The Time-Dependent Schrödinger Equation, Riccati Equation, and Airy Functions ^†

Abstract

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger 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 the 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 established. The relevant bibliography, to the best of our knowledge, is addressed.

1. Introduction

In this paper, we discuss explicit solutions of the Cauchy initial value problem for the one-dimensional Schrödinger equations

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\pm t{x}^2\psi =0,\psi \left(x,0\right)=\phi \left(x\right),$$

(1)

for the wave function $\psi$ and its time (t) and space (x) derivatives with suitable initial data on the entire real line $\mathbb{R}$. The corresponding Green functions are found in terms of compositions of elementary and Airy functions in the coordinate and momentum representations. It is known that the Airy equation describes the motion of a quantum particle in the neighborhood of the turning point on the basis of the stationary, or time-independent, Schrödinger equation in the WKB (Wentzel–Kramers–Brillouin) approximation [1,2,3,4,5,6,7]. Basic potentials of nonrelativistic and relativistic quantum mechanics, which can be integrated in the Nikiforov–Uvarov paradigm using a computer algebra system, are discussed in Refs. [8,9,10] (see also [11,12] for a traditional approach). On the contrary, here we consider an application of Airy functions to the time-dependent Schrödinger equations for certain parametric oscillators.

It is worth noting that Green’s functions for the Schrödinger equation are known explicitly only in a few special cases. An important example of this source is the forced harmonic oscillator originally considered by Richard Feynman in his path integral approach to nonrelativistic quantum mechanics [13,14,15,16,17] (see also [18,19,20]). Since then, this problem and its special and limiting cases have been extensively discussed; see Refs. [6,21,22,23,24,25] for the simple harmonic oscillator and Refs. [26,27,28,29,30] for a particle in a constant external field, and references therein.

On one occasion, Dick had to correct a mathematics education colleague who mistakenly claimed at a joint Mathematical Association of America and American Mathematical Society meeting that a simple Riccati equation (see Equation (A31)) cannot be explicitly solved, despite an extensive bibliography on this topic (see, e.g., [31,32,33,34,35,36,37,38,39]). This “sad story” was our original motivation, in the late 2000s, to discuss the complete integrability of the corresponding parametric harmonic oscillator problems in quantum mechanics—the 100th anniversary of the birth of Schrödinger’s wave mechanics is coming this year [8].

The case of the Schrödinger equation with a general variable quadratic Hamiltonian, the so-called generalized harmonic oscillator, was investigated in Refs. [40,41]. Detailed reviews with computerized proofs are presented in Ref. [42], among other points (see also [19,43,44,45,46,47,48,49,50,51] and references therein for invariants, eigenstates, statistics, and other aspects). In our general approach, all exactly solvable quadratic models mentioned are classified in terms of explicit solutions of a certain characterization equation. Here, we present extra examples that are integrable in terms of Airy functions, and all our calculations are independently verified.

These exactly solvable models may be of interest in a general treatment of linear and nonlinear evolution equations (see [20,52,53,54,55,56,57,58,59,60,61,62,63,64,65] and references therein). Moreover, these explicit solutions can also be useful when testing numerical methods of solving the time-dependent Schrödinger equations with variable coefficients. The explicit formula for the transition amplitudes of the quantum parametric oscillator problem derived in this paper is also relevant. Applications to motion in variable perpendicular magnetic and electric fields are discussed in Refs. [19,40]. Our results here provide yet another integrable case, in terms of Airy functions.

2. Green’s Function: “Increasing Case”

The fundamental solution of the time-dependent Schrödinger equation

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}+t{x}^2\psi =0$$

(2)

has the following closed form,

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{\pi ia\left(t\right)}}}exp\left(i{\displaystyle \frac{a^{\prime }\left(t\right)x^2-2xy+b\left(t\right)y^2}{a\left(t\right)}}\right),t>0,$$

(3)

which is a Green’s function in terms of elementary and Airy functions and the prime denotes the time derivative; see Appendix A and Appendix C for more details.

It is worth noting that a more general particular solution has the form

$$\psi =K\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{2\pi i\mu \left(t\right)}}}{e}^{i\left(\alpha \left(t\right)x^2+\beta \left(t\right)xy+\gamma \left(t\right)y^2\right)},$$

(4)

where $\mu =c_{1}a\left(t\right)+c_{2}b\left(t\right)$, with $\mu \left(0\right)=c_2\ne 0,$ ${\mu }^{\prime }\left(0\right)=c_1$, and

$$\begin{array}{ccc}& & \alpha ={\displaystyle \frac{c_1{a}^{\prime }\left(t\right)+c_2{b}^{\prime }\left(t\right)}{c_{1}a\left(t\right)+c_{2}b\left(t\right)}},\alpha \left(0\right)={\displaystyle \frac{c_1}{c_2}},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & \beta ={\displaystyle \frac{c_2\beta \left(0\right)}{c_{1}a\left(t\right)+c_{2}b\left(t\right)}},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & \gamma =\gamma \left(0\right)-{\displaystyle \frac{c_2{\beta }^2\left(0\right)a\left(t\right)}{4\left(c_{1}a\left(t\right)+c_{2}b\left(t\right)\right)}}.\hfill \end{array}$$

(7)

This can be straightforwardly verified by a direct substitution into the system (A31)–(A33) (see [41,42] for a general case).

3. Initial Value Problem: “Increasing Case”

The solution of the Cauchy initial value problem

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}+t{x}^2\psi =0,\psi \left(x,0\right)=\phi \left(x\right)$$

(8)

is given by the superposition principle in an integral form

$$\psi \left(x,t\right)={\int }_{-\infty }^{\infty }G\left(x,y,t\right)\phi \left(y\right)dy,$$

(9)

where one should justify the interchange of differentiation and integration for a suitable initial function $\phi$ on $\mathbb{R}$ (a rigorous proof for a general variable quadratic Hamiltonian will be given elsewhere; we proceed further formally).

The special case $\phi \left(y\right)=K\left(z,y,0\right)$ of the time evolution operator (9) is

$$K\left(x,y,t\right)={\int }_{-\infty }^{\infty }G\left(x,z,t\right)K\left(z,y,0\right)dz$$

(10)

and its inversion is given by

$$G\left(x,y,t\right)=\mu \left(0\right)\left|\beta \left(0\right)\right|{\int }_{-\infty }^{\infty }K\left(x,z,t\right)K^{*}\left(y,z,0\right)dz,$$

(11)

where the star denotes the complex conjugate. The Euler–Gaussian–Fresnel integral [66,67],

$${\int }_{-\infty }^{\infty }{e}^{i\left(a{z}^2+2bz\right)}dz=\sqrt{{\displaystyle \frac{\pi i}{a}}}{e}^{-i{b}^2/a},Ima\ge 0,$$

(12)

(here a and b are complex-valued constants) allows us to obtain the following transformation,

$$\begin{array}{ccc}& & \mu \left(t\right)=2\mu \left(0\right){\mu }_0\left(t\right)\left(\alpha \left(0\right)+{\gamma }_0\left(t\right)\right),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \alpha \left(t\right)={\alpha }_0\left(t\right)-{\displaystyle \frac{{\beta }_0^2\left(t\right)}{4\left(\alpha \left(0\right)+{\gamma }_0\left(t\right)\right)}},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & \beta \left(t\right)=-{\displaystyle \frac{\beta \left(0\right){\beta }_0\left(t\right)}{2\left(\alpha \left(0\right)+{\gamma }_0\left(t\right)\right)}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & \gamma \left(t\right)=\gamma \left(0\right)-{\displaystyle \frac{{\beta }^2\left(0\right)}{4\left(\alpha \left(0\right)+{\gamma }_0\left(t\right)\right)}},\hfill \end{array}$$

(16)

and its inverse

$$\begin{array}{ccc}& & {\mu }_0\left(t\right)={\displaystyle \frac{2\mu \left(t\right)}{\mu \left(0\right){\beta }^2\left(0\right)}}\left(\gamma \left(0\right)-\gamma \left(t\right)\right),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & {\alpha }_0\left(t\right)=\alpha \left(t\right)+{\displaystyle \frac{{\beta }^2\left(t\right)}{4\left(\gamma \left(0\right)-\gamma \left(t\right)\right)}},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & {\beta }_0\left(t\right)=-{\displaystyle \frac{\beta \left(0\right)\beta \left(t\right)}{2\left(\gamma \left(0\right)-\gamma \left(t\right)\right)}},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & {\gamma }_0\left(t\right)=-\alpha \left(0\right)+{\displaystyle \frac{{\beta }^2\left(0\right)}{4\left(\gamma \left(0\right)-\gamma \left(t\right)\right)}}\hfill \end{array}$$

(20)

in the cases (10) and (11), respectively. Referring to Appendix C, direct calculation shows that the solutions (A36) and (5)–(7) satisfy these transformation rules. It is worth noting that the transformation (A49)–(A52) allows us to derive the Green’s function considered from any regular solution of the system (A31)–(A33).

