Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

TL;DR

This paper investigates how symmetric and asymmetric confinement affect a quantum harmonic oscillator using information entropy measures, providing accurate eigenvalues and analyzing localization and energy behaviors.

Contribution

It introduces a variation-induced diagonalization method for asymmetric confinement and explores the impact of confinement asymmetry on uncertainty measures and energy spectra.

Findings

Uncertainty measures converge to free oscillator values at large confinement lengths.

Asymmetric confinement causes excited-state energies to pass through a maximum.

Localization increases with asymmetry parameter.

Abstract

Information-based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of \emph{symmetric and asymmetric} confinement in a quantum harmonic oscillator. Also, the transformation of Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ($x_c$). In symmetric case, a wide range of $x_{c}$ has been taken up, whereas asymmetric confinement is dealt by shifting the minimum of potential from origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary time propagation scheme. For asymmetric confinement, a variation-induced exact diagonalization procedure is also introduced, which produces very high-quality results. One finds that, in symmetric confinement, after a certain characteristic $x_{c}$, all these properties converge to respective values of free harmonic oscillator. In asymmetric situation, excited-state energies always pass through a maximum. For this potential, the classical turning-point decreases, whereas well depth increases with the strength of asymmetry. Study of these uncertainty measures reveals that, localization increases with an increase of asymmetric parameter.

Tables

Table 1. Table 1: Energy spectrum for first five energy states of SCHO potential at low η 𝜂 \eta values, namely 0.001, 0.01, 0.1. PT stands for Perturbation Theory. See text for details.

Energy	$\eta =0.001$		$\eta =0.01$		$\eta =0.1$
	PT§	Exact†	PT§	Exact†	PT§	Exact†
${ϵ}_0$	1.2337658956	1.2337658950	1.23435400	1.23435394	1.240235	1.240229
${ϵ}_1$	4.9349435369	4.9349435363	4.93621556	4.93621550	4.948935	4.948930
${ϵ}_2$	11.103460359	11.103460360	11.10485904	11.10485905	11.118846	11.118847
${ϵ}_3$	19.7393691362	19.7393691364	19.74081214	19.74081215	19.755242	19.755243
${ϵ}_4$	30.8426763672	30.8426763673	30.84413989	30.84413990	30.868775	30.858776

Table 2. Table 2: S x ′ subscript 𝑆 superscript 𝑥 ′ S_{x^{\prime}} and S p ′ subscript 𝑆 superscript 𝑝 ′ S_{p^{\prime}} for first three states of SCHO potential, at six different η 𝜂 \eta , namely, 0.001, 0.01, 0.1, 25, 50, 100. See text for details.

$\eta$	$S_{x^{\prime }}^0$	$S_{p^{\prime }}^0$	$S_{x^{\prime }}^1$	$S_{p^{\prime }}^1$	$S_{x^{\prime }}^2$	$S_{p^{\prime }}^2$
0.001	0.386286	1.825737	0.386293	2.220674	0.386294	2.366773
0.01	0.386215	1.825769	0.386289	2.220676	0.386294	2.366793
0.1	0.385506	1.826088	0.386244	2.220701	0.386291	2.366998
25	0.209026	1.946894	0.353513	2.258814	0.382511	2.425027
50	0.080943	2.065966	0.296874	2.337153	0.374755	2.488682
100	-0.080239	2.225139	0.179801	2.488431	0.350346	2.630012

Table 3. Table 3: I x ′ subscript 𝐼 superscript 𝑥 ′ I_{x^{\prime}} and I p ′ subscript 𝐼 superscript 𝑝 ′ I_{p^{\prime}} for first three states of SCHO potential, at six different η 𝜂 \eta , namely, 0.001, 0.01, 0.1, 25, 50, 100. See text for details.

$\eta$	$I_{x^{\prime }}^0$	$I_{p^{\prime }}^0$	$I_{x^{\prime }}^1$	$I_{p^{\prime }}^1$	$I_{x^{\prime }}^2$	$I_{p^{\prime }}^2$
0.001	9.869604	0.522753	39.478417	1.125516	88.826439	1.243259
0.01	9.869605	0.522767	39.478418	1.125436	88.826439	1.243276
0.1	9.869651	0.521816	39.478464	1.124636	88.826430	1.243445
25	11.678374	0.357351	41.880599	0.926130	89.290266	1.224696
50	14.637965	0.275574	47.436915	0.776019	92.551459	1.139451
100	20.063762	0.199485	60.996032	0.589550	106.634933	0.950170

Table 4. Table 4: E x ′ subscript 𝐸 superscript 𝑥 ′ E_{x^{\prime}} and E p ′ subscript 𝐸 superscript 𝑝 ′ E_{p^{\prime}} for first three states of SCHO potential, at six different η 𝜂 \eta , namely, 0.001, 0.01, 0.1, 25, 50, 100. See text for details.

$\eta$	$E_{x^{\prime }}^0$	$E_{p^{\prime }}^0$	$E_{x^{\prime }}^1$	$E_{p^{\prime }}^1$	$E_{x^{\prime }}^2$	$E_{p^{\prime }}^2$
0.001	0.750007	0.186731	0.7500005	0.126261	0.7500001	0.115063
0.01	0.750076	0.186723	0.750005	0.126259	0.750001	0.115059
0.1	0.750769	0.186637	0.750048	0.126237	0.750010	0.115026
25	0.930001	0.163488	0.782014	0.118493	0.7599928	0.105239
50	1.060491	0.146313	0.837179	0.109491	0.778344	0.096475
100	1.262383	0.125699	0.952923	0.095041	0.835886	0.0831471

Table 5. Table 5: S x ′ subscript 𝑆 superscript 𝑥 ′ S_{x^{\prime}} ,  I x ′ subscript 𝐼 superscript 𝑥 ′ I_{x^{\prime}} and E x ′ subscript 𝐸 superscript 𝑥 ′ E_{x^{\prime}} for first three states of SCHO potential at small η 𝜂 \eta , namely, 0.001, 0.01, 0.1. PT implies Perturbation Theory. See text for details.

Quantity	$\eta =0.001$		$\eta =0.01$		$\eta =0.1$
	PT§	Exact†	PT§	Exact†	PT§	Exact†
$S_{x^{\prime }}^0$	0.386287	0.386386	0.386216	0.386293	0.385508	0.386294
$S_{x^{\prime }}^1$	0.386246	0.386215	0.386289	0.386289	0.386246	0.386294
$S_{x^{\prime }}^2$	0.386294	0.385506	0.386294	0.386244	0.386294	0.386291
$I_{x^{\prime }}^0$	9.869604	9.869604	9.869605	9.869605	9.869652	9.869651
$I_{x^{\prime }}^1$	39.478417	39.478417	39.478418	39.478418	39.478462	39.478464
$I_{x^{\prime }}^2$	88.826439	88.826439	88.826439	88.826439	88.826429	88.826430
$E_{x^{\prime }}^0$	0.750008	0.750007	0.750077	0.750000	0.750771	0.750000
$E_{x^{\prime }}^1$	0.750000	0.750076	0.750005	0.750005	0.750048	0.750001
$E_{x^{\prime }}^2$	0.750001	0.750769	0.750048	0.750048	0.750004	0.750010

Table 6. Table 6: Calculated S x n , E x n superscript subscript 𝑆 𝑥 𝑛 superscript subscript 𝐸 𝑥 𝑛 S_{x}^{n},E_{x}^{n} of a SCHO at three chosen x c subscript 𝑥 𝑐 x_{c} ’s; 0.25, 2 and 5, for five lowest states ( n = 0 − 4 𝑛 0 4 n\!=\!0-4 ). Reference results are obtained by taking exact analytical wave functions, reported in montgomery07 . PR implies Present Result. See text for more details.

	$x_c=0.25$		$x_c=2$		$x_c=5$
Property	PR	Reference	PR	Reference	PR	Reference
$S_x^0$	$-$1.0000307	$-$1.0000307	0.9603491	0.9603491	1.0767572	1.0767573
$S_x^1$	$-$1.0000019	$-$1.0000019	1.0625421	1.0625421	1.3427276	1.3427277
$S_x^2$	$-$1.0000003	$-$1.0000003	1.0749785	1.0749785	1.4986081	1.4986082
$S_x^3$	$-$1.0000001	$-$1.0000001	1.0776239	1.0776239	1.6097006	1.6097006
$S_x^4$	$-$0.9999999	$-$0.9999999	1.0786018	1.0786018	1.6964625	1.6964627
$E_x^0$	3.0001200	3.0001200	0.4347289	0.4347291	0.3989422	0.3989422
$E_x^1$	3.0000070	3.0000075	0.3832620	0.3832622	0.2992065	0.2992067
$E_x^2$	3.0000010	3.0000014	0.3775775	0.3775772	0.2555729	0.2555724
$E_x^3$	3.0000005	3.0000005	0.3761693	0.3761693	0.2290809	0.2290810
$E_x^4$	3.0000001	3.0000001	0.3755618	0.3755620	0.2106059	0.2106061

Table 7. Table 7: Comparison of energies of first six states of ACHO potential at four distinct d m subscript 𝑑 𝑚 d_{m} . Length of the box is fixed at 2. PR implies Present Result. See text for details.

