On the possible emergence of nonstatic quantum waves in a static environment

Abstract

A noticeable research topic in optics is optical phenomena associated with the nonstatic waves. If the parameters of a medium vary by a periodic or randomly exerted perturbation, the waves become nonstatic, leading to a variation in their shapes. The decay of wave amplitudes through dissipation is also an outcome of wave nonstaticity. In this research, we demonstrate that Schrödinger equation allows nonstatic quantum waves even in the situation where the waves neither suffer perturbation nor undergo dissipation. The time behavior of nonstatic waves in such a static environment is investigated in detail in the Fock state at first. And then, we extend our development for such a peculiar wave characteristic to the case of a Gaussian wave evolution which resembles the behavior of classical states. It is shown how to define a quantitative nonstaticity measure which can be used universally beyond the Fock-state nonstatic waves. Understanding the characteristics of nonstatic-wave phenomena may not only provide deeper insight into the essence of nature, but is helpful for practical applications of the wave nonstaticity in science and technology of electromagnetic wave modulations.

Appendices

Appendix A: Evaluation of the Gaussian wave function

In order to depict the probability density associated with the Gaussian wave function given in Eq. (8), it is necessary to derive its analytical expression. For this purpose, we first evaluate $c_n$ using the basic relation which is

$$\begin{aligned} c_n = \int _{-\infty }^{\infty } \langle \psi _n (0)|q \rangle \langle q |\psi (0)\rangle {\text {d}}q. \end{aligned}$$

(A1)

A minor calculation after inserting the initial wave functions $\langle q |\psi _n(0) \rangle$ and Eq. (7) into the above equation leads to

$$\begin{aligned} c_n= & {} [ {W_\mathrm{R}(0)K_\mathrm{R}} ]^{1/4}\frac{\sqrt{2}}{\sqrt{2^n n!}} \frac{[K-W(0)]^{n/2}}{[K+W^*(0)]^{(n+1)/2}}\nonumber \\&\quad \times \exp \bigg ( \frac{K^2 \xi ^2}{2[W^*(0)+K]}\bigg ) \nonumber \\&\quad \times H_n \Bigg ( \frac{\sqrt{W_\mathrm{R}(0)}K \xi }{\{[K-W(0)][K+W^*(0)]\}^{1/2}} \Bigg ) e^{-i\gamma _n (0)}e^{-\frac{1}{2}K\xi ^2}. \nonumber \\ \end{aligned}$$

(A2)

Here, we have used the integral formula of the form [34]

$$\begin{aligned}&\int _{-\infty }^\infty e^{-(x-y)^2} H_n (\alpha x) {\text {d}}x \nonumber \\&\quad = \pi ^{1/2}(1-\alpha ^2)^{n/2} H_n \bigg ( \frac{\alpha y}{(1-\alpha ^2)^{1/2}} \bigg ). \end{aligned}$$

(A3)

Now, from the substitution of Eq. (A2) into Eq. (8) in the text, we have

$$\begin{aligned} \langle q |\psi (t) \rangle= & {} \bigg ( \frac{W_\mathrm{R}(0)W_\mathrm{R}(t)}{\pi } \bigg )^{1/4}\bigg ( \frac{2K_\mathrm{R}^{1/2}}{ g(t)\exp [-2i\Theta (t) ]} \bigg )^{1/2}\nonumber \\&\quad \times \exp \bigg [-\frac{1}{2}\Big (K \xi ^2 +i\Theta (t) \Big ) \bigg ] \nonumber \\&\quad \times \exp \bigg [ \frac{K^2 \xi ^2}{K+W^*(0)} \bigg ( \frac{1}{2} - \frac{W_\mathrm{R}(0)}{g(t)} \bigg ) \bigg ]\nonumber \\&\quad \times \exp \Bigg [ - \frac{\mathcal {W}(t)}{2}q^2 + R(t) q \Bigg ], \end{aligned}$$

(A4)

where $\Theta (t) = \omega \int _0^t f^{-1}(t'){\text {d}}t'$, and

$$\begin{aligned} {{\mathcal {W}}}(t)= & {} W(t)+ \frac{2W_\mathrm{R}(t)[K-W(0)]}{g(t)}, \end{aligned}$$

(A5)

$$\begin{aligned} g(t)= & {} [K+W^*(0)] \exp [2i\Theta (t) ]-[K-W(0)], \nonumber \\ \end{aligned}$$

(A6)

$$\begin{aligned} R(t)= & {} \frac{2K \xi \sqrt{W_\mathrm{R}(0)W_\mathrm{R}(t)} }{g(t) \exp [-i\Theta (t)]}. \end{aligned}$$

(A7)

