15;

Supersymmetric Hamiltonians

Order Duality 0 and oc

Dmax=MaxOrder;Nmax=DualOrder;(*Order0*)FSP[0,L_,k_,r_]:=

k-1

∑

u=1

r-1

∑

s=1

If[r-s≥u||s≥k-u,0,blkr[[1,u,r-s]]blkr[[1,k-u,s]]]-(2L+r+3)blkr[[1,k,r+1]];FI[0,L_,k_]:=-

(-1)

L+1

Factorial[k]

;blkr=Table[0,{k,1,Nmax},{i,1,Dmax},{j,1,Dmax}];For[k=2,k<=Dmax,k++,blkr[[1,k,k-1]]=FI[0,L,k]];Fori=3,i<=Dmax,i++,Forj=i-2,j≥1,j--,blkr[[1,i,j]]=-

L+1

FSP[0,L,i,j]//Simplify;(*Orderoc*)FSP[m_,L_,k_,r_]:=

k-1

∑

u=1

r-1

∑

s=1

If[r-s≥u||s≥k-u,0,blkr[[m+1,u,r-s]]blkr[[m+1,k-u,s]]]-(2L+2m+r+3)blkr[[m+1,k,r+1]];FI[m_,L_,k_]:=-

(-1)

L+m+1

Factorial[k]

;Foroc=1,oc<Nmax,oc++,For[k=2,k<=Dmax,k++,blkr[[oc+1,k,k-1]]=FI[oc,L,k]];Fori=3,i<=Dmax,i++,Forj=i-2,j≥1,j--,blkr[[oc+1,i,j]]=-

(L+oc+1)

FSP[oc,L,i,j]-2(j+1)

∑

oc1=1

blkr[[oc1,i,j+1]]//Simplify;(*SuperpotencialsandFactorizationConstantCL*)Clear[k];B[x_,L1_,oc_,k_]:=

k-1

∑

r=1

blkr[[oc+1,k,r]]

/.LL1ω[x_,L_,δ_,oc_,k_]:=

∑

s=1

B[x,L,oc,s]

;W[x_,L_,δ_,oc_,k_]:=

L+oc+1

+ω[x,L,δ,oc,k];Cl[L1_,δ_,oc_,k_]:=-

(L1+oc+1)

+2δ+

∑

r=2

2

∑

oc1=1

blkr[[oc1,r,1]]+(2L1+2oc+3)*blkr[[oc+1,r,1]]

/.LL1Cl[L1_,δ_,0,k_]:=-

(L1+1)

+2δ+

∑

r=2

((2*L1+3)*blkr[[1,r,1]])*

/.LL1

Eigenfunctions & Eigenvalues

Energies

En[n_,L_,δ_,k_]:=Cl[L,δ,n-L-1,k];

Eigenfunciones

ϕ[x_,L_,δ_,oc_,k_,0]:=

L+oc+1

Exp-

L+oc+1

*Exp[-Integrate[ω[x,L,δ,oc,k],x]]ϕ[x_,L_,δ_,oc_,k_,r_]:=-D[ϕ[x,L,δ,oc,k,r-1],x]+W[x,L,δ,oc-r,k]ϕ[x,L,δ,oc,k,r-1]u[x_,n_,L_,δ_,k_]:=ϕ[x,L,δ,n-L-1,k,n-L-1]

Normalization

Norm2[n_,L_,δ_,k_]:=Normal[Series[Module[{x},Integrate[Normal[

(Series[u[x,n,L,δ,k],{δ,0,k}])

],{x,0,∞}]],{δ,0,2*k}]//Simplify];un[x_,n_,L_,δ_,k_]:=u[x,n,L,δ,k]Sqrt[

Norm2[n,L,δ,k]];uns[x_,n_,L_,δ_,k_]:=Normal[Series[un[x,n,L,δ,k],{δ,0,k}]];

Eigenvalues, eigenstates and phenomenology

Energies

The energies are given by

En[n,l,δ,k]

where:
n=principal quantum number
l=angular momentum number
k=delta expansion order, less than or equal to MaxOrder
You must use numeric values for n, l and k.

En[1,0,δ,31]

-1+2δ-

145

757

69433

1536

321449

2304

2343967

5120

24316577

15360

2536041607

442368

47860811537

2211840

145923785051

1720320

159957248809633

464486400

42949294634584421

29727129600

46466864975430973

7431782400

236707218502703471

8493465600

2532503612322014333

19818086400

572761545510214991993

951268147200

2154903296646828026987

739875225600

3175701402265890412257977

219742942003200

6460837466357014142516729

87897176801280

1077124814774677420120851827

2812709657640960

1077106412854448519892862394327

527383060807680000

5556455763967436473731350818903

498616712036352000

15344972048407680571904214989889581

246815272457994240000

3266322590080908603607725231964280071

9214436838431784960000

19022474359491942077284802980167651521

9214436838431784960000

727090287686525516767769990699387693327

59235665389918617600000

61761404321733830903082674090545487890633

829299315458860646400000

Eigenfunctions

The eigenfunctions normalized to order

are given by

uns[x,n,l,δ,k]

You must use numeric values for n, l and k.

uns[x,3,0,δ,5]//Simplify

194400

-x/3



x(1248

+12

(-250+789δ)+4

(400+10800δ-110673

)+2700

(40-184δ-747

+14067

)-90

(80-160δ-990

+14823

)-405(320-19440

+196560

-3250125

+57158703

)+270x(320-19440

+196560

-3250125

+57158703

)-30

(320-9720

+146880

-3386205

+60293403

))