$d_m$	${ϵ}_0$ (PR)	${ϵ}_0$ (Ref.)	${ϵ}_1$ (PR)	${ϵ}_1$(Ref.)
0.36	2.7177633960054	2.59691966†, 2.7177633960054‡	10.283146010610	10.28314602†, 10.283146010610‡
1.92	6.0383021056781	6.03830195†, 6.0383021056781‡	13.901445986629	13.90144582†, 13.901445986629‡
5.00	26.065225076406	26.065225076406§	35.462261039378	35.462261039378§
10.0	97.474035270680	97.474035270680§	110.51944554927	110.51944554927§
	${ϵ}_2$ (PR)	${ϵ}_2$ (Ref.)	${ϵ}_3$ (PR)	${ϵ}_3$(Ref.)
0.36	22.648848755052	22.64884877†, 22.648848755052‡	39.929984298830	32.92998431†, 39.929984298830‡
1.92	26.249310409373	26.24931024†, 26.249310409373‡	43.513981920357	43.51398176†, 43.513981920357‡
5.00	47.817024422796	47.817024422796§	64.900200447511	64.900200447511§
10.0	123.593144939095	123.593144939095§	140.555432078323	140.555432078323§
	${ϵ}_4$ (PR)	${ϵ}_4$ (Ref.)	${ϵ}_5$ (PR)	${ϵ}_5$(Ref.)
0.36	62.140768627508	62.140768627508‡	89.284409553063	89.284409553063‡
1.92	65.715672311936	65.715672311936‡	92.854029622882	92.854029622882‡
5.00	87.137790461503	87.137790461503§	114.244486402564	114.244486402564§
10.0	162.519960161732	162.519960161732§	189.515389275133	189.515389275133§

Table 8. Table 8: S x , S p , S subscript 𝑆 𝑥 subscript 𝑆 𝑝 𝑆 S_{x},S_{p},S for n = 0 − 2 𝑛 0 2 n\!=\!0-2 states of ACHO at five specific d m subscript 𝑑 𝑚 d_{m} . See text for details.

$d_m$	$S_x^0$	$S_p^0$	$S^0$	$S_x^1$	$S_p^1$	$S^1$	$S_x^2$	$S_p^2$	$S^2$
0.12	0.3783	1.8296	2.2079	0.3857	2.2212	2.6069	0.3862	2.3692	2.7554
2.04	0.3428	1.8779	2.2208	0.3807	2.2512	2.6319	0.3850	2.3852	2.7703
5.0	0.2184	2.0238	2.2422	0.3636	2.3390	2.7027	0.3782	2.4512	2.8295
8.0	0.093	2.1576	2.2506	0.3360	2.4313	2.7674	0.3695	2.5429	2.9124
10.0	0.0233	2.2294	2.2527	0.3108	2.4900	2.8009	0.3611	2.6057	2.9668

Table 9. Table 9: I x ′ , I p ′ ; S x ′ , S p ′ ; E x ′ , E p ′ subscript 𝐼 superscript 𝑥 ′ subscript 𝐼 superscript 𝑝 ′ subscript 𝑆 superscript 𝑥 ′ subscript 𝑆 superscript 𝑝 ′ subscript 𝐸 superscript 𝑥 ′ subscript 𝐸 superscript 𝑝 ′ I_{x^{\prime}},I_{p^{\prime}};S_{x^{\prime}},S_{p^{\prime}};E_{x^{\prime}},E_{p^{\prime}} values for ground state of ACHO potential at three different η 𝜂 \eta , namely 0.001, 0.01, 0.1, considering d m = 1 subscript 𝑑 𝑚 1 d_{m}=1 .

$\eta$	$I_{x^{\prime }}$	$I_{p^{\prime }}$	$S_{x^{\prime }}$	$S_{p^{\prime }}$	$E_{x^{\prime }}$	$E_{p^{\prime }}$
5	13.883826	0.32456	0.224258	1.997965	0.908473	0.182345
10	19.016519	0.224533	0.072132	2.162765	1.065115	0.179876
20	26.833658	0.172067	-0.099461	2.346781	1.265513	0.168523

Equations

$$A_n=\int \sqrt{(V(x)-{ϵ}_n)}dx,$$

$$i\frac{\partial }{\partial t}\psi (x,t)=H\psi (x,t)=\left[-\frac{1}{2}\frac{d^2}{d{x}^2}+v(x)+v_c(x)\right]\psi (x,t).$$

$$v_c(x)=\{\begin{array}{l}0,-x_c<x<+x_c\\ +\infty ,|x|\ge x_c.\end{array}$$

$$-\frac{\partial \psi (x,\tau )}{\partial \tau }=H\psi (x,\tau ),$$

$$\psi (x,\tau )=\sum _{\kappa =0}^{\infty }{c}_{\kappa }{\varphi }_{\kappa }(x)\exp (-{ϵ}_{\kappa }\tau ).$$

$$\underset{\tau \to \infty }{\lim }\psi (x,\tau )\approx c_0{\psi }_0(x)e^{-{ϵ}_0\tau }.$$

$$\psi (x,\tau +\Delta \tau )=e^{-\Delta \tau H}\psi (x,\tau ).$$

$$e^{(\Delta \tau /2)H_j}{\psi }_j^{{}^{\prime }(l+1)}=e^{-(\Delta \tau /2)H_j}{\psi }_j^l,$$

$${\alpha }_j{\psi }_{j-2}^{{}^{\prime }(l+1)}+{\beta }_j{\psi }_{j-1}^{{}^{\prime }(l+1)}+{\gamma }_j{\psi }_j^{{}^{\prime }(l+1)}+{\delta }_j{\psi }_{j+1}^{{}^{\prime }(l+1)}+{\zeta }_j{\psi }_{j+2}^{{}^{\prime }(l+1)}={\xi }_j^l.$$

$$\left[\begin{array}{ccccccc}{\gamma }_1& {\delta }_1& {\zeta }_1& & & & (0)\\ {\beta }_2& {\gamma }_2& {\delta }_2& {\zeta }_2& & & \\ {\alpha }_3& {\beta }_3& {\gamma }_3& {\delta }_3& {\zeta }_3& & \\ & \ddots & \ddots & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & \ddots & \ddots & {\zeta }_{N-2}\\ & & & {\alpha }_{N-1}& {\beta }_{N-1}& {\gamma }_{N-1}& {\delta }_{N-1}\\ (0)& & & & {\alpha }_N& {\beta }_N& {\gamma }_N\end{array}\right]\left[\begin{array}{c}{\psi }_1^{\prime (l+1)}\\ {\psi }_2^{\prime (l+1)}\\ {\psi }_3^{\prime (l+1)}\\ ⋮\\ {\psi }_{N-2}^{\prime (l+1)}\\ {\psi }_{N-1}^{\prime (l+1)}\\ {\psi }_N^{\prime (l+1)}\end{array}\right]=\left[\begin{array}{c}{\xi }_1^l\\ {\xi }_2^l\\ {\xi }_3^l\\ ⋮\\ {\xi }_{N-2}^l\\ {\xi }_{N-1}^l\\ {\xi }_N^l\end{array}\right]$$

$$\varphi (p)=\frac{1}{\sqrt{2\pi }}\int \psi (x)e^{-ipx/\hslash }dx.$$

$$S_x=-\int \rho (x)\text{mbox}ln\left[\rho (x)\right]dx,S_p=-\int \rho (p)\text{mbox}ln\left[\rho (p)\right]dp.$$

$$S=S_x+S_p\ge (1+\text{mbox}ln\pi ).$$

$$I_x=\int \left[\frac{|\nabla \rho (x){|}^2}{\rho (x)}\right]dx,I_p=\int \left[\frac{|\nabla \rho (p){|}^2}{\rho (p)}\right]dp.$$

$$I=I_x{I}_p\ge 4.$$

$$E_x=\int \left[|\rho (x){|}^2\right]dx,E_p=\int \left[|\rho (p){|}^2\right]dp.$$

$$E=E_x{E}_p\le \frac{1}{2\pi }.$$

$$\begin{array}{r}-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}k{x}^2\psi +V\theta (x^2-x_c^2)\psi =E\psi ,\\ \theta (x^2-x_c^2)=0,\mathrm{at}x\le \text{abs}{x}_c\\ \theta (x^2-x_c^2)=1,\mathrm{at}x>\text{abs}{x}_c.\end{array}\}$$

