Quantum mechanical virial-like theorem for confined quantum systems

TL;DR

This paper derives a unified virial-like theorem applicable to both free and confined quantum systems, providing a general equation involving energy expectation values that remains valid under different boundary conditions and confinement types.

Contribution

It introduces a general virial-like equation for confined quantum systems that is unaffected by boundary changes and includes contributions from confining potentials.

Findings

Derived a virial-like equation valid for free and confined systems

Numerically demonstrated the theorem on harmonic oscillators and hydrogen atoms

Discussed applicability to various confinement geometries

Abstract

Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.

Tables

Table 1. Table 1: ℰ n , ( Δ ​ V n ) 2 , ( Δ ​ T n ) 2 , ⟨ T ⟩ n ​ ⟨ V ⟩ n subscript ℰ 𝑛 superscript Δ subscript 𝑉 𝑛 2 superscript Δ subscript 𝑇 𝑛 2 subscript delimited-⟨⟩ 𝑇 𝑛 subscript delimited-⟨⟩ 𝑉 𝑛 \mathcal{E}_{n},\left(\Delta V_{n}\right)^{2},\left(\Delta T_{n}\right)^{2},\langle T\rangle_{n}\langle V\rangle_{n} − - ⟨ T ​ V ⟩ n , ⟨ T ⟩ n ​ ⟨ V ⟩ n subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 subscript delimited-⟨⟩ 𝑇 𝑛 subscript delimited-⟨⟩ 𝑉 𝑛 \langle TV\rangle_{n},\langle T\rangle_{n}\langle V\rangle_{n} − - ⟨ V ​ T ⟩ n subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 \langle VT\rangle_{n} values for n = 0 , 1 𝑛 0 1 n=0,1 states in 1D CHO at six selected values of x c subscript 𝑥 𝑐 x_{c} , namely 0.1 , 0.5 , 1 , 3 , 5 , ∞ 0.1 0.5 1 3 5 0.1,0.5,1,3,5,\infty . See text for detail.

$n$	Property	$x_c=0.1$	$x_c=0.5$	$x_c=1$	$x_c=3$	$x_c=5$	$x_c=\infty$
	${ℰ}_0^{†}$	123.3707084678	4.9511293232	1.2984598320	0.5003910829	0.50000007	0.4999999999
	${\left(\Delta V_0\right)}^2$	0.000000600468	0.0003747558	0.0058688193	0.1215456043	0.124999	0.1299999999
0	${\left(\Delta T_0\right)}^2$	0.000000600466	0.0003747558	0.0058688193	0.1215456043	0.124999	0.1299999999
	${⟨T⟩}_0{⟨V⟩}_0-{⟨TV⟩}_0$	0.000000600466	0.0003747558	0.0058688193	0.1215456043	0.124999	0.1299999999
	${⟨T⟩}_0{⟨V⟩}_0-{⟨VT⟩}_0$	0.000000600466	0.0003747558	0.0058688193	0.1215456043	0.124999	0.1299999999
	${ℰ}_1^{‡}$	493.481633417	19.7745341792	5.0755820152	1.5060815272	1.5000000036	1.499999999
	${\left(\Delta V_1\right)}^2$	0.00000085445	0.00053374630	0.0084865378	0.3353761814	0.3749997486	0.374999999
1	${\left(\Delta T_1\right)}^2$	0.00000085434	0.00053374630	0.0084865378	0.3353761814	0.3749997486	0.374999999
	${⟨T⟩}_1{⟨V⟩}_1-{⟨TV⟩}_1$	0.00000085434	0.00053374630	0.0084865378	0.3353761814	0.3749997486	0.374999999
	${⟨T⟩}_1{⟨V⟩}_1-{⟨VT⟩}_1$	0.00000085434	0.00053374630	0.0084865378	0.3353761814	0.3749997486	0.374999999