Checking Schrodinger equation

We write the Schrodinger equation as:

-1



L(L+1)

2*Exp[-δ*x]

-ϵ

and we check that it vanishes to order k

SchEc[x_,n_,L_,δ_,k_]:=NormalSeries-

D[u[x,n,L,δ,k],{x,2}]

u[x,n,L,δ,k]

L(L+1)

2*Exp[-δ*x]

-En[n,L,δ,k],{δ,0,k};SchEc[x,8,0,δ,5]

Critical Screening

Critical screening is determined using Pade’s approximants

EnPadeMm1M Is the Pade’s approximant [M+1,M]

EnPadeMM Is the Pade’s approximant [M,M]

The critical screening lies between the first positive zeros of these approximantst

M=14;EnPadeMm1M=PadeApproximant[En[2,1,δ,2*M+1],{δ,0,{M+1,M}}];EnPadeMM=PadeApproximant[En[2,1,δ,2*M+1],{δ,0,{M,M}}];NSolve[EnPadeMm1M0,δ,Reals,WorkingPrecision20]NSolve[EnPadeMM0,δ,Reals,WorkingPrecision20]Plot[{EnPadeMm1M,EnPadeMM},{δ,0.2,0.3}]

{{δ-1.3174479486069275645},{δ0.6980602143962254501},{δ0.3922052512567456889},{δ-0.3669091969847330551},{δ0.28646548678206990427},{δ0.23966938223504799538},{δ0.21989862925331728444},{δ-0.20445418697180269923},{δ-0.14032571438849857118},{δ-0.10701327259877170132},{δ-0.08696232934759496153},{δ-0.07325043590827443527},{δ-0.06254730153814269979},{δ-0.05347630608419904364},{δ-0.04521889866810545039}}

{{δ0.6075230663065127198},{δ-0.4606048319768178204},{δ0.3645120454176600586},{δ0.27618954529464645945},{δ0.23655299977615848688},{δ-0.22982728229689627089},{δ0.21997911271397763045},{δ-0.15116324451086655805},{δ-0.11263182182599526477},{δ-0.09025656152690614128},{δ-0.07544457788695117790},{δ-0.06415785728615235930},{δ-0.05468015761593800895},{δ-0.04610044319092197069}}

Probabilities

P10Pad77[x_,δ_]=PadeApproximant[uns[x,1,0,δ,16],{δ,0,{8,8}}]/.a01;

P105[x_,1,0,δ_]=

(uns[x,1,0,δ,5]/.a01)

;Plot10=Plot[{P105[x,1,0,0],P105[x,1,0,1/5],P105[x,1,0,1/2],P105[x,1,0,3/4],

(P10Pad77[x,3/4])

},{x,0,10},PlotRange{0,1.4},FrameTrue,AxesFalse,FrameLabel"r/

","|r

",PlotStyle{Directive[Black],Directive[Dashed,Red],Directive[Dotted,Blue],Directive[DotDashed,Purple],Directive[Dashed,Black]},AspectRatio0.75,PlotLegendsPlaced[{"δ=0","δ=1/5","δ=1/2","δ=3/4","δ=3/4:[8,8]"},{0.8,0.7}],ImageSize400,LabelStyleDirective[Bold,10],PlotPoints4]

Clear[PadeEn10]PadeEn10[δ_,M_,N_]:=PadeApproximant[En[1,0,δ,31],{δ,0,{M,N}}];Pade10=Plot[{Evaluate[PadeEn10[δ,11,10]],Evaluate[PadeEn10[δ,10,10]],Evaluate[PadeEn10[δ,16,15]],Evaluate[PadeEn10[δ,15,15]],0},{δ,1.188,1.1905},PlotRange{-3*

-6

,4*

-7

},AxesFalse,FrameTrue,PlotStyle{Directive[Red,Dashing[{0.05,0.01}]],Directive[Red,Dashed],Directive[Blue],Directive[Blue,Dotted],Black},AspectRatio1,PlotLegendsPlaced[{"[11/10]","[10/10]","[16/15]","[15/15]"},{0.8,0.3}],ImageSize450,LabelStyleDirective[Bold,12],FrameLabel{"δ","

(δ)"}]Export["PadeE10.pdf",Pade10]

PadeE10.pdf

E10=Plot[{Evaluate[PadeEn10[δ,5,5]],En[1,0,δ,3],En[1,0,δ,6],En[1,0,δ,9]},{δ,0,1.2},PlotRange{-1,0},AxesFalse,FrameTrue,PlotStyle{Directive[Blue],Directive[Red,Dashing[{0.05,0.01}]],Directive[Red,Dashed],Directive[Blue,Dotted],Black},AspectRatio1,PlotLegendsPlaced[{"[5/5]","k=3","k=6","k=9"},{0.8,0.7}],ImageSize450,LabelStyleDirective[Bold,12],FrameLabel{"δ","

(δ)"}]Export["E10.pdf",E10]

E10.pdf

Cite this as: Mauro Napsuciale, Simón Rodríguez, "Bound states of the Yukawa Potential from Hidden Supersymmetry" from the Notebook Archive (2021), https://notebookarchive.org/2021-05-7ea9cb6