$$\displaystyle\epsilon\equiv\epsilon\bigg{(}\frac{\hbar^{2}}{m},k,x_{c}\bigg{)}\quad\mathrm{and}\quad\psi\equiv\psi\bigg{(}\frac{\hbar^{2}}{m},k,x_{c},x\bigg{)}.$$

$$-\frac{1}{2}\frac{d^2\psi (x^{\prime })}{d{x^{\prime }}^2}+\frac{1}{2}\frac{m}{{\hslash }^2}k{x}_c^4{x^{\prime }}^2\psi (x^{\prime })+V\theta ({x^{\prime }}^2-1)\psi (x^{\prime })=\frac{m{x}_c^2}{{\hslash }^2}ϵ\psi (x^{\prime }),$$

$$\displaystyle\left.\begin{aligned} \epsilon\bigg{(}\frac{\hbar^{2}}{m},k,x_{c}\bigg{)}=\frac{\hbar^{2}}{mx_{c}^{2}}\ \epsilon\bigg{(}1,\frac{mkx_{c}^{4}}{\hbar^{2}},1\bigg{)}\\ \psi\bigg{(}\frac{\hbar^{2}}{m},k,x_{c},x\bigg{)}=\frac{1}{\sqrt{\lambda}}\ \psi\bigg{(}1,\eta,1,x^{\prime}\bigg{)}\\ \eta=\frac{mkx_{c}^{4}}{\hbar^{2}}.\end{aligned}\right\}$$

$$\epsilon_{n+1}(\eta)=\epsilon_{n+1}(0)+\eta\bigg{[}\frac{1}{6}-\frac{1}{(n+1)^{2}\pi^{2}}\bigg{]},\quad n=0,1,2,3,\cdots$$

$$S_x^n=\log 4x_c-1,E_x^n=\frac{3x_c}{4},I_x^n=\frac{n^2{\pi }^2}{x_c^2},n=1,2,3,\cdots$$

$${\psi }_n(\eta ;x^{\prime })={\psi }_n^{(0)}(0;x^{\prime })+\sum _{m\ne n}^{\infty }\frac{V_{mn}^{\eta }}{{ϵ}_n^0-{ϵ}_m^0}{\psi }_m^{(0)}(0;x^{\prime })$$

$$I_{x^{\prime }}^0$$

$$E_{x^{\prime }}^0$$

$${\psi }_e(x)=e^{-\frac{x^2}{2}}{}_1{F}_1\left(\frac{1}{4}-\frac{{ϵ}_n}{2},\frac{1}{2},x^2\right),{\psi }_o(x)=x{e}^{-\frac{x^2}{2}}{}_1{F}_1\left(\frac{3}{4}-\frac{{ϵ}_n}{2},\frac{3}{2},x^2\right).$$

$${}_1{F}_1\left(\frac{1}{4}-\frac{{ϵ}_n}{2},\frac{1}{2},x_c^2\right)=0,{}_1{F}_1\left(\frac{3}{4}-\frac{{ϵ}_n}{2},\frac{3}{2},x_c^2\right)=0$$

$$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}k(x-d_m{)}^2\psi +V\theta (x^2-x_c^2)\psi =ϵ\psi ,$$

$$\displaystyle\epsilon\equiv\epsilon\bigg{(}\frac{\hbar^{2}}{m},k,d_{m},x_{c}\bigg{)}\quad\mathrm{and}\quad\psi\equiv\psi\bigg{(}\frac{\hbar^{2}}{m},k,d_{m},x_{c},x\bigg{)}.$$

$$-\frac{1}{2}\frac{d^2\psi (x^{\prime })}{d{x}^{\prime 2}}+\frac{1}{2}\frac{m}{{\hslash }^2}k{x}_c^4{\left(x^{\prime }-\frac{d_m}{\lambda }\right)}^2\psi (x^{\prime })+V\theta (x^{\prime 2}-1)\psi (x^{\prime })=\frac{m{x}_c^2}{{\hslash }^2}ϵ\psi (x^{\prime })$$

Full text

PACS Numbers: 03.65-w, 03.65Ca, 03.65Ta, 03.65.Ge, 03.67-a, 89.70.Cf

Keywords: Confined harmonic oscillator, asymmetric confinement, Shannon entropy, Fisher information, Onicescu energy.

I Introduction

In recent years, instancy in studying spatially confined quantum systems has increased momentously. In such small spatial dimensions, they exhibit many fascinating physical and chemical phenomena, in contrast to their corresponding free unconfined counterparts. This occurs mostly due to their complex energy spectra. These have potential applications in a wide range of problems, e.g., cell-model of liquid state, high-pressure physics, study of impurities in semiconductor materials, matrix isolated molecules, endohedral complexes of fullerenes, zeolites cages, helium droplets, nanobubbles, etc. Recent progress in nanotechnology has also inspired extensive research activity to explore and understand confined quantum systems (on a scale comparable to their de Broglie wave length). Their unique properties have been realized in a large array of quantum systems such as quantum wells, quantum wires, quantum dots as well as nanosized circuits (as in a quantum computer), and so forth, employing a wide variety of confining potentials. The literature is quite vast; interested reader is referred to a special issue and a book sabin09 ; sen14 , and references therein.

A significant amount of confinement work exists for model systems such as particle in a box (PIB), harmonic oscillator, as well as real systems like H, He and other atoms; H${}_{2}^{+}$, H2 and other molecules. At this point, it may be worthwhile to mention a few theoretical methods vawter68 ; vawter73 ; navarro80 ; fernandez81 ; arteca83 ; taseli93 ; vargas96 ; sinha99 ; aquino01 ; campoy02 ; montgomery07 ; montgomery10 ; roy15 employed for a 1D quantum harmonic oscillator (QHO) enclosed inside an impenetrable box, which is the subject matter of our current work. Some prominent ones are: a semi-classical WKB approximation vawter68 ; sinha99 , a series analytical solution vawter73 , perturbative, asymptotic and Padé approximant solutions navarro80 , diagonal hypervirial fernandez81 , hypervirial perturbative arteca83 , Rayleigh-Ritz variation method with trigonometric basis functions taseli93 , numerical method vargas96 , perturbation method aquino01 , power-series expansion campoy02 , exact montgomery07 ; montgomery10 as well as highly accurate power-series solution montgomery10 , imaginary-time propagation (ITP) technique roy15 , etc. While most of the works deal with effect of boundary on energy levels, several other important properties such as dipole moment aquino01 , Einstein’s A, B coefficients aquino01 ; campoy02 , magnetic effects li1991 ; buchholz2006 , effect of confinement size on non-classical statistical properties of coherent states harouni2008 under high pressure, etc., were also considered. Most of these works focus largely on symmetrically confined harmonic oscillator (SCHO); while the asymmetrically confined harmonic oscillator (ACHO) situation is treated only in a few occasions (such as campoy02 ; roy15 ).

It is well-known that, information entropy (IE)-based uncertainty measures, such as, $S_{x}+S_{p}\geq 1+\ln\pi$, with $S_{x}$, $S_{p}$ denoting Shannon entropies in position ($x$) and momentum ($p$) space, provide more rigorous, stronger bound than conventional uncertainty product, $\Delta x\Delta p\geq\frac{\hbar}{2}$ (symbols have usual significance). In quantum mechanics, $x$ and $p$ spaces are connected through uncertainty relation. Localization in $x$ space leads to delocalization in $p$ space and vice versa. In case of IE, measurements are carried out in both $x$ and $p$ space. The most striking point in this respect, however, is that extent of localization is not exactly same as delocalization in $p$ space and vice versa. Thus an inspection of IE in composite space gives a more complete information of net localization-delocalization in a quantum system.

In the last decade, much light has been shed on the topic of IE in a multitude of systems. This continues to go unabated, as evidenced by a large volume of literature available on this topic. For example, information measures, especially, Shannon entropy ($S$) and Fisher information ($I$) in a decent number of physically, chemically important potentials have been reported lately. Some notable ones are: Pöschl-Teller sun2013quantum , Rosen-Morse sun2013quantum1 , squared tangent well dong2014 , hyperbolic valencia2015 , infinite circular well song2015 , hyperbolic double-well (DW) sun2015 potential, position-dependent mass Schrödinger systems (falaye16, ; serrano16, ; hua15, ) etc. In a recent publication (mukherjee15, ), two of the present authors employed entropy measures like $S$, $I$, Onicescu energy ($E$) and Onicescu-Shannon information ($OS$) to analyze competing effect of localization-delocalization in a symmetric DW potential, represented by, $V(x)=\alpha x^{4}+\beta x^{2}+\frac{\beta^{2}}{4\alpha}$. It was found that quasi-degeneracy exists for certain values of parameters $\alpha,\beta$. Further, it was realized that, while traditional uncertainty relation and $I$ were unable to explain such dual effects, measures like $S$, $E$ were quite successful. Such measures in a 1D “Landau” system undergoing phase transition, have been studied recently song16 . In an analogous work mukherjee16 , oscillation of a particle between larger and smaller wells were followed through information analysis in an asymmetric DW potential, given by, $V(x)=\alpha x^{4}-\beta x^{2}+\gamma x$. In this case, it was possible to frame some simple rules to predict quasi-degeneracy occurring only for some characteristic values of parameters present in potential.