4. “Oscillatory Case”

The time-dependent Schrödinger equation

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}-t{x}^2\psi =0$$

(21)

has a fundamental solution (Green’s function) given by

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{-\pi ia\left(-t\right)}}}exp\left(-i{\displaystyle \frac{a^{\prime }\left(-t\right)-2xy+b\left(-t\right)y^2}{a\left(-t\right)}}\right),t>0$$

(22)

and the solution of the initial value problem is given by the integral (9). For details on a more general particular solution see Appendix C.

5. Momentum Representation

The Schrödinger equation (2) takes the form

$$i{\displaystyle \frac{\partial \psi }{\partial t}}-t{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}-{\displaystyle \frac{1}{4}}{x}^2\psi =0$$

(23)

in the momentum representation (see, e.g., Ref. [46] for more details). The Green function is given by

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{-4\pi i{b}^{\prime }\left(t\right)}}}exp\left({\displaystyle \frac{b\left(t\right)x^2-2xy+a^{\prime }\left(t\right)y^2}{4i{b}^{\prime }\left(t\right)}}\right),t>0.$$

(24)

For details on a more general particular solution see Appendix C. The “oscillatory” case is similar (see Appendix C).

6. Gauge Transformation

The time-dependent Schrödinger equation

$$i{\displaystyle \frac{\partial \psi }{\partial t}}=\left({\displaystyle \frac{1}{4}}{\left(p-A\left(x,t\right)\right)}^2+V\left(x,t\right)\right)\psi ,$$

(25)

where $p=i^{-1}\partial /\partial x$ is the linear momentum operator, can be gauge-transformed using

$$\psi =e^{-if\left(x,t\right)}{\psi }^{\prime }$$

(26)

into the following form,

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}+t{x}^2\psi +{\displaystyle \frac{i}{2t}}\left(2x{\displaystyle \frac{\partial \psi }{\partial x}}+\psi \right)=0$$

(27)

with $V=t{x}^2$ and choosing the gauge field $A=0$ and gauge function $f=-x^2/t$, as shown in Equation (A89) in Appendix C. The time-dependent Schrödinger equation (27) has a solution of the form

$$\psi \left(x,t\right)=e^{-i{x}^2/t}{\int }_{-\infty }^{\infty }G\left(x,y,t\right)\phi \left(y\right)dy,$$

(28)

where the Green function $G\left(x,y,t\right)$ is given by Equation (3), and one is solving an initial value problem similar to Equation (8). Although this solution is not continuous when $t\to 0^{+}$, the singularity is isolated into the phase and satisfies the following modified initial condition,

$$\underset{t\to 0^{+}}{lim}{e}^{i{x}^2/t}\psi \left(x,t\right)=\phi \left(x\right),$$

(29)

which reveals the structure of the singularity of the corresponding wave function at the origin and shows that one can recover the initial data as $t\to 0^{+}$. Equation (27) makes the squeezing term explicit, illuminating how squeezing dynamics play out at different time scales. We leave further details to the reader.

7. Particular Solutions of Nonlinear Schrödinger Equations

One can find solutions of the corresponding nonlinear Schrödinger equations following Refs. [40,46]. For example, consider the case

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}+t{x}^2\psi =h\left(t\right){\left|\psi \right|}^{2s}\psi ,s\ge 0,$$

(30)

where $h\left(t\right)$ represents the (real-valued) time-dependent nonlinearity coefficient, and s is the nonlinearity exponent (i.e., the power index of the nonlinearity), and look for a particular solution of the form

$$\psi =\psi \left(x,t\right)=K_h\left(x,y,t\right)={\displaystyle \frac{e^{i\varphi }}{\sqrt{\mu \left(t\right)}}}{e}^{i\left(\alpha \left(t\right)x^2+\beta \left(t\right)xy+\gamma \left(t\right)y^2+\kappa \left(t\right)\right)},\varphi =\mathrm{constant}.$$

(31)

Then, Equations (A31)–(A33) hold, with the general solution given by Equations (5)–(7). In addition,

$${\displaystyle \frac{d\kappa }{dt}}=-{\displaystyle \frac{h\left(t\right)}{{\mu }^s\left(t\right)}},\kappa \left(t\right)=\kappa \left(0\right)-{\int }_0^t{\displaystyle \frac{h\left(\tau \right)}{{\mu }^s\left(\tau \right)}}d\tau .$$

(32)

The integral in Equation (32) can be explicitly evaluated in some special cases, for example, when $h\left(t\right)=\lambda {\mu }^{\prime }\left(t\right):$

$$\kappa \left(t\right)=\left\{\begin{array}{cc}\kappa \left(0\right)-{\displaystyle \frac{\lambda }{1-s}}\left({\mu }^{1-s}\left(t\right)-{\mu }^{1-s}\left(0\right)\right),\hfill & \mathrm{when}s\ne 1,\hfill \\ \kappa \left(0\right)-\lambda ln\left({\displaystyle \frac{\mu \left(t\right)}{\mu \left(0\right)}}\right),\hfill & \mathrm{when}s=1.\hfill \end{array}\right.$$

(33)

Here, $\mu \left(0\right)\ne 0$ (see [40,46] for more details). An example of a discontinuity of the initial data can be constructed by the method of Ref. [46]. Other cases are investigated in a similar way (see also [63] and references therein for more general solutions).

8. Quantum Parametric Oscillator and Airy Functions

The time-dependent Schrödinger equation for a parametric oscillator can be written in the form

$$i\hslash {\displaystyle \frac{\partial \Psi }{\partial t}}=H\Psi$$

(34)

with the Hamiltonian

$$H={\displaystyle \frac{p^2}{2m}}+{\displaystyle \frac{m{\omega }^2\left(t\right)}{2}}{x}^2,p={\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial x}},$$

(35)

where ℏ is the reduced Planck constant, m is the mass of the particle, and $\omega \left(t\right)$ is the time-dependent oscillation frequency. The initial value problem of the form

$$i\hslash {\displaystyle \frac{\partial \Psi }{\partial t}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2\Psi }{\partial x^2}}+{\displaystyle \frac{m{\omega }^2}{2}}\left(\omega t+\delta \right)x^2\Psi ,\Psi \left(x,0\right)=\Phi \left(x\right),$$

(36)

where $\delta$ represents the dimensionless initial phase (constant offset) of the time-dependent frequency modulation, setting the trap frequency at $t=0$, can be solved by the technique from the previous Sections in terms of Airy functions. The substitution

$$\Psi \left(x,t\right)={\epsilon }^{1/2}\psi \left(\xi ,\tau \right)$$

(37)

with

$$\tau =\omega t+\delta ,\xi =\epsilon x,\epsilon =\sqrt{{\displaystyle \frac{m\omega }{2\hslash }}}$$

(38)

results in

$$i{\displaystyle \frac{\partial \psi }{\partial \tau }}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial {\xi }^2}}-\tau {\xi }^2\psi =0,\psi \left(\xi ,\delta \right)=\phi \left(\xi \right)={\epsilon }^{-1/2}\Phi \left(x\right).$$

(39)

The Green function has the form

$$G\left(x,y,t\right)=\sqrt{{\displaystyle \frac{m\omega }{4\pi i\hslash \mu \left(\tau \right)}}}exp\left[i{\displaystyle \frac{m\omega }{2\hslash }}\left(\alpha \left(\tau \right)x^2+\beta \left(\tau \right)xy+\gamma \left(\tau \right)y^2\right)\right]$$

