$title An Application from Quantum Mechanics (QUANTUM,SEQ=300)

$onText
An application from quantum mechanics:
Find energy eigenvalues of the anharmonic oscillator with g = 1
in the Gaussian and Post-Gaussian variational methods.

Erwin Kalvelagen, May 2004

Ogura, A, Post-Gaussian variational method for quantum anharmonic
oscillator, 1999. Laboratory of Physics, College of Science and
Technology, Nihon University,arXiv:physics/9905056 v1 28 May 1999

Keywords: nonlinear programming, discontinuous derivatives, quantum mechanics,
          statistics, energy eigenvalues, quantum anharmonic oscillator
$offText

Variable
   ham   'expected value of hamiltonian'
   alpha 'variational parameter'
   n     'variational parameter (n = 1: Gaussian trial function)';

Equation hamiltonian;

Scalar g / 1 /;

hamiltonian..
   ham =e= (sqr(n)/2)*(gamma(2 - 1/(2*n))/gamma(1/(2*n)))*(alpha**(1/n))
        +  (1/2)*(gamma(3/(2*n))/gamma(1/(2*n)))*(alpha**(-1/n))
        +   g*(gamma(5/(2*n))/gamma(1/(2*n)))*(alpha**(-2/n));

alpha.lo = 0.0001;
alpha.up = 10;
alpha.l  = 1;

* gaussian variational method
n.fx = 1;

Model m / hamiltonian /;

solve m minimizing ham using dnlp;

Parameter results(*,*);
results('Gaussian','Ground') = ham.l;
results('Gaussian','alpha')  = alpha.l;
results('Gaussian','n')      = n.l;

* post-gaussian variational method
n.lo = 0.001;
n.up = 10;

solve m minimizing ham using dnlp;

results('Post-Gaussian','Ground') = ham.l;
results('Post-Gaussian','alpha')  = alpha.l;
results('Post-Gaussian','n')      = n.l;

option  decimals = 6;
display results;