A vast majority of IE-related works, referred above or otherwise, deal with a respective free or unconfined system. Such reports on quantum confined systems have been rather scarce and hence, it would be highly desirable to inspect these quantities in an enclosed system as they may open up some new windows to explore. In a recent publication laguna2014 , $S$ and traditional uncertainty relations were calculated and contrasted (significant differences were found in their behavior) in a SCHO model in $x$, $p$ and phase space. In this follow-up, we wish to extend our previous IE analysis to two celebrated model confined quantum systems, viz., SCHO and ACHO inside an impenetrable 1D box. Moreover, we have transformed our original Hamiltonian into a dimensionless form (patil07, ) to make the results more universal and appealing from the perspectives of experimentalists (zawadzki87, ; buttiker88, ). As it turns out, this leads to a dimensionless parameter ($\eta=\frac{mkx_{c}^{4}}{\hbar^{2}}$), which depends on the product of force constant and forth power of confinement length; this is derived later in Sec. III. Clearly, the variation of $x_{c}$ predominates over $k$. Thus, at first, we examine variation of energy as well as $S,I$, $E$ in a SCHO for small, intermediate, large regions of $\eta$ and $x_{c}$, in both $x$ and $p$ space. Next we monitor analogous changes in an ACHO, by varying $\eta$ and shifting the potential minimum ($d_{m}$), while keeping the box length stationary. As mentioned earlier, the only IE analysis for a CHO, to the best of our knowledge, is $S$ in case of SCHO. No such attempts are known for $I$, $E$, however. And so far, no work has been reported for analogous entropic analysis in an ACHO. So the present communication aims to provide a more satisfactory picture of localization of bound stationary states of an enclosed QHO in $x$ space and delocalization of same in $p$ space, and vice versa. Additionally, classical turning points are calculated for ACHO as functions of $d_{m}$ invoking the well-known semi-classical concept $|V(x)-\epsilon_{n}(d_{m})|=0$ griffiths2004 , which can throw some light on the localization of particle in $x$ space.

In order to facilitate further understanding, companion calculations are performed for phase-space area ($A_{n}$), defined as,

[TABLE]

as a function of $x_{c}$ in SCHO and $d_{m}$ in case of an ACHO. While IE gives a purely quantum mechanical viewpoint, this presents a semi-classical picture correlating IE and phase-space results. This may enable us to explain how $x_{c}$ and $d_{m}$ impact tunneling and shape (nature) of phase space in such potentials.

In both cases, accurate eigenvalues and eigenfunctions of ground and excited states are calculated by employing an imaginary-time evolution method roy02 ; roy02a ; roy05 ; roy14 ; roy15 . It has been found to be quite successful for a variety of problems, including the confinement situation. The paper is organized as follows. In section II, a brief outline of ITP method and its implementation is presented; Section III offers a detailed discussion on IE for a boxed-in SCHO and ACHO in 1D. Finally Section V makes a few concluding remarks and future prospect.

II Method of calculation

This section provides a brief account of the ITP method, which is applied here to obtain eigenvalues and eigenfunctions of a caged-in quantum system inside an 1D hard, impenetrable box. It involves a transformation of respective time-dependent Schrödinger equation in imaginary time, to a non-linear diffusion equation. The latter is then solved numerically in conjunction with minimization of an expectation value to reach the lowest-energy state. By maintaining the orthogonality requirement with all lower states with same space and spin symmetry, higher states could be generated sequentially. Since the method has been discussed at length earlier, here we summarize only essential details. For various other features, the reader may consult references dey99 ; roy99 ; roy02 ; roy02a ; gupta02 ; wadehra03 ; roy05 ; roy14 ; roy15 and therein.

The method is in principle, exact and was originally proposed several decade ago. It was first utilized in the context of a few physical, chemical problems, e.g., random-walk simulation for ab initio Schrödinger equation for electronic systems, like H ${}^{2}P$, H${}_{3}^{+}$ (D3h) ${}^{1}A_{1}$, H2 ${}^{3}\Sigma_{u}^{+}$, H4 ${}^{1}\Sigma_{g}^{+}$, Be ${}^{1}S$ anderson75 ; garmer87 , etc. Later, it was invoked for Morse potential, Hénon-Heiles system and weakly bound states of He on a Pt surface, by representing the Hamiltonian in grid through a relaxation method kosloff86 . Thereafter, it was applied in direct calculation of ground-state densities and other properties of rare gas atoms, molecules (H2, HeH+, He${}_{2}^{++}$) through a time-dependent quantum fluid dynamical density functional theory dey99 ; roy99 ; roy02a , ground, excited states of various 1D anharmonic DW roy02 , multiple-well gupta02 , self-interacting wadehra03 , 2D DW potentials roy05 , etc. Also a finite difference time domain approach was proposed for numerical solution of diffusion equation, which was applied to infinite square potential, harmonic oscillators in 1D, 2D, 3D; H atom, a charged particle in magnetic field, etc., sudiarta07 ; sudiarta08 ; sudiarta09 . It was also successfully used in discretization of linear and non-linear Schrödinger equation by means of split-operator method lehtovaara2007 , large-scale 2D eigenvalue problems in presence of magnetic field luukko2013 , and in several other situations aichinger2005 ; chin2009 ; strickland2010 .

Without any loss of generality, the non-relativistic time-dependent Schrödinger equation for a particle trapped inside an 1D impenetrable box (atomic unit employed unless mentioned otherwise) can be written as,

[TABLE]

All the terms have usual significance. In present work, we mainly focus on case of a harmonic oscillator, $v(x)=\frac{1}{2}k(x-d_{m})^{2}$ (force constant $k$ being kept fixed at unity throughout). For a SCHO, $d_{m}=0$, whereas for an ACHO, $d_{m}\neq 0$. Our desired confinement is achieved by enclosing the system inside two infinitely high hard walls:

[TABLE]

Figure (1), panel (a) illustrates a SCHO scenario for two box lengths 1, 2 respectively. It is well-known that, SCHO can be understood as an intermediate between a PIB and a QHO gueorguiev06 . Next, we note that asymmetric confinement in a QHO can be accomplished in two ways: (i) by changing the box boundary, keeping box length and $d_{m}$ fixed at zero (ii) the other way is to adjust $d_{m}$ by keeping length and boundary fixed. We have adopted the second option; where a rise in $d_{m}$ shifts the minimum towards right of origin, keeping box length, $b_{2}-b_{1}=2$ constant, while left, right boundaries are located at $b_{1}=-1$ and $b_{2}=1$. Furthermore, since $\frac{1}{2}k(x\pm d)^{2}$ represents a mirror-image pair potential confined within $-b_{1}$ and $b_{2}$, their eigenspectra, expectation values are same for all states. As respective wave functions are mirror images of each other, in this occasion, it suffices to consider the behavior of any one of them; other one automatically follows from it. Right panel (b) of Fig. (1) shows a schematic representation of an ACHO potential, at five specific $d_{m}$ values.

One can introduce a Wick rotational transformation from real time to imaginary time ($\tau=it$, where $t$ denotes real time), to write Eq. (2) in a form given below,

[TABLE]

whose formal solution can be written as below,

[TABLE]

Thus, taking $\epsilon_{0}<\epsilon_{1}<\epsilon_{2}<\cdots$, for large imaginary time, wave function $\psi(x,\tau)$ will contain the lowest-energy (ground) state as dominating, i.e.,

[TABLE]

Hence if a initial trial wave function $\psi(x,\tau)$ at $\tau=0$ is evolved for sufficiently long $\tau$, one can reach the desired lowest-state energy. In other words, provided $c_{0}\neq 0$, apart from a normalization constant, the global minimum corresponding to an expectation value $\langle\psi(x,\tau)|H|\psi(x,\tau)\rangle$ could be attained.