In the derivation of Eq. (A4), Mehler’s formula [35] has been used:

$$\begin{aligned} \sum _{n=0}^{\infty } \frac{(z/2)^n}{n!} H_n (x) H_n (y)= & {} \frac{1}{(1-z^2)^{1/2}}\nonumber \\&\quad \times \exp \left[ \frac{2xyz-(x^2+y^2)z^2}{1-z^2}\right] . \nonumber \\ \end{aligned}$$

(A8)

If we confine the range of $\varphi$ within $-\pi /2 \le \varphi < \pi /2$ for convenience, $\Theta (t)$ can be evaluated, leading for $t\ge 0$ to

$$\begin{aligned} \Theta (t) = \tan ^{-1} Z(t) - \tan ^{-1} Z(0)+{{\mathcal {G}}}(t), \end{aligned}$$

(A9)

where

$$\begin{aligned} Z(t)= & {} C+A\tan (\omega t +\varphi ), \end{aligned}$$

(A10)

$$\begin{aligned} {{\mathcal {G}}}(t)= & {} \pi \sum _{m=0}^{\infty }u[t-(2m+1)\pi /(2\omega )+\varphi /\omega ]. \end{aligned}$$

(A11)

In Eq. (A11), u[t] is the step function (the Heaviside step function).

We can also have the probability density associated with the wave function given in Eq. (A4) from a straightforward evaluation as

$$\begin{aligned} |\langle q |\psi (t) \rangle |^2= & {} \sqrt{\frac{W_\mathrm{R}(0)W_\mathrm{R}(t)}{\pi }} \frac{2K_\mathrm{R}^{1/2}}{\sqrt{G(t)}}\nonumber \\&\quad \times \exp \{-X(t)[q-\eta (t)]^2\}, \end{aligned}$$

(A12)

where

$$\begin{aligned} X(t)= & {} W_\mathrm{R}(t) [1-Z(t)/G(t)], \end{aligned}$$

(A13)

$$\begin{aligned} Z(t)= & {} g(t)[W^*(0)-K^*]+g^*(t)[W(0)-K], \nonumber \\ \end{aligned}$$

(A14)

$$\begin{aligned} G(t)= & {} g(t)g^*(t), \end{aligned}$$

(A15)

$$\begin{aligned} \eta (t)= & {} \frac{\xi }{\sqrt{W_\mathrm{R}(0)W_\mathrm{R}(t)}} \{W_\mathrm{R}(0) \cos [\Theta (t)] \nonumber \\&+W_\mathrm{I}(0)\sin [\Theta (t)]\}. \end{aligned}$$

(A16)

Equation (A12) with Eqs. (A13)–(A16) has been used in the plot of Fig. 5 in the text.

Appendix B: Evaluation of the measure of nonstaticity for the Gaussian wave

For the Gaussian wave, the measure of nonstaticity is determined from ${{\mathcal {W}}}$. Let us divide ${{\mathcal {W}}}$ as the real and imaginary parts, such that

$$\begin{aligned} {{\mathcal {W}}} = {{\mathcal {W}}}_\mathrm{R} +i {{\mathcal {W}}}_\mathrm{I}. \end{aligned}$$

(B1)

A straightforward evaluation using Eq. (A5) gives

$$\begin{aligned} {{\mathcal {W}}}_\mathrm{R}= & {} \frac{4}{G(t)}K_\mathrm{R} W_\mathrm{R}(0)W_\mathrm{R}(t), \end{aligned}$$

(B2)

$$\begin{aligned} {{\mathcal {W}}}_\mathrm{I}= & {} \frac{2}{G(t)} [\mu (t) W_\mathrm{R}(t) + \nu (t) W_\mathrm{I}(t) ], \end{aligned}$$

(B3)

where

$$\begin{aligned} \mu (t)= & {} [W_\mathrm{R}^2(0)-W_\mathrm{I}^2(0) + 2W_\mathrm{I}(0)K_\mathrm{I} -K_\mathrm{R}^2 -K_\mathrm{I}^2]\nonumber \\&\quad \times \sin [2\Theta (t)]+2W_\mathrm{R}(0)[K_\mathrm{I}-W_\mathrm{I}(0)]\cos [2\Theta (t)], \nonumber \\ \end{aligned}$$

(B4)

$$\begin{aligned} \nu (t)= & {} 2\{ \{W_\mathrm{R}(0) \cos [\Theta (t)] + W_\mathrm{I}(0)\sin [\Theta (t)] \}^2 \nonumber \\&\quad +\,(K_\mathrm{R}^2 + K_\mathrm{I}^2)\sin ^2[\Theta (t)] -2K_\mathrm{I} \{W_\mathrm{R}(0) \cos [\Theta (t)] \nonumber \\&\quad +\,W_\mathrm{I}(0) \sin [\Theta (t)]\}\sin [\Theta (t)] \}. \end{aligned}$$

(B5)

From these expressions, we easily have ${{{\mathcal {W}}}_\mathrm{I}}/{{{\mathcal {W}}}_\mathrm{R}}$ given in Eq. (9) in the text.