(40)

with $\tau =\omega t+\delta ,$ where $\mu \left(\delta \right)=0,$ ${\mu }^{\prime }\left(\delta \right)=-\left(1/2\right)W\left(ai,bi\right)=1/2$, W is the Wronskian (A17), and $ai$ and $bi$ are the Airy functions, and

$$\begin{array}{ccc}\hfill \mu \left(\tau \right)& =& {\displaystyle \frac{1}{2}}\left(ai\left(-\delta \right)bi\left(-\tau \right)-bi\left(-\delta \right)ai\left(-\tau \right)\right),\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \alpha \left(\tau \right)& =& -{\displaystyle \frac{a{i}^{\prime }\left(-\tau \right)bi\left(-\delta \right)-ai\left(-\delta \right)b{i}^{\prime }\left(-\tau \right)}{ai\left(-\tau \right)bi\left(-\delta \right)-ai\left(-\delta \right)bi\left(-\tau \right)}},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill \beta \left(\tau \right)& =& {\displaystyle \frac{2}{ai\left(-\tau \right)bi\left(-\delta \right)-ai\left(-\delta \right)bi\left(-\tau \right)}},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill \gamma \left(\tau \right)& =& -{\displaystyle \frac{a{i}^{\prime }\left(-\delta \right)bi\left(-\tau \right)-ai\left(-\tau \right)b{i}^{\prime }\left(-\delta \right)}{ai\left(-\tau \right)bi\left(-\delta \right)-ai\left(-\delta \right)bi\left(-\tau \right)}}.\hfill \end{array}$$

(44)

This can be derived using transformation (A49)–(A52). Thus,

$$\mu \sim {\displaystyle \frac{1}{2}}\omega t,\alpha \sim {\displaystyle \frac{1}{\omega t}},\beta \sim -{\displaystyle \frac{2}{\omega t}},\gamma \sim {\displaystyle \frac{1}{\omega t}}$$

(45)

as $t\to 0^{+}$ and the corresponding asymptotic formula is

$$G\left(x,y,t\right)\sim \sqrt{{\displaystyle \frac{m}{2\pi i\hslash t}}}exp\left({\displaystyle \frac{im{\left(x-y\right)}^2}{2\hslash t}}\right),t\to 0^{+},$$

(46)

where the expression on the right-hand side is a free-particle propagator. The solution of the initial value problem (36) is given by

$$\Psi \left(x,t\right)={\int }_{-\infty }^{\infty }G\left(x,y,t\right)\Phi \left(y\right)dy.$$

(47)

Omitting the detailed calculations, we proceed to consider an application.

The time-dependent quadratic potential of the form

$$V\left(x,t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}m{\omega }_0^2{x}^2,\hfill & t\le 0,\hfill \\ {\displaystyle \frac{1}{2}}m{\omega }^2\left(\omega t+\delta \right)x^2,\hfill & 0\le t\le T,\hfill \\ {\displaystyle \frac{1}{2}}m{\omega }_1^2{x}^2,\hfill & t\ge T,\hfill \end{array}\right.$$

(48)

describes a parametric oscillator that changes its frequency from ${\omega }_0$ to ${\omega }_1$ during the time interval $T.$ The continuity at $t=0$ and $t=T$ defines the transition parameters $\omega$ and $\delta$ as follows,

$$\omega ={\left({\displaystyle \frac{{\omega }_1^2-{\omega }_0^2}{T}}\right)}^{1/3},\delta ={\omega }_0^2{\left({\displaystyle \frac{T}{{\omega }_1^2-{\omega }_0^2}}\right)}^{2/3}$$

(49)

in terms of the initial ${\omega }_0$ and terminal ${\omega }_1$ oscillator frequencies. This model is integrable in terms of Airy functions with the help of the Green function found in this section as follows.

When $t<0$, the normalized wave function for a state with the definite energy $E_n^{\left(0\right)}=\hslash {\omega }_0\left(n+1/2\right)$ is [5,6]

$${\Psi }_n^{\left(0\right)}\left(x,t\right)={\displaystyle \frac{e^{-i{\omega }_0\left(n+1/2\right)t}}{\sqrt{2^{n}n!}}}{\left({\displaystyle \frac{m{\omega }_0}{\pi \hslash }}\right)}^{1/4}exp\left(-{\displaystyle \frac{m{\omega }_0}{2\hslash }}{x}^2\right)H_n\left(\sqrt{{\displaystyle \frac{m{\omega }_0}{\hslash }}}x\right),$$

(50)

where $H_n\left(\xi \right)$ are the Hermite polynomials [68,69,70,71,72,73,74,75]. When $0\le t\le T$, the corresponding transition wave function is given by the time evolution operator $U\left(t\right)$:

$${\Psi }_n\left(x,t\right)=U\left(t\right){\Psi }_n^{\left(0\right)}={\int }_{-\infty }^{\infty }G\left(x,y,t\right){\Psi }_n^{\left(0\right)}\left(y,0\right)dy$$

(51)

with the Green function (40)–(44). Finally, for $t\ge T$ the wave function is a linear combination

$${\Psi }_n\left(x,t\right)=\sum _{k=0}^{\infty }{c}_{kn}\left(T\right){\Psi }_k^{\left(1\right)}\left(x,t\right)$$

(52)

of the eigenfunctions

$${\Psi }_k^{\left(1\right)}\left(x,t\right)={\displaystyle \frac{e^{-i{\omega }_1\left(k+1/2\right)\left(t-T\right)}}{\sqrt{2^{k}k!}}}{\left({\displaystyle \frac{m{\omega }_1}{\pi \hslash }}\right)}^{1/4}exp\left(-{\displaystyle \frac{m{\omega }_1}{2\hslash }}{x}^2\right)H_k\left(\sqrt{{\displaystyle \frac{m{\omega }_1}{\hslash }}}x\right)$$

(53)

corresponding to the new eigenvalues $E_k^{\left(1\right)}=\hslash {\omega }_1\left(k+1/2\right)$, with $k=0,1,2,....$ Thus, function $c_{kn}\left(T\right)$ gives the quantum mechanical amplitude for the oscillator, initially in state $\left({\omega }_0,n\right)$, to be found at time T in state $\left({\omega }_1,k\right).$

For the transition period $0\le t\le T$, use the integral

$${\int }_{-\infty }^{\infty }{e}^{-{\lambda }^2{\left(x-y\right)}^2}{H}_n\left(ay\right)dy={\displaystyle \frac{\sqrt{\pi }}{{\lambda }^{n+1}}}{\left({\lambda }^2-a^2\right)}^{n/2}{H}_n\left({\displaystyle \frac{\lambda ax}{{\left({\lambda }^2-a^2\right)}^{1/2}}}\right),\mathrm{Re}{\lambda }^2>0,$$

(54)

which is equivalent to Equation (30) in Ref. [76] (Vol. II, p. 195) (the Gauss transform of Hermite polynomials), or Equation (17) in Ref. [77] (Vol. II, p. 290). The initial wave function evolves as follows:

$$\begin{array}{ccc}\hfill {\Psi }_n\left(x,t\right)& =& i^n{\left({\displaystyle \frac{m{\omega }_0}{\pi \hslash }}\right)}^{1/4}\sqrt{{\displaystyle \frac{\omega }{i\mu 2^{n+1}n!\left({\omega }_0-i\gamma \omega \right)}}}{\left({\displaystyle \frac{{\omega }_0+i\gamma \omega }{{\omega }_0-i\gamma \omega }}\right)}^{n/2}\hfill \\ & & \times exp\left[i{\displaystyle \frac{m\omega }{2\hslash }}\left(\alpha -{\displaystyle \frac{{\omega }^2{\beta }^2\gamma }{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}\right)x^2\right]\hfill \\ & & \times exp\left[-{\displaystyle \frac{m{\omega }_0{\omega }^2{\beta }^2{x}^2}{8\hslash \left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}\right]H_n\left(\sqrt{{\displaystyle \frac{m{\omega }_0}{4\hslash \left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}}\omega \beta x\right),\hfill \end{array}$$

(55)

where the time-dependent coefficients $\mu$, $\alpha$, $\beta$, and $\gamma$ are given by Equations (41)–(44) in terms of Airy functions, with the argument $\tau =\omega t+\delta$ during the time interval $0\le t\le T.$ The asymptotics (45) imply that ${\Psi }_n\left(x,t\right)\to {\Psi }_n^{\left(0\right)}\left(x,0\right)$ as $t\to 0^{+}$ with the choice of the principal branch of the radicals. A direct integration shows that

$${\int }_{-\infty }^{\infty }{\left|{\Psi }_n\left(x,t\right)\right|}^{2}dx=1,0\le t\le T$$

(56)

by the orthogonality relation of the Hermite polynomials. The normalization of the wave function holds, due to the unitarity of the time evolution operator.