For numerical solution of Eq. (4), one needs to follow the time propagation of $\psi(x,\tau)$; this is achieved by invoking a Taylor series expansion of $\psi(x,\tau+\Delta\tau)$ around time $\tau$,

[TABLE]

This gives a prescription to advance the diffusion function $\psi(x,\tau)$ at an initial time $\tau$ to a future function $\psi(x,\tau+\Delta\tau)$ at time $\tau+\Delta\tau$. This is accomplished through the right-hand side evolution operator $e^{-\Delta\tau H}$, which is non-unitary and thus normalization of $\psi(x,\tau)$ at a given instant does not necessarily preserve the same at a forward time $\tau+\Delta\tau$. At this stage, it is convenient to write above equation in to an equivalent, symmetrical form,

[TABLE]

where $j,l$ signify space and time indices, whereas a prime indicates an unnormalized diffusion function. Now one can (i) expand the exponentials in both sides (ii) truncate them after second term (ii) make use of five-point finite-difference formula for spatial derivative, to obtain a set of $N$ simultaneous equations,

[TABLE]

Quantities $\alpha_{j},\beta_{j},\gamma_{j},\delta_{j},\zeta_{j},\xi_{j}^{l}$ can be derived from straightforward algebra roy15 ; so not repeated here. Primes in left-hand side signify unnormalized diffusion function at $(l+1)$th time. There may be some cancellation of errors as discretization and truncation occurs in both sides. Equation (9) can be recast in to a pentadiagonal matrix eigenvalue problem,

[TABLE]

which can be readily solved by standard routines to obtain $\{\psi^{\prime(l+1)}\}$; this work uses an algorithm provided in pentaroutine . Thus, an initial trial function $\psi^{0}_{j}$ at zeroth time step is guessed to launch the calculation. Then the desired propagation is completed via a series of instructions through (as detailed in roy15 ) Eq. (7) and satisfying boundary condition that $\psi_{1}^{l}=\psi_{N}^{l}=0$ for all $l$. Further at any given time step also, another sequence of operations need to be carried out, viz., (a) normalization of $\psi^{\prime(l+1)}$ to $\psi^{(l+1)}$ (b) maintaining orthogonality requirement with all lower states (c) computation of relevant expectation value, such as, $\epsilon_{(l+1)}=\langle\psi^{(l+1)}|\hat{H}|\psi^{(l+1)}\rangle$, etc. Initial trial functions are chosen as $H_{m}(x)e^{-x^{2}}$ for $m$-th state. In principle, one can choose arbitrary functions; however a good guess reduces computational time and reaches convergence in lesser propagation steps. For more details, see roy15 .

The $p$-space wave functions are obtained from Fourier transform of their $x$-space counterparts; this is accomplished through standard, available routines fftw3 ,

[TABLE]

In the current work, we deal with three information measures. First of them is Shannon entropy (shannon1951, ), that signifies the expectation value of logarithmic probability density function. In $x$ and $p$ space, this is given by,

[TABLE]

Total Shannon entropy ($S$), defined below, obeys the bound given below,

[TABLE]

Second one is Fisher information (cover2006, ), which in $x$ and $p$ space, reads as,

[TABLE]

Net Fisher information, $I$, given as product of $I_{x}$ and $I_{p}$, satisfies the following bound,

[TABLE]

The last one, Onicescu energy (chatzisavvas2005, ; alipour2012, ; agop2015, ) is a rather recent development. This is expressed in $x$ and $p$ space as,

[TABLE]

The corresponding total quantity, $E$ is defined as,

[TABLE]

III Results and Discussion

III.1 Symmetric confinement

The original SCHO Hamiltonian can be represented as,

[TABLE]

Here, $\theta(x^{2}-x_{c}^{2})$ is a Heaviside theta function and $V$ is a constant, having very large value. The effect of localization and delocalization depends on $x_{c}$ and $k$. It has been observed (zawadzki87, ; buttiker88, ) that the above Hamiltonian can be generalized into a dimensionless form, so that one can correlate experimental observation with theoretical results. Further, it is established (zawadzki87, ) that $k$ is proportional to square root of magnetic field parallel to the gradient of confining potential. Now, it seems more appropriate to study the composite effect of $x_{c}$ and $k$ with the aid of a single dimensionless parameter. This will make our current observation more interesting and appealing from an experimental viewpoint. It follows that,

[TABLE]

After substitution of $x=\lambda x^{\prime}$ and $x_{c}=\lambda$, into Eq. (18), the modified dimensionless Hamiltonian can be written as,

[TABLE]

where $x^{\prime}$ is a dimensionless variable. The above conversion leads to,

[TABLE]

Equation (20) implies that $\eta$ depends on product of $k$, $m$ and quartic power of $x_{c}$. However, if we choose $m=\hbar=1$, then effective dependence remains on product of $x_{c}^{4}$ and $k$.

It is well-known that SCHO represents an intermediate situation between a PIB and QHO model. The two limiting values of smaller ($\ll 1$) and larger ($\gg 1$) $\eta$, would correspond to these two model systems respectively. So, at small-$\eta$ region, actually one can use Rayleigh-Schrödinger perturbation theory to get an approximate analytical expression for energy. The Hamiltonian is constructed by choosing the PIB as unperturbed system, and quadratic potential ($\frac{1}{2}\eta{x^{\prime}}^{2}$) as perturbed term. After some straightforward algebra and considering upto only first-order correction, the energy expression becomes,

[TABLE]

where $\epsilon_{n+1}(0)$ denotes zeroth-order PIB energy spectrum. Table I reports estimated energy values for $n=0-4$, in three small representative $\eta$. Column labeled PT gives results corresponding to perturbation expression, Eq. (21), while “Exact” results refer to those obtained from the “exact” analytical SCHO wave functions montgomery07 . As expected, PT energies deviate from actual values more and more as $\eta$ advances. Of late, an analogous study has been made by olendski15 for a particle confined in a well under the influence of an electric field.

Since wave function, energy, position expectation values in a SCHO were presented earlier (see, e.g., navarro80 ; campoy02 ; montgomery10 ; roy15 ) in some detail, we do not discuss them in this work. Further, a thorough discussion on accuracy, convergence of ITP eigenvalues, eigenfunctions with respect to grid parameters, in context of confinement, has been published in roy15 ; hence omitted here to avoid repetition. Thus, our primary focus is on information analysis. For that, at first we explore the influence of $\eta$, followed by the effect of $x_{c}$ on IE. At the outset, it may be mentioned that, for symmetric confinement, some information theoretic ($S$ only) measures in $x$, $p$ and Wigner phase-space have been published laguna2014 . However, to the best of our knowledge, no such attempt is known for $I$, $E$ for a SCHO. In this sense, present work takes some inspiration from laguna2014 and extends it further.

Let us consider the variation of IE as a function of $n$ (state index) for two special cases: (a) $\eta=0$ which corresponds to a PIB model, (b) $\eta\rightarrow\infty$ leads to a QHO. IE measures in $x$-space, for a PIB are as follows,

[TABLE]

One notices that $S_{x}^{n}$ and $E_{x}^{n}$ remain stationary with $n$, but $I_{x}^{n}$ varies as $n^{2}$. More information about $S$ can be found in (majernik97, ; sanchez97, ; laguna2014, ; olendski15, ). The IE measures for QHO in $x,p$ space, on the other hand, were discussed in a number of works yanez94 ; majernik96 including a recent one from our laboratory (mukherjee15, ). It was found that $I_{x}^{n}$ changes with $\eta$ (linearly) and $n$ ($I_{x}=4(2n+1)\eta$), whereas $I_{p}^{n}$ varies inversely with $\eta$ and increases with $n$ ($I_{p}=\frac{2n+1}{\eta}$). Further, $S_{x}^{n}$ decreases with increase in $\eta$ but increases with $n$, whereas $S_{p}^{n}$ increases with both $\eta$ and $n$. And $E_{x}^{n}$, $E_{p}^{n}$ show opposite dependence of $S_{x}^{n}$, $S_{p}^{n}$ with both $\eta$ and $n$. Tables II-IV illustrate these behaviors of $S,I,E$ respectively, for arbitrary $\eta$, which also includes the limiting case of $\eta\rightarrow 0$. Note that, now onwards (both in SCHO and ACHO), all information measures are suffixed with a a prime to denote the respective quantities for an arbitrary $\eta$, whereas unprimed variables are used for extreme cases (like PIB or QHO). All these calculations were performed by taking exact analytical wave functions of SCHO montgomery07 (see, Eqs. (25) and (26) later). A detailed analysis of IE from these tables confirms a PIB behavior at small $\eta$. But it is not so obvious to conclude any kind of QHO behavior at large $\eta$ region. This requires a much larger $\eta$ than considered here and not approached; rather it is straightforward to demonstrate this from an analysis of $x_{c}$ (Figs. 3, 4, as well as Table VI later illustrate this).

