The Hagedorn–Hermite Correspondence

Abstract

We investigate the relationship between the semiclassical wave packets of Hagedorn and the Hermite functions by establishing a relationship between their ladder operators. This Hagedorn-Hermite correspondence provides a unified view as well as simple proofs of some essential results on the Hagedorn wave packets. Particularly, we show that Hagedorn's ladder operators are a natural set of ladder operators obtained from the position and momentum operators using the symplectic group. This construction reveals an algebraic structure of the Hagedorn wave packets, and explains the relative simplicity of Hagedorn's parametrization compared to the rather intricate construction of the generalized squeezed states. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets, generalizing Hagedorn's one-dimensional result to multi-dimensions. The Hagedorn-Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets based on the generating function for the Hermite functions. This result, in turn, reveals the relationship between the Hagedorn polynomials and the Hermite polynomials.

Approximation of Semiclassical Expectation Values by Symplectic Gaussian Wave Packet Dynamics

“…We drop the time and spatial dependence for brevity here. Just as we have done in the above subsection, rewriting the cubic term in (22) for α using (10), we obtain…”

“…What if one uses the classical Hamiltonian system for (q, p) as in (4) of Hagedorn? As discussed in Remark 2.5, we have α as shown in (22). Then, as shown in Appendix A.3, we have…”

“…Overview. We first give a brief review of the Hagedorn wave packets [6][7][8][9] (see also [2] and [22]), and then derive the evolution equation satisfied by the Gaussian wave packet (3) where the parameters evolve in time according to our equations (5). The evolution equation resembles the Schrödinger equation (1a) but differs by a residual term.…”

“…In fact, one may think of them as a generalization of the Hermite functions, and also can find a unitary operator on L 2 (R d ) that relates each element of the Hagedorn wave packet with the Hermite function of the same index [22]. Because of this correspondence, we refer to φ n as an |n|-th excited state with |n| := d i=1 n i for any multi-index n ∈ N d 0 .…”

Computing quantum dynamics in the semiclassical regime

“…Hagedorn's semiclassical wave packets are closely related to, or coincide with, generalized coherent states or generalized squeezed states, as studied by Combescure (1992), Robert (2007) and Combescure and Robert (2012). The precise relationship was expounded by Lasser and Troppmann (2014) and more recently by Ohsawa (2019). Ohsawa (2018) derived differential equations for the variational approximation by a single Hagedorn function ϕ ε k of arbitrary index k, for an approximate Hamiltonian.…”