Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform

Abstract

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.

Nonlinear dynamics from linear quantum evolutions

“…Therefore, starting with a normalized vector lying on the unit sphere S(H), equation (1.3) defines an immersion of V into S(H) which is, again, injective. However, it is worth noticing that even though V is a vector space, the immersion does not respect linearity, i.e., ψ v 1 + ψ v 2 = ψ v 1 +v 2 in general, and i(V) is not a linear subspace of H. In summary, generalized coherent states allow us to define injective maps from a given set to the Hilbert space H associated with a quantum system, every point of the set labelling a given vector in H. This is the main property we will exploit in the first part of the paper in order to study how to induce one parameter groups of transformations on subsets of states starting from unitary maps on H. Using coherent states it is possible to interpret these induced maps as classical-like dynamics in the framework of classical-to quantum transition (see [2,3,4]). However, it is worth stressing that the following discussion can be extended to submanifolds which do not possess classical-like properties, representing, therefore, generic constraints.…”

Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

“…There are many interesting applications of this simple observation [4,5,33,36,37], in particular, Proposition 3.3 ( [33,36]). Let G be a Lie group with a Lie algebra g and ρ be a representation of G on a Hilbert space L 2 (R n ).…”

“…The approach is illustrated here by the crucial example of the Heisenberg group H 1 , however the technique is not limited to this case, cf. [5,33,36]. The topics of coherent states and covariant transform (also known under many other names) are extensively covered in the existing literature, e.g., [28,Section 13], [30,Appendix V.2], and [9,18,19,22,34,37,40], and we refer to authoritative surveys [3,15,20] for further references.…”