ラゲールの陪関数の再定義

Lnk:=(x,n,l)->expand(diff(E^x*diff(x^n*E^(-x),x$n),x$l));
Lnk2:=(x,n,l)->expand(diff(E^x*diff(x^(n+l)*E^(-x),x $ (n+l)),x $ (2*l+1)));
for n from 1 to 3 do
  for l from 0 to n do
    if n-l-1>=0 then
      print(hold(Lnk)(x,n+l,2*l+l)=Lnk(x,n+l,2*l+1));
      print(hold(Lnk2)(x,n,l)=Lnk2(x,n,l));     
    end_if
  end_for
end_for;
Lnk3:=(Z,r,n,l)->Simplify(subs(Lnk2(x,n,l),x=2*r*Z/(n*a0)));

(x, n, l) -> expand(diff(E^x*diff(x^n/E^x, x $ n), x $ l))
(x, n, l) -> expand(diff(E^x*diff(x^(n + l)/E^x, x $ l + n), x $ 2*l + 1))
Lnk(x, 1, 0) = -1
Lnk2(x, 1, 0) = -1
Lnk(x, 2, 0) = 2*x - 4
Lnk2(x, 2, 0) = 2*x - 4
Lnk(x, 3, 3) = -6
Lnk2(x, 2, 1) = -6
Lnk(x, 3, 0) = - 3*x^2 + 18*x - 18
Lnk2(x, 3, 0) = - 3*x^2 + 18*x - 18
Lnk(x, 4, 3) = 24*x - 96
Lnk2(x, 3, 1) = 24*x - 96
Lnk(x, 5, 6) = -120
Lnk2(x, 3, 2) = -120
(Z, r, n, l) -> Simplify(subs(Lnk2(x, n, l), x = (2*r*Z)/(n*a0)))

水素様原子の動径関数

delete a0;//Bohr radius 0.529*1E-10 [m]
R:=(Z,r,n,l)->-(2*Z/(n*a0))^(3/2)*sqrt((n-l-1)!/(2*n*(n+l)!))*E^(-r*Z/(n*a0))*r^l*Lnk3(Z,r,n,l);

(Z, r, n, l) -> -(((2*Z)/(n*a0))^(3/2)*((n - l - 1)!/(2*n*(n + l)!))^(1/2)*r^l*Lnk3(Z, r, n, l))/E^((Z*r)/(a0*n))

動径関数の式表示

for n from 1 to 7 do
  for l from 0 to n do
    if n-l-1>=0 then
      print(hold(R)(1,r,n,l)=Simplify(R(1,r,n,l)));
    end_if
  end_for
end_for;