Then, in view of the orthogonality of eigenfunctions (53), the transition amplitudes are

$$c_{kn}\left(T\right)={\int }_{-\infty }^{\infty }{\left({\Psi }_k^{\left(1\right)}\left(x,T\right)\right)}^{*}{\Psi }_n\left(x,T\right)dx,$$

(57)

where one can use another classical integral evaluated by Wilfrid Norman Bailey [78]:

$$\begin{array}{ccc}& & {\int }_{-\infty }^{\infty }{e}^{-{\lambda }^2{x}^2}{H}_m\left(ax\right)H_n\left(bx\right)dx\hfill \\ & & ={\displaystyle \frac{2^{m+n}}{{\lambda }^{m+n+1}}}\Gamma \left({\displaystyle \frac{m+n+1}{2}}\right){\left(a^2-{\lambda }^2\right)}^{m/2}{\left(b^2-{\lambda }^2\right)}^{n/2}\hfill \\ & & \times {}_{2}F_1\left(\begin{array}{c}-m,-n\\ {\displaystyle \frac{1}{2}}\left(1-m-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{ab}{\sqrt{\left(a^2-{\lambda }^2\right)\left(b^2-{\lambda }^2\right)}}}\right)\right),\mathrm{Re}{\lambda }^2>0,\hfill \end{array}$$

(58)

if $m+n$ is even; the integral vanishes by symmetry if $m+n$ is odd (see Refs. [78,79] and references therein for earlier works on these integrals and their special cases and extensions).

The result is $c_{kn}\left(T\right)=0,$ if $k+n$ is odd, and

$$\begin{array}{ccc}\hfill c_{kn}\left(T\right)& =& i^n\Gamma \left({\displaystyle \frac{k+n+1}{2}}\right){\left({\displaystyle \frac{{\omega }_0{\omega }_1}{{\pi }^2}}\right)}^{1/4}\sqrt{{\displaystyle \frac{2^{k+n}}{i\mu k!n!\left({\omega }_0-i\gamma \omega \right)}}}{\left({\displaystyle \frac{{\omega }_0+i\gamma \omega }{{\omega }_0-i\gamma \omega }}\right)}^{n/2}\hfill \\ & & \times {\left[{\displaystyle \frac{{\omega }_1}{\omega }}-{\displaystyle \frac{{\omega }_0\omega {\beta }^2}{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}+i\left(\alpha -{\displaystyle \frac{{\omega }^2{\beta }^2\gamma }{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}\right)\right]}^{k/2}\hfill \\ & & \times {\left[({\displaystyle \frac{{\omega }_0\omega {\beta }^2}{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}-{\displaystyle \frac{{\omega }_1}{\omega }}+i\left(\alpha -{\displaystyle \frac{{\omega }^2{\beta }^2\gamma }{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}\right)\right]}^{n/2}\hfill \\ & & \times {\left[{\displaystyle \frac{{\omega }_1}{\omega }}+{\displaystyle \frac{{\omega }_0\omega {\beta }^2}{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}-i\left(\alpha -{\displaystyle \frac{{\omega }^2{\beta }^2\gamma }{4\left({\omega }_0^2+{\gamma }^2{\omega }^2\right)}}\right)\right]}^{-\left(k+n+1\right)/2}\hfill \\ & & \times {}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right),\hfill \end{array}$$

(59)

where

$$\zeta ={\displaystyle \frac{\omega \beta \sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1+\alpha \gamma {\omega }^2-{\beta }^2{\omega }^2/4\right)}^2}}},$$

(60)

if $k+n$ is even. The terminating hypergeometric function is transformed as follows:

$$\begin{array}{ccc}& & {}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right)\hfill \\ & & =\left\{\begin{array}{cc}{\displaystyle \frac{{\left(1/2\right)}_r{\left(1/2\right)}_s}{{\left(1/2\right)}_{r+s}}}{}_{2}F_1\left(\begin{array}{c}-r,-s\\ 1/2\end{array};-{\zeta }^2\right),\hfill & \mathrm{if}k=2r,n=2s,\hfill \\ -{\displaystyle \frac{{\left(3/2\right)}_r{\left(3/2\right)}_s}{{\left(3/2\right)}_{r+s}}}i\zeta {}_{2}F_1\left(\begin{array}{c}-r,-s\\ 3/2\end{array};-{\zeta }^2\right),\hfill & \mathrm{if}k=2r+1,n=2s+1,\hfill \end{array}\right.\hfill \end{array}$$

(61)

and is valid in the entire complex plane; the details are given in Appendix B. The transformation (61) completes the evaluation of the Bailey integral (58); see Equation (A26).

The function $c_{kn}\left(T\right)$ explicitly gives the quantum mechanical amplitude for the oscillator, initially in state $\left({\omega }_0,n\right)$, to be found at time T in the state $\left({\omega }_1,k\right).$ The unitarity of the time evolution operator implies the discrete orthogonality relation

$$\sum _{k=0}^{\infty }{c}_{kn}^{*}\left(T\right)c_{kp}\left(T\right)={\delta }_{np}$$

(62)

for ${}_{2}F_1$ functions under consideration, where ${\delta }_{np}$ is the Kronecker delta. The known orthogonal systems at this level are Jacobi, Kravchuk, Meixner, and Meixner–Pollaczek polynomials (see, e.g., [69,70,71,72,74,75,76,80,81,82,83,84,85,86,87] and references therein). This particular ${}_{2}F_1$ orthogonal system is reduced by the transformation (61) to the Meixner polynomials. A group theoretical interpretation of the transition amplitudes and their relation with Bargmann’s functions is discussed in Section 10.

In the limit $T\to 0^{+},$ when the oscillator frequency changes instantaneously from ${\omega }_0$ to ${\omega }_1,$ the transition amplitudes are essentially simplified. As a result, $c_{kn}\left(0\right)=0,$ if $k+n$ is odd, and

$$\begin{array}{ccc}\hfill c_{kn}\left(0\right)& =& i^n\Gamma \left({\displaystyle \frac{k+n+1}{2}}\right){\left({\displaystyle \frac{{\omega }_0{\omega }_1}{{\pi }^2}}\right)}^{1/4}\sqrt{{\displaystyle \frac{2^{k+n+1}}{k!n!\left({\omega }_0+{\omega }_1\right)}}}{\left({\displaystyle \frac{{\omega }_1-{\omega }_0}{{\omega }_1+{\omega }_0}}\right)}^{\left(k+n\right)/2}\hfill \\ & & \times {}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+2i{\displaystyle \frac{\sqrt{{\omega }_0{\omega }_1}}{\left|{\omega }_0-{\omega }_1\right|}}\right)\right),\hfill \end{array}$$

(63)

if $k+n$ is even. The discrete orthogonality relation (62) and transformation (61) hold. The limit ${\omega }_1\to {\omega }_0$ is interesting from the view point of perturbation theory.

If the oscillator is in the ground state $\left({\omega }_0,0\right)$ before the start of interaction, the transition probability of finding the oscillator in the nth excited energy eigenstate $\left({\omega }_1,n\right)$ with the new frequency is given by ${\left|c_{2k+1,0}\left(T\right)\right|}^2=0$ and

$$\begin{array}{ccc}\hfill {\left|c_{2k,0}\left(T\right)\right|}^2& =& {\displaystyle \frac{\left|\beta \right|\omega \sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}}\hfill \\ & & \times {\displaystyle \frac{{\left(1/2\right)}_k}{k!}}{\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1+\alpha \gamma {\omega }^2-{\beta }^2{\omega }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}\right)}^k,\hfill \end{array}$$

(64)

where $k=0,1,2,...$ and ${\sum }_{k=0}^{\infty }{\left|c_{2k,0}\left(T\right)\right|}^2=1$ using the binomial theorem. If the oscillator is in the first excited state $\left({\omega }_0,1\right),$ the transition probability of finding the oscillator in the nth excited state $\left({\omega }_1,n\right)$ is given by ${\left|c_{2k,1}\left(T\right)\right|}^2=0$ and

$$\begin{array}{ccc}\hfill {\left|c_{2k+1,1}\left(T\right)\right|}^2& =& {\left({\displaystyle \frac{{\beta }^2{\omega }^2{\omega }_0{\omega }_1}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}\right)}^{3/2}\hfill \\ & & \times {\displaystyle \frac{{\left(3/2\right)}_k}{k!}}{\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1+\alpha \gamma {\omega }^2-{\beta }^2{\omega }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}\right)}^k,\hfill \end{array}$$

(65)

where $k=0,1,2,...$ and ${\sum }_{k=0}^{\infty }{\left|c_{2k+1,1}\left(T\right)\right|}^2=1.$ These probabilities can be recognized as two special cases of the negative binomial distribution, or Pascal distribution, which gives the normalized weight function for the Meixner polynomials of a discrete variable [69,70,71,72,75,76].

