Novel quantum description of time-dependent molecular interactions obeying a generalized non-central potential

Abstract

Quantum features of time-dependent molecular interactions are investigated by introducing a time-varying Hamiltonian that involves a generalized non-central potential. According to the Lewis–Riesenfeld theory, quantum states (wave functions) of such dynamical systems are represented in terms of the eigenstates of the invariant operator. Hence, we have derived the eigenstates of the invariant operator of the system using elegant mathematical manipulations known as the unitary transformation method and the Nikiforov–Uvarov method. Based on full wave functions that are evaluated by considering such eigenstates, quantum properties of the system are analyzed. The time behavior of probability densities which are the absolute square of the wave functions are illustrated in detail. This research provides a novel method for investigating quantum characteristics of complicated molecular interactions. The merit of this research compared to conventional ones in this field is that time-varying parameters, necessary for the actual description of intricate atomic and molecular behaviors with high accuracy, are explicitly considered.

Appendices

Appendix A: Methods

To solve the quantum problem of the system, several mathematical methods have been used. They are as follows.

1. Invariant operator method

This method is useful when the Hamiltonian of a system depends on time on account of the possible existence of time-varying parameters or driving forces. According to its definition, the invariant operator can be established from the Liouville-von Neumann equation. By solving the eigenvalue equation of the invariant operator, we can get the eigenfunctions of the system. It is well known that such eigenfunctions are different from wave functions of the system by only some time-dependent phase factors [10]. Hence, it is possible to obtain analytical wave functions which are complete quantum solutions of the system by multiplying appropriate phase factors on the eigenfunctions, that can be obtained with the help of the Schrödinger equation.

2. Nikiforov–Uvarov method

When we solve second-order differential equations such as the Schrö dinger equation, the NU method [32, 45,46,47,48] is useful. Through a coordinate transformation $\xi =\xi (r)$ [45], we can express a general type of a second-order differential equation for an arbitrary function $f(\xi )$ as

$$\begin{aligned} \frac{\hbox {d}^2 f(\xi )}{\hbox {d} \xi ^2}+\frac{\tilde{\tau }(\xi )}{\sigma (\xi )} \frac{\hbox {d} f(\xi )}{\hbox {d} \xi }+\frac{\tilde{\sigma }(\xi )}{\sigma ^{2}(\xi )}f(\xi )=0, \end{aligned}$$

(A1)

where $\sigma (\xi )$, $\tilde{\sigma }(\xi )$, and $\tilde{\tau }(\xi )$ are polynomials of $\xi$. We suppose that $\sigma (\xi )$ and $\tilde{\sigma }(\xi )$ are allowed up to a second degree and another polynomial $\tilde{\tau }(\xi )$ up to a first degree. A class of special orthogonal functions [45] that are described by Eq. (A1) can be simple forms through a mathematical manipulation after denoting $f_{n}(\xi )=h_{n}(\xi )y_{n}(\xi )$, where $h_{n}(\xi )$ are appropriately chosen functions. If we represent Eq. (A1) in terms of $y_{n}(\xi )$, the equation reduces to

$$\begin{aligned} \frac{\hbox {d}^2 y_{n}}{\hbox {d} \xi ^2}+\frac{\tau (\xi )}{\sigma \left( \xi \right) }\frac{\hbox {d} y_{n}}{\hbox {d} \xi }+\frac{\lambda _n}{\sigma \left( \xi \right) } y_{n}=0, \end{aligned}$$

(A2)

where we can represent $\tau (\xi )$ as $\tau (\xi )=\tilde{\tau }(\xi )+2\Pi (\xi )$ with the condition that the derivative of $\tau (\xi )$ is negative, while $\lambda _n$ are constants of the form [48]

$$\begin{aligned} \lambda _{n}=-\frac{n(n-1)}{2}\frac{\hbox {d}^2 \sigma }{\hbox {d} \xi ^2}-n\frac{\hbox {d} \tau }{\hbox {d} \xi }. \end{aligned}$$

(A3)

Notice that $\lambda _n$ can be written as a particular solution of the equation for $y_{n}(\xi )$. In fact, this is an nth-order degree of the polynomial. It is possible to represent hypergeometric functions $y_{n}(\xi )$ by means of the Rodrigues relation of the form [48]

$$\begin{aligned} y_{n}(\xi )={B_{n}}{\rho ^{-1} (\xi )}\frac{\hbox {d}^{n}\left[ \sigma ^{n}(\xi )\rho (\xi )\right] }{\hbox {d}\xi ^{n}} , \end{aligned}$$