R(1, r, 1, 0) = 2*exp(-r/a0)*(1/a0)^(3/2)
R(1, r, 2, 0) = (2^(1/2)*exp(-r/(2*a0))*(1/a0)^(5/2)*(2*a0 - r))/2
R(1, r, 2, 1) = (6^(1/2)*r*exp(-r/(2*a0))*(1/a0)^(3/2))/2
R(1, r, 3, 0) = (4*3^(1/2)*exp(-r/(3*a0))*(1/a0)^(7/2)*(27*a0^2 - 18*a0*r + 2*r^2))/81
R(1, r, 3, 1) = (8*6^(1/2)*r*exp(-r/(3*a0))*(1/a0)^(5/2)*(6*a0 - r))/27
R(1, r, 3, 2) = (4*30^(1/2)*r^2*exp(-r/(3*a0))*(1/a0)^(3/2))/9
R(1, r, 4, 0) = -(exp(-r/(4*a0))*((r*(12*a0 - r)^2)/(2*a0^3) - 96)*(1/a0)^(3/2))/16
R(1, r, 4, 1) = (15^(1/2)*r*exp(-r/(4*a0))*(1/a0)^(3/2)*(80*a0^2 - 20*a0*r + r^2))/(16*a0^2)
R(1, r, 4, 2) = (3*5^(1/2)*r^2*exp(-r/(4*a0))*(1/a0)^(5/2)*(12*a0 - r))/4
R(1, r, 4, 3) = (3*35^(1/2)*r^3*exp(-r/(4*a0))*(1/a0)^(3/2))/2
R(1, r, 5, 0) = (16*5^(1/2)*exp(-r/(5*a0))*(1/a0)^(3/2)*(9375*a0^4 - 7500*a0^3*r + 1500*a0^2*r^2 - 100*a0*r^3 + 2*r^4))/(15625*a0^4)
R(1, r, 5, 1) = (16*30^(1/2)*r*exp(-r/(5*a0))*(1/a0)^(3/2)*(3750*a0^3 - 1125*a0^2*r + 90*a0*r^2 - 2*r^3))/(3125*a0^3)
R(1, r, 5, 2) = (24*70^(1/2)*r^2*exp(-r/(5*a0))*(1/a0)^(3/2)*(525*a0^2 - 70*a0*r + 2*r^2))/(625*a0^2)
R(1, r, 5, 3) = (96*70^(1/2)*r^3*exp(-r/(5*a0))*(1/a0)^(5/2)*(20*a0 - r))/125
R(1, r, 5, 4) = (144*70^(1/2)*r^4*exp(-r/(5*a0))*(1/a0)^(3/2))/25
R(1, r, 6, 0) = (6^(1/2)*exp(-r/(6*a0))*(1/a0)^(13/2)*(174960*a0^5 - 145800*a0^4*r + 32400*a0^3*r^2 - 2700*a0^2*r^3 + 90*a0*r^4 - r^5))/4374
R(1, r, 6, 1) = (210^(1/2)*r*exp(-r/(6*a0))*(1/a0)^(11/2)*(68040*a0^4 - 22680*a0^3*r + 2268*a0^2*r^2 - 84*a0*r^3 + r^4))/1458
R(1, r, 6, 2) = (4*105^(1/2)*r^2*exp(-r/(6*a0))*(1/a0)^(3/2)*(9072*a0^3 - 1512*a0^2*r + 72*a0*r^2 - r^3))/(243*a0^3)
R(1, r, 6, 3) = (4*35^(1/2)*r^3*exp(-r/(6*a0))*(1/a0)^(3/2)*(648*a0^2 - 54*a0*r + r^2))/(9*a0^2)
R(1, r, 6, 4) = (40*7^(1/2)*r^4*exp(-r/(6*a0))*(1/a0)^(5/2)*(30*a0 - r))/3
R(1, r, 6, 5) = 40*77^(1/2)*r^5*exp(-r/(6*a0))*(1/a0)^(3/2)
R(1, r, 7, 0) = (32*7^(1/2)*exp(-r/(7*a0))*(1/a0)^(3/2)*(37059435*a0^6 - 31765230*a0^5*r + 7563150*a0^4*r^2 - 720300*a0^3*r^3 + 30870*a0^2*r^4 - 588*a0*r^5 + 4*r^6))/(5764801*a0^6)
R(1, r, 7, 1) = (256*21^(1/2)*r*exp(-r/(7*a0))*(1/a0)^(3/2)*(3529470*a0^5 - 1260525*a0^4*r + 144060*a0^3*r^2 - 6860*a0^2*r^3 + 140*a0*r^4 - r^5))/(823543*a0^5)
R(1, r, 7, 2) = (384*105^(1/2)*r^2*exp(-r/(7*a0))*(1/a0)^(3/2)*(21*a0 - r)*(21609*a0^3 - 3087*a0^2*r + 105*a0*r^2 - r^3))/(117649*a0^4)
R(1, r, 7, 3) = (960*42^(1/2)*r^3*exp(-r/(7*a0))*(1/a0)^(3/2)*(61740*a0^3 - 6615*a0^2*r + 210*a0*r^2 - 2*r^3))/(16807*a0^3)
R(1, r, 7, 4) = (1440*154^(1/2)*r^4*exp(-r/(7*a0))*(1/a0)^(3/2)*(2695*a0^2 - 154*a0*r + 2*r^2))/(2401*a0^2)
R(1, r, 7, 5) = (5760*231^(1/2)*r^5*exp(-r/(7*a0))*(1/a0)^(5/2)*(42*a0 - r))/343
R(1, r, 7, 6) = (2880*3003^(1/2)*r^6*exp(-r/(7*a0))*(1/a0)^(3/2))/49

動径関数のグラフ表示

DIGITS:=3;//3桁表示
a0:=0.529*1E-10;
for n from 1 to 5 do
  for l from 0 to n do
    if n-l-1>=0 then
      f:=float(Simplify(R(1,r,n,l)));
      print(hold(R)(1,r,n,l)=f);
      plot(plot::Function2d(f,r=0..4e-9));
    end_if
  end_for
end_for;

