  For low dimension systems admitting a moment equation representation (MER),
the development of an effective eigenenergy bounding theory applicable to all
discrete states had remained elusive, until now. Whereas Handy et al (1988
Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it Moment
Problem} based, Eigenvalue Moment Method (EMM), for generating arbitrarily
tight bounds to the multidimensional, positive, bosonic ground state, its
extension to arbitrary excited states seemed intractable. We have discovered a
new, moment representation based, quantization formalism that achieves this.
Unlike EMM, no convex optimization methods are required. The entire formulation
is algebraic. As a result of our preliminary investigation, we are able to
match, or surpass, the excellent, but intricate, analysis of Kravchenko et al
(1996 Phys. Rev. A 54 287) with respect to the quadratic Zeeman effect, for a
broad range of magnetic field strengths. Unlike their analysis, the proposed
method is simple, involves no truncations, and the projection of the quantum
operator is exact, within each moment subspace. Our new approach, the
Orthogonal Polynomial Projection Quantization-Bounding Method (OPPQ-BM),
exploits the implicit bounding capabilities of a previous method developed by
Handy and Vrinceanu (2013 J. of Phys. A: Math. Theor. 46 135202). What emerges
is a completely new type of analysis (i.e. constrained quadratic form
minimization) that validates the importance of moment equation representations
for quantizing physical systems. Whereas the underlying principles of EMM
guarantee it to be more efficient than OPPQ-BM, the ability to implement
algebraic computations, as opposed to pursuing nonlinear convex optimization
methods (which can be relaxed through linear programming alternatives)
recommends OPPQ-BM. We give an overview of the new method with applications.