Similarly, the probability that the oscillator initially in eigenstate $\left({\omega }_0,n\right)$ is found at time T after the transition in state $\left({\omega }_1,k\right)$ is given by ${\left|c_{kn}\left(T\right)\right|}^2=0,$ if $k+n$ is odd, and

$$\begin{array}{ccc}\hfill {\left|c_{kn}\left(T\right)\right|}^2& =& {\displaystyle \frac{\left|\beta \right|\omega \sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}}\hfill \\ & & \times {\displaystyle \frac{2^{k+n}}{k!n!\pi }}{\Gamma }^2\left({\displaystyle \frac{k+n+1}{2}}\right)\hfill \\ & & \times {\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1+\alpha \gamma {\omega }^2-{\beta }^2{\omega }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}}\right)}^{\left(k+n\right)/2}\hfill \\ & & \times {\left|{}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right)\right|}^2,\hfill \end{array}$$

(66)

if $k+n$ is even. The transformation (61) is, certainly, valid, but the square of the hypergeometric function can be simplified to a single positive sum with the help of the quadratic transformation (A21) followed by the Clausen formula (A27):

$$\begin{array}{ccc}& & {\left|{}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right)\right|}^2\hfill \\ & & ={}_{3}F_2\left(\begin{array}{c}-k,-n,-\left(k+n\right)/2\\ \left(1-k-n\right)/2,-k-n\end{array};z\right)\hfill \end{array}$$

(67)

with

$$z={\displaystyle \frac{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1-\alpha \gamma {\omega }^2+{\beta }^2{\omega }^2/4\right)}^2}{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2{\omega }^2+{\left({\omega }_0{\omega }_1+\alpha \gamma {\omega }^2-{\beta }^2{\omega }^2/4\right)}^2}}.$$

(68)

More details are given in Section 9 just below.

Thus, we have determined the complete dynamics of the quantum parametric oscillator transition from the initial state with the frequency ${\omega }_0$ to the terminal one with the frequency ${\omega }_1$ by explicitly solving the time-dependent Schrödinger equation with variable potential (48) for all times.

9. Quantum Parametric Oscillator: General Case

The general case of the parametric oscillator with a variable frequency of the form

$$\omega \left(t\right)=\left\{\begin{array}{cc}{\omega }_0,\hfill & t\le 0,\hfill \\ \omega \left(t\right),\hfill & 0\le t\le T,\hfill \\ {\omega }_1,\hfill & t\ge T,\hfill \end{array}\right.{\omega }_0=\omega \left(0\right),\omega \left(T\right)={\omega }_1,$$

(69)

(Figure 1) can be investigated in a similar way. By the method of Ref. [40], the corresponding transition Green function is given by

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{2\pi i\mu \left(t\right)}}}exp\left(i{\displaystyle \frac{m}{2\hslash }}\left(\alpha \left(t\right)x^2+\beta \left(t\right)xy+\gamma \left(t\right)y^2\right)\right),0\le t\le T,$$

(70)

where $\mu =\mu \left(t\right)$ is a solution of the equation of motion for the classical parametric oscillator [88,89]:

$${\mu }^{″}+{\omega }^2\left(t\right)\mu =0,$$

(71)

which satisfies the initial conditions $\mu \left(0\right)=0$ and ${\mu }^{\prime }\left(0\right)=$ $\hslash /m.$ The coefficients of the quadratic form are

$$\begin{array}{ccc}& & \alpha \left(t\right)={\displaystyle \frac{{\mu }^{\prime }\left(t\right)}{\mu \left(t\right)}},\beta \left(t\right)=-{\displaystyle \frac{2\hslash }{m\mu \left(t\right)}},\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& & \gamma \left(t\right)={\displaystyle \frac{{\hslash }^2}{m^2}}\left({\displaystyle \frac{1}{\mu \left(t\right){\mu }^{\prime }\left(t\right)}}-{\int }_0^t{\displaystyle \frac{{\omega }^2\left(\tau \right)}{{\left({\mu }^{\prime }\left(\tau \right)\right)}^2}}d\tau \right).\hfill \end{array}$$

(73)

The asymptotics (46) hold as $t\to 0^{+}$.

A similar calculation gives the wave function (51) during the transition period $0<t\le T$:

$$\begin{array}{ccc}\hfill {\Psi }_n\left(x,t\right)& =& {\left({\displaystyle \frac{\hslash {\omega }_0}{\pi m}}\right)}^{1/4}{\displaystyle \frac{1}{\sqrt{\mu 2^{n}n!\left(\gamma +i{\omega }_0\right)}}}{\left({\displaystyle \frac{\gamma -i{\omega }_0}{\gamma +i{\omega }_0}}\right)}^{n/2}\hfill \\ & & \times exp\left[i{\displaystyle \frac{m}{2\hslash }}\left(\alpha -{\displaystyle \frac{{\beta }^2\gamma }{4\left({\gamma }^2+{\omega }_0^2\right)}}\right)x^2\right]\hfill \\ & & \times exp\left(-{\displaystyle \frac{m{\omega }_0{\beta }^2{x}^2}{8\hslash \left({\gamma }^2+{\omega }_0^2\right)}}\right)H_n\left(\sqrt{{\displaystyle \frac{m{\omega }_0}{4\hslash \left({\gamma }^2+{\omega }_0^2\right)}}}\beta x\right),\hfill \end{array}$$

(74)

which satisfies the normalization condition (56). The continuity property ${\Psi }_n\left(x,t\right)\to {\Psi }_n^{\left(0\right)}\left(x,0\right)$ holds as $t\to 0^{+}$ for the principal branch of the radicals.

The transition amplitudes (57) are $c_{kn}\left(T\right)=0,$ if $k+n$ is odd, and

$$\begin{array}{ccc}\hfill c_{kn}\left(T\right)& =& \Gamma \left({\displaystyle \frac{k+n+1}{2}}\right){\left({\displaystyle \frac{{\omega }_0{\omega }_1}{{\pi }^2}}\right)}^{1/4}\sqrt{{\displaystyle \frac{2^{k+n+1}\hslash }{\mu k!n!\left(\gamma +i{\omega }_0\right)m}}}{\left({\displaystyle \frac{\gamma -i{\omega }_0}{\gamma +i{\omega }_0}}\right)}^{n/2}\hfill \\ & & \times {\left[{\omega }_1-{\displaystyle \frac{{\omega }_0{\beta }^2}{4\left({\gamma }^2+{\omega }_0^2\right)}}+i\left(\alpha -{\displaystyle \frac{{\beta }^2\gamma }{4\left({\gamma }^2+{\omega }_0^2\right)}}\right)\right]}^{k/2}\hfill \\ & & \times {\left[-{\omega }_1+{\displaystyle \frac{{\omega }_0{\beta }^2}{4\left({\gamma }^2+{\omega }_0^2\right)}}+i\left(\alpha -{\displaystyle \frac{{\beta }^2\gamma }{4\left({\gamma }^2+{\omega }_0^2\right)}}\right)\right]}^{n/2}\hfill \\ & & \times {\left[{\omega }_1+{\displaystyle \frac{{\omega }_0{\beta }^2}{4\left({\gamma }^2+{\omega }_0^2\right)}}-i\left(\alpha -{\displaystyle \frac{{\beta }^2\gamma }{4\left({\gamma }^2+{\omega }_0^2\right)}}\right)\right]}^{-\left(k+n+1\right)/2}\hfill \\ & & \times {}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right),\hfill \end{array}$$

(75)

where

$$\zeta ={\displaystyle \frac{\beta \sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}}},$$

(76)

if $k+n$ is even. The transformation (61) is applied, and the unitarity of the time evolution operator implies the discrete orthogonality relation (62) for the ${}_{2}F_1$ functions. Their relations with Meixner polynomials and Bargmann’s functions are discussed in Section 10.

For the oscillator initially in the ground state $\left({\omega }_0,0\right)$, the transition probability of finding the oscillator in the nth excited energy eigenstate $\left({\omega }_1,n\right)$ with the new frequency is given by ${\left|c_{2k+1,0}\left(T\right)\right|}^2=0$ and

$$\begin{array}{ccc}\hfill {\left|c_{2k,0}\left(T\right)\right|}^2& =& {\displaystyle \frac{\left|\beta \right|\sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}}\hfill \\ & & \times {\displaystyle \frac{{\left(1/2\right)}_k}{k!}}{\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}\right)}^k,\hfill \end{array}$$

(77)

where $k=0,1,2,...$ and ${\sum }_{k=0}^{\infty }{\left|c_{2k,0}\left(T\right)\right|}^2=1.$ For the oscillator initially in the first excited state $\left({\omega }_0,1\right)$, the transition probability of finding the oscillator in the nth excited state $\left({\omega }_1,n\right)$ is given by ${\left|c_{2k,1}\left(T\right)\right|}^2=0$ and

$$\begin{array}{ccc}\hfill {\left|c_{2k+1,1}\left(T\right)\right|}^2& =& {\left({\displaystyle \frac{{\beta }^2{\omega }_0{\omega }_1}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}\right)}^{3/2}\hfill \\ & & \times {\displaystyle \frac{{\left(3/2\right)}_k}{k!}}{\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}\right)}^k,\hfill \end{array}$$

(78)

where $k=0,1,2,...$ and ${\sum }_{k=0}^{\infty }{\left|c_{2k,0}\left(T\right)\right|}^2=1.$ These probabilities are two special cases of the negative binomial distribution, which gives the normalized weight function for the Meixner polynomials of a discrete variable [69,70,71,72,75,76].