(A4)

where $B_{n}$ are normalization factors and $\rho (\xi )$ is a weight function that yields a condition of the form [45]

$$\begin{aligned} \frac{\hbox {d} [\sigma \left( \xi \right) \rho \left( \xi \right) ] }{\hbox {d} \xi } =\tau \left( \xi \right) \rho \left( \xi \right) . \end{aligned}$$

(A5)

Hence, the weight function can be obtained using Eq. (A5) together with the expression of $\Pi$ which is (see Eq. (12) of Ref. [48])

$$\begin{aligned}&\Pi (\xi )=\frac{1}{2}\bigg [\frac{\hbox {d} \sigma (\xi )}{\hbox {d} \xi }-\tilde{\tau }(\xi )\bigg ]\nonumber \\&\quad \quad \qquad \pm \bigg \{ \bigg [ \frac{1}{2}\bigg (\frac{\hbox {d} \sigma (\xi )}{\hbox {d} \xi }-\tilde{\tau }(\xi )\bigg )\bigg ] ^{2}- \tilde{\sigma }(\xi )+k\sigma (\xi )\bigg \}^{1/2}.\nonumber \\ \end{aligned}$$

(A6)

Let us regard that the mathematical representation given inside the square root in Eq. (A6) can be rewritten as a square of a function provided that its discriminant is zero. It would then be possible to obtain an equation for k. Such a value of k is necessary in the development of the NU method. We can easily verify that k is represented as $k=\lambda -{\hbox {d}}\Pi (\xi )/{\hbox {d}} \xi$. Finally, we derive the representation of $h_{n}(\xi )$ from

$$\begin{aligned} \frac{\hbox {d} h_n(\xi )}{\hbox {d} \xi }=\frac{\Pi (\xi )}{\sigma (\xi )}h_n(\xi ). \end{aligned}$$

(A7)

Apparently, this equation can be solved by the separation of variables method. Thus, by multiplying $y_n(\xi )$ obtained from Eq. (A4) and $h_n(\xi )$ obtained from Eq. (A7), we have the complete solutions for $f_n(\xi )$.

Appendix B: Mathematical relations

Some important mathematical relations used in the text are given by

$$\begin{aligned}&U_{\mathrm {II}}(t)rU_{\mathrm {II}}^\dagger (t)=\left( \frac{ \beta _{2}(t)}{\alpha _{2}}\right) ^{\frac{1}{2}}r, \end{aligned}$$

(B1)

$$\begin{aligned}&U_{\mathrm {II}}(t)pU_{\mathrm {II}}^\dagger (t)=\left( \frac{\beta _{2}(t)}{\alpha _{2}}\right) ^{-\frac{1}{2} }\left( p-\frac{\beta _{3}(t)}{\alpha _{2}}r\right) , \end{aligned}$$

(B2)

$$\begin{aligned}&\exp \left[ \left( \ln \frac{\beta _2(t)}{\alpha _2}\right) r \frac{\partial }{\partial r} \right] f(r) = f\left( \frac{\beta _2(t)}{\alpha _2} r\right) . \end{aligned}$$

(B3)

The first two relations can be obtained by straightforward evaluations using Eq. (23) [or using Eqs. (24) and (25)] [13]. For the last relation, you can refer to Ref. [44].

Appendix C: Derivation of Eq. (68)

From Eq. (16), we replace $| \tilde{\phi }_{nn^{\prime }m}(t)\rangle$ with $U_{\mathrm {II} }^{-1}| \tilde{\phi }_{nn^{\prime }m}^{0}\rangle$ according to Eq. (66):

$$\begin{aligned} \hbar \frac{\hbox {d}}{\hbox {d}t}\gamma _{n}(t)=\big \langle \tilde{\phi }_{nn^{\prime }m}^{0}\mid U_{\mathrm {II}}\left( i\hbar \frac{ \partial }{\partial t}-\tilde{H}(r,\theta ,\varphi ,t)\right) U_{\mathrm {II} }^{-1}\mid \tilde{\phi }_{nn^{\prime }m}^{0}\big \rangle . \end{aligned}$$

(C1)

Now we easily see that a straightforward evaluation gives

$$\begin{aligned} U_{\mathrm {II}}\left( i\hbar \frac{\partial }{\partial t}-\tilde{H}(r,\theta ,\varphi ,t)\right) U_{\mathrm {II}}^{-1} = -\frac{a(t)\tilde{I}_{0}}{\beta _{2}(t)}. \end{aligned}$$

(C2)

Then, we finally have Eq. (68) in the text.