In order to probe further, at small $\eta$ limit, a perturbation treatment as in energy, can be employed. Considering up to first order, the perturbed wave functions can be expressed as,

[TABLE]

Here, we only retain only two consecutive corrected terms in the summation and $V_{mn}^{\eta}=\langle\psi_{m}^{(0)}|\frac{1}{2}\eta x^{\prime 2}|\psi_{n}^{(0)}\rangle$. After some tedious algebraic manipulation, one can write the following expressions for $I_{x^{\prime}}$ and $E_{x^{\prime}}$,

[TABLE]

where $a_{0}=5832,b_{0}=29837,c_{0}=3293;a_{1}=2985984,b_{1}=4253353,c_{1}=1055137;a_{2}=1024,b_{2}=689,c_{2}=801,a_{3}=68024448,a_{4}=144190368,a_{5}=7085880,a_{6}=21851723$.

Table V gives a cross section of $S,I$ and $E$ in $x$ space for three low-lying states at three specific $\eta$. The four quantities, given in Eq. (24) are estimated from the perturbation expressions, whereas all others including Shannon entropies are numerically evaluated by taking approximate wave function of Eq. (23). For sake of easy comparison, we also quote respective exact results from Tables II-IV. The two results show quite good agreement in small-$\eta$ region and disagreements tend to set in as $\eta$ assumes larger values.

The general variation of IE with respect to $\eta$ is now presented graphically. For this, all three information measures $I,S,E$ were computed for $n=0-4$ states. Moderate ranges of $\eta$ were considered in all cases. From this set, here we present in Fig. (2), only $S_{x^{\prime}},S_{p^{\prime}}$ and $S$ in three lowest states. Higher states as well as $I,E$ plots are given in supplementary material. In all cases, a gain in $\eta$ leads to a consistent decrease in $S_{x^{\prime}}$, which clearly indicates the localization of particle in $x$ space. On the other hand, $S_{p^{\prime}}$ shows a reverse trend with rise in $\eta$, signifying a de-localization of particle in $p$ space. Panel c(i) of Fig. (2) shows that, $S$ for $n=0$ decreases with $\eta$ steadily, whereas, from the remaining two rightmost panels, behavior in excited states remains somehow less clear cut, because of the composite behavior of $S_{x^{\prime}}$, $S_{p^{\prime}}$. Interestingly, in $S$ plots, $n=1$ possesses a minimum and $n=3,4$ states in Fig. (S1) tend to show a steady progress throughout. Similar plots of $I$ and $E$, covering same range of $\eta$, are produced in Figs. (S2) and (S3) respectively, for $n=0-4$ states. In $n=0-2$, $I_{x^{\prime}}$ increases with $\eta$, whereas for $n>2$, it decreases. One sees that $I_{p^{\prime}}$ decreases with $\eta$ in first two states, and then at $n=2$, there appears a dome-shaped structure with a maximum. Thereafter for $n>2$, it completely reverses the trend. For all states, $E_{x^{\prime}}$ increases monotonically while $E_{p^{\prime}}$ decreases monotonically, with $\eta$. And finally, except for $n=0$, $E$ decreases with $\eta$. Hence, it can be concluded that $S$ and $E$ could be possibly used to explain the localization and de-localization behavior in a SCHO.

Up to now, we were concerned with $\eta$ variations. But since from Eq. (20), $\eta\propto kx_{c}^{4}$, these results include combined effects from both $k$ and $x_{c}$. In order to get a clear picture of confinement, these two effects need to be separated. This motivates our forthcoming analysis with respect to $x_{c}$ on IE, keeping $k=1$. For all $x_{c}$ variation throughout the article, unprimed variables are used in IE suffixes. At first Table VI presents sample results for $S_{x}$, $E_{x}$ for lowest five energy states of SCHO potential at three chosen $x_{c}$, viz., 0.25, 2 and 5 covering small, intermediate and large regions. Now onwards, all IE results on SCHO were performed following the ITP method as prescribed in Sec. (II). Several test calculations were done on various number of grid points $N$, to check convergence. Generally, quality of results improves as $N$ increases and after some value, these remain practically unchanged. Similar conclusions also hold for $I_{x}$ and hence not repeated. While one can think of ways to improve such results further by employing higher precision wave function on finer grid, present results are sufficiently accurate for the purpose at hand. Thus there is no necessity to pursue such accurate calculations, as we are primarily interested in the qualitative trend of IE. For each of these quantities we also report the respective reference results, obtained by taking exact analytical eigenfunctions montgomery07 .

For even and odd states, these are given as:

[TABLE]

Here, $\epsilon_{n}$ and ${}_{1}F_{1}\left(a,b,x\right)$ denote eigenvalues and Kummer confluent hypergeometric function. Energies are computed by putting $x=x_{c}$ and numerically solving the following equations:

[TABLE]

for even and odd case. In all cases, we notice that present results are practically coincident with those from reference. Similar conclusions hold for $I_{x}$.

Once the authenticity of ITP method is established, now we proceed for a detailed analysis on information measures with respect to $x_{c}$. First, Fig. (3) shows plots of $S_{x}$, $S_{p}$ versus $x_{c}$, for five low-lying states of SCHO. In panel (a), $S_{x}$ increases with box length (indicating delocalization of particle) and converges to a constant value ($S_{x}$ of QHO) at sufficiently large box length. At small $x_{c}$ region, $S_{x}$ changes insignificantly with state index $n$, resembling the behavior of a PIB problem (where $S_{x}$ remains stationary with $n$). Next, panel (b), suggests that, $S_{p}$ decreases with increase in box length and finally merges to corresponding QHO value. It is interesting to note that, there appears a minimum (which becomes progressively more prominent as $n$ increases) in $S_{p}$ for all excited states. Appearance of such minimum may be attributed to the competing effect in $p$ space. Actually, three possibilities could be envisaged ($l_{x}$, $l_{p}$ are box lengths in $x$, $p$ space):

1. (a)

When $l_{x}\rightarrow 0$ then $l_{p}\rightarrow\infty$. 2. (b)

When $l_{x}$ finite then $l_{p}$ is also finite. But an increase in $l_{x}$ leads to a decrease in $l_{p}$. 3. (c)

When $l_{x}\rightarrow\infty$ then also $l_{p}\rightarrow\infty$.

From above plots it is clear that, initially with increase in $l_{x}$, particle gets localized in $p$ space ($S_{p}$ decreases), but when potential starts behaving like QHO, de-localization occurs. Thus, existence of minimum in $S_{p}$ is due to balance of two conjugate forces. These variations of $S_{x}$, $S_{p}$, total $S$ (not presented here) with $x_{c}$ are in harmony with the findings of laguna2014 .

Next we move on to $I$ and $E$, in a SCHO potential, which have not been studied before. For this, first we analyze $x$ space localization, then $p$ space de-localization and finally composite space localization-de-localization by computing $I_{x},~{}E_{x}$; $I_{p},~{}E_{p}$; $I,~{}E$. These calculation will consolidate the conclusions obtained from the study of $S_{x}$, $S_{p}$, $S$ delineated earlier. Behavior of $I_{x}$, $I_{p}$ of first five stationary-states are demonstrated in two bottom panels (a), (b) of Fig. (4). One sees that initially $I_{x}$ falls sharply, extent of which is maximum for lowest state and progressively decreases with $n$. Then it assumes a constant value beyond a certain $x_{c}$. All the states remain well separated; no mixing occurs amongst them. On the other hand, in panel (b), $I_{p}$ strongly increases with $x_{c}$ at first; the extent increases with $n$; lowest state producing lowest. Finally, for all states, it again reaches a state-dependent constant value as in $I_{x}$. It is relevant to mention that, behavior of $I_{x}$, $I_{p}$ under confinement matches with those of $\Delta p$, $\Delta x$ laguna2014 . Next, from panel (c), we discern that, $E_{x}$ for all states remain very close to each other at smaller $x_{c}$; then as $x_{c}$ increases, $E_{x}$ for individual states branch out and decreases indicating localization. At the end, it assumes some constant value for all $n$. Interestingly, up to about $x_{c}\approx 1$, $E_{x}^{n}$ does not depend on $n$ quantum number. Lastly in (d), $E_{p}$s tend to increase in the beginning for all states at low $x_{c}$ region (indicating localization), eventually becoming flat after some threshold $x_{c}$.

Now, results on $I$, $E$ are given in left and right columns of Fig. (5) and Fig. (S4). Total $I$, for three states having to $n=0-2$ and remaining two excited states corresponding to $n=3,4$ are depicted, from bottom to top in a(i)–a(iii) of Fig. (5) and a(i)–a(ii) of Fig. (S4) respectively. For $n=2,3,4$, one notices maximum in $I$ plots, signifying a competition in localization-delocalization. Large $I$ values in small $x_{c}$ region correspond to PIB model, while smaller values in large $x_{c}$ signify those for a free QHO. Middle region of these plots indicates effect of confinement, implying that SCHO may be viewed as an intermediate between these two limiting cases. Further, it is verified that behavior of $I$ resembles closely that of $\Delta x\Delta p$ for a SCHO laguna2014 . For highest states, however, we notice a reversal in these limiting $I$ values. Right-hand side panels b(i)–b(iii) of Fig. (5) and b(i)–b(ii) of Fig. (S4) portray $E$ plots for same five states, which in all cases, consistently decrease with increase in $x_{c}$.