Similarly, the probability that the oscillator initially in eigenstate $\left({\omega }_0,n\right)$ is found at time T after the transition in state $\left({\omega }_1,k\right)$ is given by ${\left|c_{kn}\left(T\right)\right|}^2=0,$ if $k+n$ is odd, and

$$\begin{array}{ccc}\hfill {\left|c_{kn}\left(T\right)\right|}^2& =& {\displaystyle \frac{\left|\beta \right|\sqrt{{\omega }_0{\omega }_1}}{\sqrt{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}}\hfill \\ & & \times {\displaystyle \frac{2^{k+n}}{k!n!\pi }}{\Gamma }^2\left({\displaystyle \frac{k+n+1}{2}}\right)\hfill \\ & & \times {\left({\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}\right)}^{\left(k+n\right)/2}\hfill \\ & & \times {\left|{}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right)\right|}^2,\hfill \end{array}$$

(79)

if $k+n$ is even (see also [90] for related work). The transformation (61) is valid but the square of the hypergeometric function can be simplified to a single positive sum with the help of a quadratic transformation (A21) followed by the Clausen formula (A27):

$$\begin{array}{ccc}& & {\left|{}_{2}F_1\left(\begin{array}{c}-k,-n\\ {\displaystyle \frac{1}{2}}\left(1-k-n\right)\end{array};{\displaystyle \frac{1}{2}}\left(1+i\zeta \right)\right)\right|}^2\hfill \\ & & ={\left[{}_{2}F_1\left(\begin{array}{c}-k/2,-n/2\\ \left(1-k-n\right)/2\end{array};1+{\zeta }^2\right)\right]}^2\hfill \\ & & ={}_{3}F_2\left(\begin{array}{c}-k,-n,-\left(k+n\right)/2\\ \left(1-k-n\right)/2,-k-n\end{array};z\right),\hfill \end{array}$$

(80)

where

$$z=1+{\zeta }^2={\displaystyle \frac{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}}.$$

(81)

Thus, substituting (80)–(81) into Equation (79) one obtains the final representation of the probability ${\left|c_{kn}\left(T\right)\right|}^2$ in terms of a positive terminating ${}_{3}F_2$ generalized hypergeometric function.

For an arbitrary initial data point in $L^2\left(\mathbb{R}\right)$,

$${\Psi }^{\left(0\right)}\left(x,0\right)=\sum _{n=0}^{\infty }{c}_n^{\left(0\right)}{\Psi }_n^{\left(0\right)}\left(x,0\right),\sum _{n=0}^{\infty }{\left|c_n^{\left(0\right)}\right|}^2=1,$$

(82)

the wave function after the transition is given by

$$\begin{array}{ccc}\hfill {\Psi }^{\left(1\right)}\left(x,T\right)& =& {\int }_{-\infty }^{\infty }G\left(x,y,T\right){\Psi }^{\left(0\right)}\left(y,0\right)dy\hfill \\ & =& \sum _{n=0}^{\infty }{c}_n^{\left(0\right)}{\int }_{-\infty }^{\infty }G\left(x,y,T\right){\Psi }_n^{\left(0\right)}\left(y,0\right)dy\hfill \\ & =& \sum _{n=0}^{\infty }{c}_n^{\left(0\right)}{\Psi }_n\left(y,T\right)=\sum _{n=0}^{\infty }{c}_k^{\left(1\right)}{\Psi }_k^{\left(1\right)}\left(x,T\right)\hfill \end{array}$$

(83)

with

$$c_k^{\left(1\right)}=\sum _{n=0}^{\infty }{c}_{kn}\left(T\right)c_n^{\left(0\right)}$$

(84)

by Equation (52). A group theoretical interpretation is given in Section 10 just below. The orthogonality property of the transition amplitudes (62) implies the conservation law of the total probability

$$\sum _{k=0}^{\infty }{\left|c_k^{\left(1\right)}\right|}^2=\sum _{n=0}^{\infty }{\left|c_n^{\left(0\right)}\right|}^2=1,$$

(85)

which follows from the conservation of the norm of the wave function during the transition.

Thus, we have solved the problem of the parametric oscillator in nonrelativistic quantum mechanics, provided that the solution of the corresponding classical problem (71) is known. A more convenient form of the transition amplitudes (75) is given below in Section 10 in terms of Bargmann’s functions. Moreover, the quantum forced parametric oscillator can be investigated using the methods of Refs. [19,40,48]. We leave further details to the reader.

10. Group Theoretical Meaning of Transition Amplitudes

The group theoretical properties of the harmonic oscillator wave functions are investigated in detail. In addition to the known relation with the Heisenberg–Weyl algebra of the creation and annihilation operators, the n-dimensional oscillator wave functions form a basis of the irreducible unitary representation of the Lie algebra of the noncompact group $SU\left(1,1\right)$ corresponding to the discrete positive series ${\mathcal{D}}_{+}^j$ (see [18,48,71,91,92,93,94]). In this paper, we deal with the one-dimensional case only.

Define the creation and annihilation operators,

$$a^{†}\left(\omega \right)=\sqrt{{\displaystyle \frac{m}{2\hslash \omega }}}\left(\omega x-{\displaystyle \frac{\hslash }{m}}{\displaystyle \frac{\partial }{\partial x}}\right),a\left(\omega \right)=\sqrt{{\displaystyle \frac{m}{2\hslash \omega }}}\left(\omega x+{\displaystyle \frac{\hslash }{m}}{\displaystyle \frac{\partial }{\partial x}}\right),$$

(86)

respectively, with the commutator

$$a\left(\omega \right)a^{†}\left(\omega \right)-a^{†}\left(\omega \right)a\left(\omega \right)=1$$

(87)

and the Hamiltonian

$$H\left(\omega \right)={\displaystyle \frac{\hslash \omega }{2}}\left(a\left(\omega \right)a^{†}\left(\omega \right)+a^{†}\left(\omega \right)a\left(\omega \right)\right).$$

(88)

Their actions on “stationary” oscillator wave functions,

$${\Psi }_n\left(x\right)={\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}{\displaystyle \frac{1}{\sqrt{2^{n}n!}}}exp\left(-{\displaystyle \frac{m\omega }{2\hslash }}{x}^2\right)H_n\left(\sqrt{{\displaystyle \frac{m\omega }{\hslash }}}x\right),$$

(89)

are given by

$$a{\Psi }_n=\sqrt{n}{\Psi }_{n-1},a^{†}{\Psi }_n=\sqrt{n+1}{\Psi }_{n+1}.$$

(90)

Introducing operators

$$\begin{array}{cc}\hfill J_{+}\left(\omega \right)& ={\displaystyle \frac{1}{2}}{\left(a^{†}\left(\omega \right)\right)}^2,J_{-}\left(\omega \right)={\displaystyle \frac{1}{2}}{\left(a\left(\omega \right)\right)}^2,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill J_0\left(\omega \right)& ={\displaystyle \frac{1}{4}}\left(a\left(\omega \right)a^{†}\left(\omega \right)+a^{†}\left(\omega \right)a\left(\omega \right)\right)={\displaystyle \frac{1}{2\hslash \omega }}H\left(\omega \right),\hfill \end{array}$$

(92)

one can straightforwardly verify the following commutation relations:

$$\left[J_0,J_{\pm }\right]=\pm J_{\pm },\left[J_{+},J_{-}\right]=-2J_0.$$

(93)

For the Hermitian operators

$$J_x={\displaystyle \frac{1}{2}}\left(J_{+}+J_{-}\right),J_y={\displaystyle \frac{1}{2i}}\left(J_{+}-J_{-}\right),J_z=J_0,$$

(94)

one obtains

$$\left[J_x,J_y\right]=-i{J}_z,\left[J_y,J_z\right]=i{J}_x,\left[J_z,J_x\right]=i{J}_y.$$

(95)

The commutation rules (95) are valid for the infinitesimal operators of the non-compact group $SU\left(1,1\right)$ (see, e.g., [48,71,91,93,95,96] for more details).

