Two electron atoms

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var('a'); assume (a>0); phi1s(r)=(a^3/pi)^(1/2)*exp(-a*r) t1= integral(phi1s(r)*(-(1/2)*diff(phi1s(r),r,2) )*4*pi*r^2,r,0,oo); t2=integral(phi1s(r)*(-diff(phi1s(r),r,1) )*4*pi*r,r,0,oo);t1+t2 

1/2*a^2

1/2*a^2

var('Z,a'); assume (a>0);assume (Z>0); phi1s(r)=(a^3/pi)^(1/2)*exp(-a*r) V= integral(phi1s(r)^2*(-Z)*4*pi*r,r,0,oo);V 

-Z*a

-Z*a

var('r,a'); assume (a>0); V(r) = 1/r - ( a +1/r )*exp(-2*a*r); phi2s(r)=(a^3/(32*pi))^(1/2)*(2 - a*r)*exp(-a*r/2) #phi2s(r)=exp(-a*r); J= integral(phi2s(r)^2*V(r)*4*pi*r^2,r,0,oo); J 

17/81*a

17/81*a

var('r,a'); assume (a>0); V = 1/r - ( a +1/r )*exp(-2*a*r); phi2s=exp(-a*r); J= integral(phi2s^2*V*4*pi*r^2,r,0,oo); J 

5/8*pi/a^2

5/8*pi/a^2