In order to get further insight, next we present phase space for same lowest five states of SCHO potential for three distinct sets of $x_{c}$ (not necessarily same for all $n$). These are displayed in Fig. (6) ($n=0-2$) and Fig. (S5) ($n=3,4$). For $n=0$, phase space is examined at $x_{c}=1.25,1.5,2$ in panels a(i)–a(iii) of Fig. (6). At the first $x_{c}$, it indicates a PIB-like behavior; at remaining two $x_{c}$, however, its shape changes. Appearance of a line in phase space in a(i) of Fig. (6) (corresponding to $x_{c}=1.25$) indicates the onset of tunneling. Similarly, the first excited state phase spaces are plotted for $x_{c}=1.9,2$ and 2.5 respectively in segments b(i)–b(iii) of Fig. (6). In this occasion too, shape of phase space changes with increase in $x_{c}$. From b(i), we discern that, tunneling begins approximately at $x_{c}=1.9$. Analogous conclusions can be drawn from phase spaces of $n=2-4$, where tunneling starts at $x_{c}\approx 2.25,2.75\ \mathrm{and}\ 3$. Actually, this study only exhibits the PIB behavior in small $x_{c}$ region, but somehow insufficient to explain the large $x_{c}$ nature of SCHO potential.

Additional information on confinement could be gathered from a semi-classical interpretation. For this we computed phase-space area $A_{n}$ (left side) as well as probability $T_{n}$ (right side) of finding the particle outside classical region for $n=0-4$; these are depicted in panels a(i)–a(iii), b(i)–b(iii) of Fig. (7) for $n=0-2$, and in a(i)–a(ii), b(i)–b(ii) of Fig. (S6) for $n=3,4$. For all $n$, we observe a swift jump of $A_{n}$ from a higher to lower value at a certain characteristic $x_{c}$, marking the onset of tunneling, as encountered earlier. For $n=0-4$, this occurs roughly when $x_{c}$ falls in the range of 1.25–1.5, 1.75–2, 2.25–2, 2.75–3 and 3–3.25. Here tunneling begins at a point when particle starts sensing the presence of harmonic potential. Another fact is that $A_{n}$ values converge to 1.11, 3.33, 5.55, 7.77 and 9.99 for $n=0-4$. Appearance of such convergence in $A_{n}$ implies that at large $x_{c}$, SCHO potential behaves much like a QHO potential. Next, we present $T_{n}$ for same lowest five states in right panels b(i)–b(iii) of Fig. (7) ($n=0-2$) and b(i)–b(ii) of Fig. (S6) ($n=3,4$). For all $n$, $T_{n}$ increases steadily reaching a particular value characteristic for that given state. These values for $n=0-4$ are roughly 0.16, 0.11, 0.095, 0.085 and 0.078. This again explains that at large $x_{c}$, SCHO potential behaves like a QHO system. Here $T_{n}$ decreases with increase in $n$, as $A_{n}$ increases with $n$. In summary, small $x_{c}$ region can be understood by the nature of phase-space, whereas large $x_{c}$ domain can be interpreted in terms of $A_{n}$ and $T_{n}$.

III.2 Asymmetric confinement

Now, we move on to a discussion on ACHO case. At first we make the ACHO Hamiltonian into dimensionless form, represented as

[TABLE]

From this equation, one reads that,

[TABLE]

As in the SCHO case, again incorporation of $x=\lambda x^{\prime}$ and $x_{c}=\lambda$ into Eq. (27), leads to a modified dimensionless Hamiltonian, expressed as,

[TABLE]

where $x^{\prime}$ is a dimensionless variable. This gives the following set of equations,

[TABLE]

So far, no IE analysis has been done as yet for this system. Our current presentation is based on results of $\epsilon_{n}$; $S_{x},S_{p},S$; $I_{x},I_{p},I$; $E_{x},E_{p},E$ as functions of $d_{m}$ and $\eta$, for some low-lying states. For all forthcoming calculations box length has been kept constant at 2 and boundaries are placed at $-$1 and 1. At first, let us present the variation of classical turning point ($L_{cl}$) with $d_{m}$, in Fig. (S7) for first six $n$ values. It is easily concluded that, an increase in $d_{m}$ leads to localization in $x$ space. Thus it is expected that, there will be delocalization in $p$ space. Note that, ACHO reduces to SCHO, if $d_{m}=0$.

Before proceeding for a discussion on IE in ACHO, it is appropriate to make some comments regarding its eigenvalues and eigenfunctions. Good-quality results for ACHO have been rather scarce; two notable ones are power-series solution campoy02 and ITP method roy15 . In this section, we take this opportunity to present another simple method for ACHO potential, which, as our future discussion confirms, produces quite accurate results. Although it is aside the main objective of this work, nevertheless quite relevant and worthwhile mentioning. In this so-called variation induced exact diagonalization procedure griffiths2004 , an energy functional is minimized using a SCHO basis set. Recently this method has been used successfully in symmetric and asymmetric double-well potentials mukherjee15 ; mukherjee16 , where a QHO basis was utilized. This complete basis set corresponds to exact solution of SCHO potential, written as below ($\alpha=\sqrt{\frac{1}{8}}$; while e,o signify even, odd states):

[TABLE]

The integrals involved in Hamiltonian matrix are easy to evaluate by using these functions. Presence of a single non-linear parameter $\alpha$ permits us to adopt a coupled variation strategy, which is advantageous than purely linear variation. Further it is easy to ensure convergence of results with respect to basis dimension $N$. Thus it offers a secular equation at each $\alpha$. The kinetic energy part becomes infinitely large when $\alpha\rightarrow\infty$, whereas, on other extreme ($\alpha\rightarrow 0$), potential energy part behaves in a similar fashion. This qualitative analysis through uncertainty principle, ensures the existence of such basis. This fulfills one of the requirements of a satisfactory basis set and therefore justifies its adoption. Diagonalization of $H_{mn}=\langle m|\hat{H}|n\rangle$ leads to accurate eigenvalues and eigenfunctions, which is accomplished through MATHEMATICA package. In principle, to obtain exact solution, one needs to employ a complete basis; however for practical purposes a truncated basis of finite dimension is invoked. $N=50$ appears sufficient; with further increase in basis, result improves. A cross-section of energies generated from above scheme is produced in Table VII, for lowest six states of ACHO for four selected $d_{m}$. Best reported literature values are quoted for comparison, wherever available. For first two $d_{m}$, present energies with SCHO basis, practically coincide with ITP results roy15 , for all digits reported., while those from campoy02 show good agreement with ours. For last two $d_{m}$, no reference result could be found. Thus fresh ITP calculations were performed for them and cited, which, as expected, shows excellent agreement with present energies in all states. It is hoped that this SCHO basis may be useful in future for other confined and/or free potentials in 1D.

Our calculated ACHO energies of Table VII are portrayed in Fig. (8), for lowest six states with respect to $d_{m}$, in panels (a) through (f). Ranges of $\epsilon_{n}$ and $d_{m}$ vary for each $n$. Changes in $\epsilon_{0}$ is rather unique amongst all $\epsilon_{n}$, where, from an initial positive value at small $d_{m}$, ground-state energy continuously falls. In all excited states, however, it passes through a maximum; with increase in $n$, which shifts to higher values of $d_{m}$. Next Fig. (9) depicts our computed wave functions for $n\!=\!0-5$ states of ACHO at same four $d_{m}$. Clearly, the maximum, minimum and nodal positions of all $\psi_{n}$’s shift towards right with increase in $d_{m}$. Clearly, these are characterized by requisite number of nodes for a given $n$. These plots indicate that particle gets localized in $x$ space as $d_{m}$ increases.

Now we move on to $I_{x},I_{p}$ and $I$ variations with respect to $d_{m}$. These are shown graphically in Fig. (10) ($n=0-2$) and Fig. (S8) ($n=3,4$); left (a), middle (b) and right (c) panels display these in $x$, $p$ and composite space respectively. For $n\!=\!0-1$, $I_{x}$ tends to increase with $d_{m}$, on the whole, whereas for remaining states ($n\!\geq\!2$), the same decreases. Thus, for $n=0,1$, it suggests localization in $x$ space and delocalization for higher states. A careful observation of panels b(i)–b(iii) of Fig. (10) and b(i)–b(ii) of Fig. (S8) reveals that, $I_{p}$, for all $n$ under investigation, consistently decreases with increase of $d_{m}$, signifying a delocalization of particle in $p$ space. Further exploration of net $I$ in segments c(i)–c(iii) of Fig. (10) and c(i)–c(ii) of Fig. (S8) shows that, except for $n\!=\!0$, $I$ decreases as $d_{m}$ increases, indicating an overall delocalization in composite space. However, for $n\!=\!0$, there appears a minimum in $I$, which could be possibly due to a competition between localization-delocalization.