One can use a different notation for the oscillator wave functions (89) as follows:

$${\psi }_{j\lambda }=\left\{\begin{array}{cc}{\Psi }_{2n}\left(x\right),\hfill & \mathrm{if}j=-3/4\mathrm{and}\lambda =n+1/4,\hfill \\ {\Psi }_{2n+1}\left(x\right),\hfill & \mathrm{if}j=-1/4\mathrm{and}\lambda =n+3/4,\hfill \end{array}\right.$$

(96)

where $n=0,1,2,...$ and the inequality $\lambda \ge j+1$ holds. The operators $J_{\pm }$ and $J_0$ have an explicit form

$$J_{\pm }={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{m\omega }{\hslash }}{x}^2-{\displaystyle \frac{H}{\hslash \omega }}\mp {\displaystyle \frac{1}{2}}\mp x{\displaystyle \frac{\partial }{\partial x}}\right),J_0={\displaystyle \frac{H}{2\hslash \omega }},$$

(97)

and their actions on the oscillator wave functions are given by

$$J_{\pm }{\psi }_{j\lambda }=\sqrt{\left(\lambda \mp j\right)\left(\lambda \pm j\pm 1\right)}{\psi }_{j,\lambda \pm 1},J_0{\psi }_{j\lambda }=\lambda {\psi }_{j\lambda },$$

(98)

whence

$$J^2{\psi }_{j\lambda }=j\left(j+1\right){\psi }_{j\lambda }$$

(99)

with $J^2=J_0^2+J_0-J_{-}{J}_{+}=J_0^2-J_0-J_{+}{J}_{-}.$ These relations coincide with the formulas that define the action of the infinitesimal operators $J_{\pm }$ and $J_0$ of the Lie group $SU\left(1,1\right)$ on a basis $\left|j,\lambda \right.〉$ of the irreducible representation ${\mathcal{D}}_{+}^j$ belonging to the discrete positive series in an abstract Hilbert space [96]. Thus, the even and odd wave functions ${\Psi }_n\left(x\right)$ of the one-dimensional harmonic oscillator form, respectively, bases for the two irreducible representations ${\mathcal{D}}_{+}^j$ of the algebra $SU\left(1,1\right)$, with the moments $j=-3/4$ for the even values of n and $j=-1/4$ for odd $n.$ These correspond to the double-valued representations of the group $SU\left(1,1\right),$ or quadruple-valued representations of the $SO\left(2,1\right)$ (see [71,93] for more details). It is crucial for the purpose of this paper that these group theoretical properties are valid “instantaneously” when $\omega =\omega \left(t\right)$ is an arbitrary function of time.

As a result of an elementary but rather tedious calculation, the transition amplitudes (75) can be rewritten in the form

$$c_{kn}\left(T\right)=T_{\lambda {\lambda }^{\prime }}^j\left(\theta ,\tau ,\phi \right)$$

(100)

with the new $SU\left(1,1\right)$ quantum numbers

$$j=-{\displaystyle \frac{3}{4}},\lambda =r+{\displaystyle \frac{1}{4}},{\lambda }^{\prime }=s+{\displaystyle \frac{1}{4}},$$

(101)

if $k=2r$, $n=2s,$ (with $r,s=0,1,2,3,...$) and

$$j=-{\displaystyle \frac{1}{4}},\lambda =r+{\displaystyle \frac{3}{4}},{\lambda }^{\prime }=s+{\displaystyle \frac{3}{4}},$$

(102)

if $k=2r+1$ and $n=2s+1$ (with $r,s=0,1,2,3,...$). The matrix elements $T_{\lambda {\lambda }^{\prime }}^j\left(\theta ,\tau ,\phi \right)$ are the so-called Bargmann functions, or the generalized spherical harmonics of $SU\left(1,1\right)$ [71,95,96]:

$$T_{\lambda {\lambda }^{\prime }}^j\left(\theta ,\tau ,\phi \right)=e^{-i\lambda \theta }{t}_{\lambda {\lambda }^{\prime }}^j\left(\tau \right)e^{-i{\lambda }^{\prime }\phi }.$$

(103)

Here,

$$\begin{array}{cc}\hfill & t_{\lambda {\lambda }^{\prime }}^j\left(\tau \right)={\displaystyle \frac{{\left(-1\right)}^{\lambda -j-1}}{\Gamma \left(2j+2\right)}}\sqrt{{\displaystyle \frac{\Gamma \left(\lambda +j+1\right)\Gamma \left({\lambda }^{\prime }+j+1\right)}{\left(\lambda -j-1\right)!\left({\lambda }^{\prime }-j-1\right)!}}}{\left(sinh{\displaystyle \frac{\tau }{2}}\right)}^{-2j-2}{\left(tanh{\displaystyle \frac{\tau }{2}}\right)}^{\lambda +{\lambda }^{\prime }}\hfill \\ \hfill & \times {}_{2}F_1\left(\begin{array}{c}-\lambda +j+1,-{\lambda }^{\prime }+j+1\\ 2j+2\end{array};-{\displaystyle \frac{1}{{sinh}^2\left(\tau /2\right)}}\right)\hfill \end{array}$$

(104)

and the corresponding angles are given by

$$\begin{array}{ccc}\hfill tan\theta & =& {\displaystyle \frac{2\alpha {\omega }_0^2{\omega }_1+2\gamma {\omega }_1\left(\alpha \gamma -{\beta }^2/4\right)}{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)\left(\alpha {\omega }_0-\gamma {\omega }_1\right)-\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}},\hfill \end{array}$$

(105)

$$\begin{array}{ccc}\hfill tan\phi & =& {\displaystyle \frac{-2\gamma {\omega }_0{\omega }_1^2-2\alpha {\omega }_0\left(\alpha \gamma -{\beta }^2/4\right)}{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)\left(\alpha {\omega }_0-\gamma {\omega }_1\right)+\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}}\hfill \end{array}$$

(106)

for the two Euclidean rotations, and

$${tanh}^2\left({\displaystyle \frac{\tau }{2}}\right)={\displaystyle \frac{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}{{\left(\alpha {\omega }_0+\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1-\alpha \gamma +{\beta }^2/4\right)}^2}}$$

(107)

for the hyperbolic one. The branch of the radical is taken such that

$${\displaystyle \frac{1}{sinh\left(\tau /2\right)}}={\displaystyle \frac{\beta {\omega }_0{\omega }_1}{\sqrt{{\left(\alpha {\omega }_0-\gamma {\omega }_1\right)}^2+{\left({\omega }_0{\omega }_1+\alpha \gamma -{\beta }^2/4\right)}^2}}}.$$

(108)

The following symmetry holds: if $\alpha \leftrightarrow \gamma$ and ${\omega }_0\leftrightarrow {\omega }_1,$ then $\theta \leftrightarrow \phi .$ This interchanges the initial and terminal oscillator states.

The Formulas (100)–(108) give a straightforward group theoretical interpretation of the transition amplitudes for the parametric oscillator in quantum mechanics:

$$\begin{array}{ccc}\hfill c_{kn}\left(T\right)& =& 〈{\Psi }_k^{\left(1\right)}\left|U\left(T\right){\Psi }_n^{\left(0\right)}\right.〉\hfill \\ & =& 〈{\psi }_{j\lambda }^{\left(1\right)}\left|T\left(\theta ,\tau ,\phi \right){\psi }_{j{\lambda }^{\prime }}^{\left(0\right)}\right.〉\hfill \\ & =& 〈{\psi }_{j\lambda }^{\left(1\right)}\left|e^{-i\theta J_0\left({\omega }_1\right)}{e}^{-i\left(\tau +ln\left({\omega }_1/{\omega }_0\right)\right)J_y}{e}^{-i\phi J_0\left({\omega }_0\right)}{\psi }_{j{\lambda }^{\prime }}^{\left(0\right)}\right.〉\hfill \\ & =& e^{-i\lambda \theta }〈{\psi }_{j\lambda }^{\left(1\right)}\left|e^{-i\left(\tau +ln\left({\omega }_1/{\omega }_0\right)\right)J_y}{\psi }_{j{\lambda }^{\prime }}^{\left(0\right)}\right.〉e^{-i{\lambda }^{\prime }\phi }\hfill \\ & =& T_{\lambda {\lambda }^{\prime }}^j\left(\theta ,\tau ,\phi \right),\hfill \end{array}$$

(109)

if $k+n$ is even, in terms of the generalized spherical harmonics of the $SU\left(1,1\right)$ algebra for the discrete positive series ${\mathcal{D}}_{+}^j$, with $j=-3/4$ and $j=-1/4$ for the even and odd oscillator functions, respectively. The Formula (84) gives a transformation of the “coordinates” of a wave function from the old basis to the new one.