3
5.29*10^(-11)
R(1, r, 1, 0) = 5.2*10^15*exp(-1.89*10^10*r)
MuPAD graphics
R(1, r, 2, 0) = -1.0*exp(-9.45*10^9*r)*(3.47*10^25*r - 3.68*10^15)
MuPAD graphics
R(1, r, 2, 1) = 3.18*10^15*r*exp(-9.45*10^9*r)
MuPAD graphics
R(1, r, 3, 0) = exp(-6.3*10^9*r)*(1.59*10^35*r^2 - 7.56*10^25*r + 6.0*10^15)
MuPAD graphics
R(1, r, 3, 1) = -1.18*10^14*r*exp(-6.3*10^9*r)*(3.02*10^11*r - 96.0)
MuPAD graphics
R(1, r, 3, 2) = 6.33*10^15*r^2*exp(-6.3*10^9*r)
MuPAD graphics
R(1, r, 4, 0) = -1.0*exp(-4.73*10^9*r)*(5.49*10^44*r^3 - 6.96*10^35*r^2 + 2.21*10^26*r - 1.56*10^16)
MuPAD graphics
R(1, r, 4, 1) = r*exp(-4.73*10^9*r)*(2.25*10^35*r^2 - 2.38*10^26*r + 5.03*10^16)
MuPAD graphics
R(1, r, 4, 2) = -1.0*r^2*exp(-4.73*10^9*r)*(8.24*10^25*r - 5.23*10^16)
MuPAD graphics
R(1, r, 4, 3) = 2.31*10^16*r^3*exp(-4.73*10^9*r)
MuPAD graphics
R(1, r, 5, 0) = 9.3*10^13*exp(-3.78*10^9*r)*(1.63*10^40*r^4 - 4.32*10^31*r^3 + 3.43*10^22*r^2 - 9.07*10^12*r + 600.0)
MuPAD graphics
R(1, r, 5, 1) = -1.0*r*exp(-3.78*10^9*r)*(9.85*10^44*r^3 - 2.34*10^36*r^2 + 1.55*10^27*r - 2.73*10^17)
MuPAD graphics
R(1, r, 5, 2) = r^2*exp(-3.78*10^9*r)*(5.97*10^35*r^2 - 1.11*10^27*r + 4.38*10^17)
MuPAD graphics
R(1, r, 5, 3) = -1.0*r^3*exp(-3.78*10^9*r)*(3.16*10^26*r - 3.34*10^17)
MuPAD graphics
R(1, r, 5, 4) = 1.25*10^17*r^4*exp(-3.78*10^9*r)
MuPAD graphics

動径分布

DIGITS:=3;//3桁表示
a0:=0.529*1E-10;
for n from 1 to 5 do
  for l from 0 to n do
    if n-l-1>=0 then
      f:=float(Simplify(R(1,r,n,l)))^2*r^2;
      print(hold(R)(1,r,n,l)=f);
      plot(plot::Function2d(f,r=0..4e-9));
    end_if
  end_for
end_for;

3
5.29*10^(-11)
R(1, r, 1, 0) = 2.7*10^31*r^2*exp(-3.78*10^10*r)
MuPAD graphics
R(1, r, 2, 0) = 1.0*r^2*exp(-1.89*10^10*r)*(3.47*10^25*r - 3.68*10^15)^2
MuPAD graphics
R(1, r, 2, 1) = 1.01*10^31*r^4*exp(-1.89*10^10*r)
MuPAD graphics
R(1, r, 3, 0) = r^2*exp(-1.26*10^10*r)*(1.59*10^35*r^2 - 7.56*10^25*r + 6.0*10^15)^2
MuPAD graphics
R(1, r, 3, 1) = 1.39*10^28*r^4*exp(-1.26*10^10*r)*(3.02*10^11*r - 96.0)^2
MuPAD graphics
R(1, r, 3, 2) = 4.0*10^31*r^6*exp(-1.26*10^10*r)
MuPAD graphics
R(1, r, 4, 0) = 1.0*r^2*exp(-9.45*10^9*r)*(5.49*10^44*r^3 - 6.96*10^35*r^2 + 2.21*10^26*r - 1.56*10^16)^2
MuPAD graphics
R(1, r, 4, 1) = r^4*exp(-9.45*10^9*r)*(2.25*10^35*r^2 - 2.38*10^26*r + 5.03*10^16)^2
MuPAD graphics
R(1, r, 4, 2) = 1.0*r^6*exp(-9.45*10^9*r)*(8.24*10^25*r - 5.23*10^16)^2
MuPAD graphics
R(1, r, 4, 3) = 5.32*10^32*r^8*exp(-9.45*10^9*r)
MuPAD graphics
R(1, r, 5, 0) = 8.65*10^27*r^2*exp(-7.56*10^9*r)*(1.63*10^40*r^4 - 4.32*10^31*r^3 + 3.43*10^22*r^2 - 9.07*10^12*r + 600.0)^2
MuPAD graphics
R(1, r, 5, 1) = 1.0*r^4*exp(-7.56*10^9*r)*(9.85*10^44*r^3 - 2.34*10^36*r^2 + 1.55*10^27*r - 2.73*10^17)^2
MuPAD graphics
R(1, r, 5, 2) = r^6*exp(-7.56*10^9*r)*(5.97*10^35*r^2 - 1.11*10^27*r + 4.38*10^17)^2
MuPAD graphics
R(1, r, 5, 3) = 1.0*r^8*exp(-7.56*10^9*r)*(3.16*10^26*r - 3.34*10^17)^2
MuPAD graphics
R(1, r, 5, 4) = 1.57*10^34*r^10*exp(-7.56*10^9*r)
MuPAD graphics