Eliminating the Second-Order Time Dependence from the Time Dependent Schrödinger Equation Using Recursive Fourier Transforms

Abstract

A strategy is developed for writing the time-dependent Schrödinger Equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the first without using the time ordering operator. The energy distribution is calculated for a number of standard perturbation theory examples at first- and second-order. Possible applications include characterization of photonic spectra for bosonic sampling and four-wave mixing in quantum computation and Bardeen tunneling amplitude in quantum mechanics.

28 pages,10 figures. Revision: minor modification of title. Inserted additional discussion on four-wave mixing. Revision: minor notational clarifications and corrections. Revision: title updated per reviewer suggestions. Figures improved for clarity. Minor adjustments to ordering of sections

References in corpus (8)

• Heralded Generation of Ultrafast Single Photons in Pure Quantum States
• Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber
• Multi-qubit gates and Schrödinger cat states in an optical clock
• Multiboson Correlation Interferometry with arbitrary single-photon pure states
• Attosecond electron microscopy by free-electron homodyne detection
• Ultimate quantum sensitivity in the estimation of the delay between two interfering photons through frequency-resolving sampling
• Self-localized Solitons of a q-Deformed Quantum System
• Calculation of tunneling current across Trapezoidal potential barrier in a Scanning Tunneling Microscope