With maxima, im not able to figure out how to do this numerical integral:
quad_qags( -psi0(x, beta)*diff(psi0(x,beta), x,2 )/2 + V(x)*psi0(x,beta)**2, x, -inf,inf);
it gives back the noun form, is there a way to do this? maple gives back an answer of -0.60....
This is the context of the problem, which comes from the variational problem in QM, starting from a gaussian function psi0, and
computing the energy:
V(x):= -1/sqrt(1+x**2) ;
psi0(x, beta):= (sqrt(beta)*%e^(-((beta^2*x^2)/2)))/%pi^(1/4);
Energy(beta):= integrate(-psi0(x, beta)*diff(psi0(x,beta), x,2 )/2 + V(x)*psi0(x,beta)**2, x,-inf,inf);
This is not possible to do analytically, i think, so i tried the same with quad_qags.
EDIT Here is where i have reached with beta=1, and expand():
quad_qagi ( expand( %o28) , x, -inf, inf, 'epsrel=1d-8);
(%o35) quad_qagi(-(integrate(psi0(x,1)*('diff(psi0(x,1),x,2))+(2*psi0(x,1)^2)/sqrt(x^2+1),x,-inf,inf)/2),x,-inf,inf,epsrel=1.0*10^-8,epsabs=0.0,limit=200)
Here's what I get after fixing the call to quad_qagi and working around the problem with -inf by replacing it with minf.
(%i2) V(x):= -1/sqrt(1+x**2) ;
- 1
(%o2) V(x) := ------------
2
sqrt(1 + x )
(%i3) psi0(x, beta):= (sqrt(beta)*%e^(-((beta^2*x^2)/2)))/%pi^(1/4);
2 2
beta x
- --------
2
sqrt(beta) %e
(%o3) psi0(x, beta) := -----------------------
1/4
%pi
(%i4) Energy(beta):= quad_qagi (-psi0(x, beta)*diff(psi0(x,beta), x,2 )/2 + V(x)*psi0(x,beta)**2, x, minf, inf);
(%o4) Energy(beta) :=
- psi0(x, beta) diff(psi0(x, beta), x, 2)
quad_qagi(-----------------------------------------
2
2
+ V(x) psi0 (x, beta), x, minf, inf)
(%i5) Energy(1);
(%o5) [- 0.6098866396410092, 4.3393502531511445e-9, 270, 0]
which matches your previous result.