Next, it is necessary to consider changes in behavior of $S_{x},S_{p},S$ as function of $d_{m}$. These are offered in Fig. (11) ($n=0-2$) and Fig. (S9) ($n=3,4$), adopting an identical presentation strategy as in previous figure. It is clear from left panels a(i)–a(iii) of Fig. (11) and a(i)–a(ii) of Fig. (S9) that, $S_{x}$ for all five $n$ generally decreases monotonically as $d_{m}$ grows, signifying localization of particle in $x$ space. Likewise, a scrutiny of mid-region plots b(i)–b(iii) of Fig. (11) and b(i)–b(ii) of Fig. (S9) leads to conclusion that particle is delocalized for all $n$ in $p$ space, for $S_{p}$ increases with $d_{m}$. Finally, five net $S$ plots, c(i)–c(iii) of Fig. (11) and c(i)-c(ii) of Fig. (S9), in right-side column maintains the same trend of $S_{p}$ with $d_{m}$. This could imply that, there is net delocalization in composite $x$, $p$ space (since extent of delocalization in $p$ space is more than extent of localization in $x$ space). A sample of $S_{x},S_{p}$ and $S$ are reported in Table VIII for $n\!=\!0,1,2$ states at five particular $d_{m}$, i.e., 0.12, 2.04, 5, 7 and 10. Evidently, these entries corroborate the findings of Fig. (11). Now, we move on to an analysis of $E$, in Fig. (12) ($n=0-2$) and Fig. (S10) ($n=3,4$), adopting same presentation format of $I$ and $S$. For all $n$ in a(i)–a(iii) of Fig. (12) and a(i)–a(ii) of Fig. (S10), there is an overall increase of $E_{x}$ with $d_{m}$ leading to localization in $x$ space. Panels b(i)–b(iii) of Fig. (12) and b(i)–b(ii) of Fig. (S10) in middle shows that, general trend of $E_{p}$ is a gradual decline with an increase in $d_{m}$; thus it is able to account for delocalization in $p$ space. At last, c(i)–c(iii) of Fig. (12) and c(i)-c(ii) of Fig. (S10) clearly show that, for all $n$ except zero, $E$ decreases with increasing $d_{m}$, interpreting a net delocalization in combined $x,p$ space. Interestingly, in the ground state, while $E_{x},E_{p}$ maintain same pattern of other states, net measure $E$ for $n\!=\!0$ deviates from that for excited states.

On the basis of above discussion, it is clear that, only $S_{x},S_{p},S$ can satisfactorily explain localization-delocalization phenomena in an ACHO. Study of $E_{x},E_{p},E$ can also offer valuable insight in to the dual nature of $E$ in composite $x$, $p$ space, except for ground state. But, $I_{x},I_{p},I$ appear to be inadequate in explaining the contrasting phenomena in ACHO.

As done in SCHO case, before passing, we would like to mention a few words about the dependency of IE with dimensionless variable $\eta$. Table IX shows results for all three measures $I,S,E$, in both $x$ and $p$ space, in an ACHO potential, at three selected $\eta$, viz., 5, 10, 20. To get a more complete understanding, we consider lowest 5 states, varying $\eta$ from 1-20, which is sufficient for our purpose. Fig. (13) gives Shannon entropy for $n=0-2$ states, including $S_{x^{\prime}},S_{p^{\prime}}$ and $S$. The higher $n$ plots for $S$ are given in Fig. (S11); analogous plots for $I$ and $E$ are offered in Figs. (S12), (S13) respectively. In all these cases, $S_{x^{\prime}}$ decreases with increase of $\eta$; which clearly indicates localization of particle in $x$ space. It is interesting to note that at $\eta\rightarrow 0$ regime, $S_{x^{\prime}}$ becomes independent of $n$, because of its PIB-like behavior. Similarly, it can be seen that, $S_{p^{\prime}}$ increases with $\eta$ signifying de-localization of particle in $p$ space. The net $S$ for all $n$ increases with $\eta$. As usual, $I_{x^{\prime}}$ for $n=0-2$, increase with $\eta$, but for $n=3,4$ decrease. On the other side, $I_{p^{\prime}}$ decreases for all $n$. There appears a minimum in $n=0$ for $I$, due to a tug-of-war between $x$ and $p$-space quantities. In case of $n=1-4$, an increase in $\eta$ causes lowering of $I$. In all cases, $E_{x^{\prime}}$ increases with $\eta$, wheres $E_{p^{\prime}}$ decreases. But net $E$ decreases for $n=0,1$, and increases for remaining $n$.

Finally, Fig. (14) and Fig. (S14) offer a semi-classical viewpoint of ACHO potential from a phase-space analysis in terms of phase space characteristics, $A_{n}$ and $T_{n}$. Phase-space diagrams for $n\!=\!0-2$ are displayed in panels a(i)–a(iii); in each case, five representative $d_{m}$ is chosen. Shapes of phase space changes with $d_{m}$. Third and fourth states are given in Fig. (S14). Initially at small values of $d_{m}$, ACHO potential behaves as PIB. However, as $d_{m}$ increases, more localization tends to take place indicating deviation from a PIB situation. For $n\!=\!0$, chosen values of $d_{m}$ are 1.56, 2.52, 5, 10 and 12.5; in this case tunneling sets in at around $d_{m}\approx 0.9-1$. A similar analysis on other four states in a(ii)–a(iii) and Fig. (S14) (a(i)–a(ii)) advocates that, bounded area decreases with $d_{m}$; moreover phase spaces gradually tend to assume rectangular shape. Tunneling for $n\!=\!1-4$ occurs at around $d_{m}\approx 4.75,11.5,21$ and 33.5. From middle panels b(i)–b(iii) and Fig. (S14) (b(i)–b(ii)), we observe that $A_{n}$ decreases as $d_{m}$ increases. $A_{n}$ in $n\!=\!0$, jumps down from a higher to lower value when $d_{m}$ falls in the range 1.3–1.5. Equivalent jumps happen at approximate $d_{m}$ ranges of 3–5, 10–12.5, 20–22.5 and 32.5–35 respectively, for $n\!=\!1-4$. These characteristic $d_{m}$ values mark the beginning of tunneling in these states. Lastly, from $T_{n}$ plots in right-most panels, c(i)–c(iii) and Fig. (S14) (c(i)-c(ii)), it is seen that, tunneling begins at a state-dependent threshold value and then increases as $d_{m}$ progresses. But at larger $d_{m}$ region, rate of tunneling falls off steadily showing very insignificant changes in $T_{n}$ plots.

IV Conclusion

Information-based uncertainty measures like $I,S,E$ have been employed to understand the effect of confinement in SCHO and ACHO. SCHO may be treated as an interim model between PIB and a QHO, whereas, asymmetric confinement in ACHO may be probed starting from a SCHO ($d_{m}\!=\!0$). Accurate eigenvalues and eigenfunctions are obtained from ITP method in both cases. For ACHO model, additionally, we also introduced a simple variation method utilizing the SCHO basis set, which offers excellent results. Nature of changes of $I_{x},I_{p},I$ complement the behavior of traditional uncertainties, $\Delta x,\Delta p,\Delta x\Delta p$.

In case of ACHO, an increase in $d_{m}$ value leads to localization in $x$ space and delocalization in $p$ space. While $S_{x},S_{p},S$ can completely explain the competing phenomena in ACHO, in contrast, $I_{x},I_{p},I$ proves insufficient for such situation. Further, $E_{x},E_{p},E$ offer important information which helps us for a better understanding of such systems. Besides, from a semi-classical perspective, it is found that, $A_{n}$ decreases with $x_{c}$ in SCHO and maintains the same trend with $d_{m}$ in ACHO. In the former, one observes delocalization, whereas localization occurs in the latter; in either situation, tunneling increases leading to a decrease in $A_{n}$. Further, in SCHO potential it is found that, $x_{c}$ and $\eta$ produce opposite effect on IE measures. However, in an ACHO, $\eta$ and $d_{m}$ cause similar changes in IE.

V Acknowledgement

AG is grateful to UGC for a Junior Research Fellowship (JRF). NM acknowledges IISER Kolkata for a post-doctoral fellowship. The authors thank Dr. A. K. Tiwari for useful discussion. Financial support from DST SERB (EMR/2014/000838) is gratefully acknowledged. Effort of Mr. Suman K. Chakraborty is appreciated, for his assistance in computers. Constructive comments from three anonymous referees have been very helpful.