WentzelKramersBrillouin method in the Bargmann representation
Abstract
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the boundstate problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a selfconsistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phasespace representation of the quantum state vectors.
 Publication:

Physical Review A
 Pub Date:
 December 1989
 DOI:
 10.1103/PhysRevA.40.6814
 Bibcode:
 1989PhRvA..40.6814V
 Keywords:

 Harmonic Oscillators;
 Quantum Mechanics;
 WentzelKramerBrillouin Method;
 Degrees Of Freedom;
 Eigenvalues;
 Hamiltonian Functions;
 RayleighRitz Method;
 Schroedinger Equation;
 Physics (General);
 03.65.Sq;
 Semiclassical theories and applications