A138160 A triangular sequence of coefficients of the expansion of a Green's function for the radial Morse potential with x being the kinetic energy and t being the radius: Hamiltonian; H=K+V=x+Exp[-2*t]-2*Exp[t];G*Exp[t*x)=Exp[x*t]/(t-H); p(t,x)=Exp[t*x]/(t-x-Exp[-2*t] + 2*Exp[-t]).

1, -1, 1, -1, 4, -4, 3, -2, 1, -24, 36, -27, 13, -6, 3, -1, 182, -354, 330, -198, 85, -28, 10, -4, 1, -1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1, 19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6, 1, -264334, 850990, -1246504, 1121960, -711480, 345268, -135639, 44997, -12922, 3171, -644, 119, -28, 7, -1

Offset: 1

Author

Roger L. Bagula, May 04 2008

Keywords

• uned
• tabf
• sign

Comments

The row sums are:A133942; (-1)^n*n!;

{1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800};

Plotting the relative Kinetic curves gives even peaks and odd Serpentine curves:

s = Table[Plot[g[[n]]/(1 - x)^(n + 1), {x, 0, 10}], {n, 1, Length[g]}].

It appears this method solves the spectrum of relative excited states for a Morse potential Hamiltonian system.

Examples

			Triangle begins:
  {1},
  {-1, 1, -1},
  {4, -4, 3, -2, 1},
  {-24, 36, -27, 13, -6, 3, -1},
  {182, -354, 330, -198, 85, -28, 10, -4, 1},
  {-1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1},
  {19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6, 1},
  {-264334, 850990, -1246504, 1121960, -711480, 345268, -135639, 44997, -12922, 3171, -644, 119, -28, 7, -1},
  ...
		

References

• A. Messiah, Quantum mechanics, vol. 2, page 712, 795-800, North Holland, 1969.

Programs

Mathematica

p[t_, x_] = FullSimplify[Exp[t*x]/(t - x - Exp[ -2*t] + 2*Exp[ -t])] g = Table[ FullSimplify[ExpandAll[n!*(1 - x)^(n + 1)*SeriesCoefficient[ Series[p[t, x], {t, 0, 30}], n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[n!*(1 - x)^(n + 1)*SeriesCoefficient[Series[p[t, x], {t, 0, 30}], n]]], x], {n, 0, 10}]; Flatten[a]

Formula

p(t,x)=Exp[t*x]/(t - x - Exp[ -2*t] + 2*Exp[ -t])=sum(P(x,n)*t^n/n!,m{n,0,Infinity}); out_n,m=Coefficients(n!*(1-x)^(n+1)*P(x,n)).