Bargmann Space Methods in First-Order Optics

M. W. AlMasri

We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition $|α| < \tfrac{1}{2}$ for generalized Gaussian trial functions $ψ(z) = e^{αz^2 + βz}$ through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator ($\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4$), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions ($ψ_n(z) = z^n$) in the Segal--Bargmann space -- while providing rigorous upper bounds $E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)$ for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., $x^3 + x^4$), showing that displacement parameters are essential to capture parity breaking and stabilization effects ($E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdots$).

S. N. Karbelkar, R. S. Sirohi

Ravinder Rupchand Puri

Hong-Yi Fan

By virtue of the technique of integration within an ordered product of operators we show that the ordinary Bargmann space basis function can be extended to two-variable Hermite polynomials, the latter can span a generalized Bargmann space. The corresponding two-mode Hermite polynomial states are thus introduced and their properties are investigated.

J.R. Chamberlain

D. K. Sunko, Jerzy Ciosłowski

Abstract The three-dimensional harmonic oscillator is solved in Bargmann space. The treatment is pedagogically more transparent than the standard ones, at the price of introducing the Bargmann transform in the context of the one-dimensional oscillator. The standard solid harmonics are similarly derived with minimal technical effort, amounting to a complete self-contained exposition suitable for introductory courses in quantum mechanics or mathematical methods of physics. It provides an early exposure to wavelets, with important contemporary applications in signal analysis and quantum optics.