[Paper Review] The time-dependent Schroedinger equation, Riccati equation and Airy functions

Nathan A. Lanfear, Sergeĭ K. Suslov|ArXiv.org|Mar 20, 2009

Quantum Mechanics and Non-Hermitian Physics51 references20 citations

TL;DR

This paper constructs exact Green functions for time-dependent Schrödinger equations with quadratic potentials using Airy functions and Riccati equation solutions. It derives explicit propagators in coordinate and momentum representations, solves the Cauchy initial value problem, and establishes connections to hypergeometric functions and Bargmann's functions for quantum parametric oscillators.

ABSTRACT

We construct the Green functions (or Feynman's propagators) for the Schroedinger equations of the form $iψ_{t}+{1/4}ψ_{xx}\pm tx^{2}ψ=0$ in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schroedinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is stablished.

Motivation & Objective

• To solve the Cauchy initial value problem for time-dependent Schrödinger equations with variable quadratic Hamiltonians.
• To derive explicit Green functions (propagators) in both coordinate and momentum representations.
• To establish connections between the solutions and special functions such as Airy functions, hypergeometric functions, and Bargmann's functions.
• To classify integrable cases of the quantum parametric oscillator via solutions of a characterization Riccati equation.
• To provide exact solutions useful for testing numerical methods in time-dependent quantum mechanics.

Proposed method

• Employing the ansatz $ \psi = A(t) e^{iS(x,y,t)} $ with a quadratic phase function $ S(x,y,t) = \alpha(t)x^2 + \beta(t)xy + \gamma(t)y^2 $.
• Reducing the time-dependent Schrödinger equation to a system of Riccati equations for $ \alpha(t), \beta(t), \gamma(t) $, with the first being $ \frac{d\alpha}{dt} - t + \alpha^2 = 0 $.
• Transforming the Riccati equation into a second-order linear ODE $ \mu'' - t\mu = 0 $ via $ \alpha = \mu'/\mu $, solvable in terms of Airy functions.
• Using the fundamental solution $ \mu(t) $ to express the Green function as $ G(x,y,t) = \frac{1}{\sqrt{\pi i a(t)}} \exp\left(i \frac{a'(t)x^2 - 2xy + b(t)y^2}{a(t)} \right) $, where $ a(t), b(t) $ are Airy functions.
• Applying gauge transformations and group-theoretical methods to relate transition amplitudes to Bargmann's functions.
• Deriving transformation formulas for hypergeometric functions using Clausen's formula and reflection identities for the gamma function.

Experimental results

Research questions

• RQ1 How can the Green function for the time-dependent Schrödinger equation with a $ \pm t x^2 $ potential be constructed explicitly?
• RQ2 What is the role of Airy functions in solving the Riccati equation arising from the quadratic Schrödinger Hamiltonian?
• RQ3 How are the transition amplitudes for the quantum parametric oscillator related to hypergeometric functions and Bargmann's functions?
• RQ4 In what way do the solutions of the Riccati equation classify integrable cases of the time-dependent Schrödinger equation?
• RQ5 What is the group-theoretical significance of the transition amplitudes derived from the propagator?

Key findings

• The Green function for the Schrödinger equation $ i\psi_t + \frac{1}{4}\psi_{xx} + t x^2 \psi = 0 $ is explicitly given as $ G(x,y,t) = \frac{1}{\sqrt{\pi i a(t)}} \exp\left(i \frac{a'(t)x^2 - 2xy + b(t)y^2}{a(t)} \right) $, where $ a(t), b(t) $ are Airy functions.
• The solution to the Cauchy initial value problem is obtained in both coordinate and momentum representations using the derived propagator.
• The transition amplitudes for the quantum parametric oscillator are expressed in terms of a hypergeometric function $ {}_2F_1 $, with explicit transformation formulas derived using Clausen's formula and gamma function identities.
• The group-theoretical meaning of the transition amplitudes is established through their relation to Bargmann's functions, linking the solution to representation theory.
• The Wronskian of the Airy functions $ a(t) $ and $ b(t) $ is $ -1 $, and their derivatives satisfy $ W(a'(t), b'(t)) = t $, confirming the linear independence and normalization.
• The system of Riccati equations is solved via the Airy function solution of $ \mu'' - t\mu = 0 $, with initial conditions $ \mu(0) = 0 $, $ \mu'(0) = 1/2 $, yielding the full propagator in closed form.