A new variational approach to the hydrogen atom in magnetic fields

Abstract

We develop a simple and useful variational method to obtain approximate eigenenergies and expectation values of r 2 and p2 for a hydrogenlike atom in a magnetic field. Results agree quite well with accurate numerically computed ones in the whole range of field intensities.

Bounds to the energy of quantum-mechanical models from variational functionals

“…. (7) In principle, one can set the value of m according to the asymptotic behavior of V(x) to force F(a,x) to have a maximum or minimum. For instance, if V(x) exhibits a minimum Vo the choice m = 1 leads to the rough bound EO > VOa 2 which can be improved by a better choice of m .…”

“…For instance, if V(x) exhibits a minimum Vo the choice m = 1 leads to the rough bound EO > VOa 2 which can be improved by a better choice of m . If we substitute (7) into (4b) and solve for a in terms of x we obtain a = (m -1 )/ [ 2m (xd ) ] " which when replaced back into (7) gives us the variational functional…”

“…A variational functional is a semiclassical energy function in configuration space built in such a way that the value at one of its stationary points (typically a minimum) is an approximation to a bound-state energy of a given quantum-mechanical model [1][2][3][4][5][6][7][8][9][10][11][12]. Such functionals have been applied to a wide variety of problems: anharmonic oscillators [ 1,2,4,6,9, lo], central field models [ 4,8], and atoms (either isolated [3,5,11,12] or in magnetic fields [7,9,10]). An attempt to improve the accuracy of the functionals by means of perturbation theory has led to a method for summing strongly divergent perturbation series [ 13-1 5 1.…”