The time evolution operator has the form

$$U\left(t\right)=T\left(\theta ,\tau ,\phi \right)=e^{-i\theta J_0\left(\omega \right)}{e}^{-i\left(\tau +ln\left(\omega /{\omega }_0\right)\right)J_y}{e}^{-i\phi J_0\left({\omega }_0\right)}$$

(110)

in terms of the corresponding infinitesimal operators, where

$$-i{J}_y={\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}x{\displaystyle \frac{\partial }{\partial x}}.$$

(111)

The action of the hyperbolic rotation operator, or Lorentz boost, on a wave function is given by

$$e^{-i\tau J_y}\psi \left(x\right)=e^{\tau /4}\psi \left(e^{\tau /2}x\right).$$

(112)

Therefore,

$$\begin{array}{ccc}\hfill t_{\lambda {\lambda }^{\prime }}^j\left(\tau \right)& =& 〈{\psi }_{j\lambda }^{\left(1\right)}\left|e^{-i\tau J_y}{\psi }_{j{\lambda }^{\prime }}^{\left(1\right)}\right.〉\hfill \\ & =& e^{\tau /4}{\int }_{-\infty }^{\infty }{\Psi }_k^{\left(1\right)}\left(x\right){\Psi }_n^{\left(1\right)}\left(e^{\tau /2}x\right)dx,\hfill \end{array}$$

(113)

if $k+n$ is even, which gives an integral representation for the Bargmann function under consideration; cf. [71].

Thus, the $SU\left(1,1\right)$ symmetry suggests the following algebraic form,

$$\begin{array}{ccc}\hfill U\left(t\right)& =& exp\left[{\displaystyle \frac{i\theta }{2\hslash \omega }}\left({\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}-{\displaystyle \frac{m{\omega }^2}{2}}{x}^2\right)\right]\hfill \\ & & \times exp\left[{\displaystyle \frac{\tau +ln\left(\omega /{\omega }_0\right)}{2}}\left({\displaystyle \frac{1}{2}}+x{\displaystyle \frac{\partial }{\partial x}}\right)\right]\hfill \\ & & \times exp\left[{\displaystyle \frac{i\phi }{2\hslash {\omega }_0}}\left({\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}-{\displaystyle \frac{m{\omega }_0^2}{2}}{x}^2\right)\right]\hfill \end{array}$$

(114)

of the time evolution operator, where the group parameters $\theta ,$ $\tau ,$ and $\phi$ are governed by the oscillator transition dynamics through Formulas (105)–(108) back to the classical equation of motion for the parametric oscillator (71).

It is worth noting that Bargmann’s functions are studied in detail. The functions $t_{\lambda {\lambda }^{\prime }}^j\left(\tau \right)$ are related to the Meixner polynomials—the unitarity property of the Bargmann functions,

$$\sum _{{\lambda }^{″}=j+1}^{\infty }{t}_{\lambda {\lambda }^{″}}^j\left(\tau \right)t_{{\lambda }^{\prime }{\lambda }^{″}}^j\left(\tau \right)={\delta }_{\lambda {\lambda }^{\prime }},$$

(115)

gives the discrete orthogonality relation of these polynomials. A connection with a finite set of Jacobi polynomials orthogonal on an infinite interval is also relevant. All basic facts about the functions $T_{\lambda {\lambda }^{\prime }}^j\left(\theta ,\tau ,\phi \right)$ can be derived from the known properties of the Meixner and Jacobi polynomials (see Refs. [71,95,97] for more details).

11. Summary

The time-dependent Schrödinger equations with variable coefficients

$$i{\displaystyle \frac{\partial \psi }{\partial t}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\pm t{x}^2\psi =0$$

(116)

have Green functions of the form

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{\pm \pi ia\left(\pm t\right)}}}exp\left(\pm i{\displaystyle \frac{a^{\prime }\left(\pm t\right)-2xy+b\left(\pm t\right)y^2}{a\left(\pm t\right)}}\right),t>0,$$

(117)

where $a\left(t\right)=ai\left(t\right)$ and $b\left(t\right)=bi\left(t\right)$ are solutions of the Airy equation ${\mu }^{″}-t\mu =0$ that satisfy the initial conditions $a\left(0\right)=b^{\prime }\left(0\right)=0$ and $a^{\prime }\left(0\right)=b\left(0\right)=1$ (see Appendix A below for the construction of these solutions).

In the momentum representation, the corresponding Schrödinger equations with variable coefficients

$$i{\displaystyle \frac{\partial \psi }{\partial t}}\mp t{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}-{\displaystyle \frac{1}{4}}{x}^2\psi =0$$

(118)

have Green functions of the form

$$G\left(x,y,t\right)={\displaystyle \frac{1}{\sqrt{\mp 4\pi i{b}^{\prime }\left(\pm t\right)}}}exp\left(\mp i{\displaystyle \frac{b\left(\pm t\right)-2xy+a^{\prime }\left(\pm t\right)y^2}{4b^{\prime }\left(\pm t\right)}}\right),t>0,$$

(119)

where $a^{\prime }\left(t\right)=a{i}^{\prime }\left(t\right)$ and $b^{\prime }\left(t\right)=b{i}^{\prime }\left(t\right)$ are solutions of the equation ${\mu }^{″}-\left(1/t\right){\mu }^{\prime }-t\mu =0$ that satisfy the initial conditions $a^{\prime }\left(0\right)=1$ and $b^{\prime }\left(0\right)=0$ (see Appendix A for further properties of these functions).

The solution of the corresponding Cauchy initial value problem is given by the time evolution operator as follows,

$$\psi \left(x,t\right)={\int }_{-\infty }^{\infty }G\left(x,y,t\right)\phi \left(y\right)dy,\psi \left(x,0\right)=\phi \left(x\right)$$

(120)

for a suitable function $\phi$ on $\mathbb{R}$. Additional integrable cases are given with the help of the gauge transformation.

Particular solutions of the corresponding nonlinear Schrödinger equations are obtained using the methods of Refs. [40,46]. A special case of the quantum parametric oscillator with the Hamiltonian of the form (36) is studied in detail. The Green function is explicitly evaluated in terms of Airy functions by Equations (40)–(44) and the corresponding transition amplitudes are given in terms of a hypergeometric function by the Formula (59). A discrete orthogonality relation for certain ${}_{2}F_1$ functions is derived from the fundamentals of quantum physics. It is then identified as the orthogonality property of special Meixner polynomials with the help of a quadratic transformation. An extension to the general case of a parametric oscillator in quantum mechanics is also given. The relation of the transition amplitudes with unitary irreducible representations of the Lorentz group $SU\left(1,1\right)$ is established.

The mathematical framework discussed in this paper—solving the time-dependent Schrödinger equation with variable coefficients in terms of Airy functions and revealing an underlying $SU(1,1)$ symmetry—provides a solid foundation for understanding a variety of quantum optical phenomena. In particular, the analysis naturally connects with the theory of two-mode squeezing and parametric amplification, which are crucial for generating squeezed states of light. Such states play an essential role in advanced interferometric devices aimed at reducing quantum noise, as originally proposed by Carlton Caves [98] and later demonstrated in gravitational-wave observatories [99]. Moreover, the same parametric oscillator Hamiltonian underpins spontaneous parametric down-conversion processes used to generate entangled photon pairs [100], thereby linking the formal Airy function solutions to the evolution of quantum optical states in these systems. (For further background on SU(1,1) symmetry in quantum optics, see, e.g., [90,96,101].)

The techniques developed herein also extend naturally to other areas of modern physics. For instance, the formalism can be applied to Bose–Einstein condensates, where the non-autonomous Gross–Pitaevskii equation—describing condensate dynamics in time-dependent traps or with modulated interaction strengths—shares mathematical similarities with the equations considered here [102]. Such approaches have led to analytical soliton solutions and have been used to stabilize condensate dynamics via Feshbach resonance management [103]. Additionally, the same mathematical structure is at the heart of the dynamical Casimir effect, where time-dependent boundary conditions amplify vacuum fluctuations into real photons [104] (see also [13] for a discussion of related time-dependent quantum effects). Together, these applications highlight the broad impact of the Airy function approach to time-dependent quantum oscillators, offering deep insights into both fundamental and applied quantum phenomena. Further extension to the quantum forced parametric oscillator can be obtained accordingly.