Table 2. Table 2: ℰ n r , ℓ , ( Δ ​ V n r , ℓ ) 2 , ( Δ ​ T n r , ℓ ) 2 , ⟨ T ⟩ n r , ℓ ​ ⟨ V ⟩ n r , ℓ − ⟨ T ​ V ⟩ n r , ℓ , ⟨ T ⟩ n r , ℓ ​ ⟨ V ⟩ n r , ℓ − ⟨ V ​ T ⟩ n r , l subscript ℰ subscript 𝑛 𝑟 ℓ superscript Δ subscript 𝑉 subscript 𝑛 𝑟 ℓ 2 superscript Δ subscript 𝑇 subscript 𝑛 𝑟 ℓ 2 subscript delimited-⟨⟩ 𝑇 subscript 𝑛 𝑟 ℓ subscript delimited-⟨⟩ 𝑉 subscript 𝑛 𝑟 ℓ subscript delimited-⟨⟩ 𝑇 𝑉 subscript 𝑛 𝑟 ℓ subscript delimited-⟨⟩ 𝑇 subscript 𝑛 𝑟 ℓ subscript delimited-⟨⟩ 𝑉 subscript 𝑛 𝑟 ℓ subscript delimited-⟨⟩ 𝑉 𝑇 subscript 𝑛 𝑟 𝑙 \mathcal{E}_{n_{r},\ell},(\Delta V_{n_{r},\ell})^{2},(\Delta T_{n_{r},\ell})^{2},\langle T\rangle_{n_{r},\ell}\langle V\rangle_{n_{r},\ell}-\langle TV\rangle_{n_{r},\ell},\langle T\rangle_{n_{r},\ell}\langle V\rangle_{n_{r},\ell}-\langle VT\rangle_{n_{r},l} for 1 ​ s , 1 ​ p , 2 ​ s 1 𝑠 1 𝑝 2 𝑠 1s,~{}1p,~{}2s states in 3DCHO at six selected r c subscript 𝑟 𝑐 r_{c} ’s, namely 0.1 , 0.5 , 1 , 2 , 5 , ∞ 0.1 0.5 1 2 5 0.1,0.5,1,2,5,\infty . See text for detail.

State	Property	$r_c=0.1$	$r_c=0.5$	$r_c=1$	$r_c=2$	$r_c=5$	$r_c=\infty$
	${ℰ}_{1,0}^{\P }$	493.4816334599	19.774534179	5.0755820153	1.7648164388	1.5000000003	1.499999999
	${\left(\Delta V_{1,0}\right)}^2$	0.00000085434	0.0005337463	0.0084865378	0.1211110138	0.3749999628	0.374999999
$1s$	${\left(\Delta T_{1,0}\right)}^2$	0.00000085434	0.0005337463	0.0084865378	0.1211110138	0.3749999628	0.374999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}$ $-$ ${⟨TV⟩}_{1,0}$	0.00000085434	0.0005337463	0.0084865378	0.1211110138	0.3749999628	0.374999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}$ $-$ ${⟨VT⟩}_{1,0}$	0.00000085434	0.0005337463	0.0084865378	0.1211110138	0.3749999628	0.374999999
	${ℰ}_{1,1}^{\S }$	1009.53830080	40.428276496	10.282256939	3.246947098	2.5000000584	2.499999999
	${\left(\Delta V_{1,1}\right)}^2$	0.0000008424	0.00052642239	0.0084064867	0.129302864	0.6249963610	0.624999999
$1p$	${\left(\Delta T_{1,1}\right)}^2$	0.00000084238	0.00052642239	0.0084064867	0.129302864	0.6249963610	0.624999999
	${⟨T⟩}_{1,1}{⟨V⟩}_{1,1}-{⟨TV⟩}_{1,1}$	0.00000084238	0.00052642239	0.0084064867	0.129302864	0.6249963610	0.624999999
	${⟨T⟩}_{1,1}{⟨V⟩}_{1,1}-{⟨VT⟩}_{1,1}$	0.00000084238	0.00052642239	0.0084064867	0.129302864	0.6249963610	0.624999999
	${ℰ}_{2,0}^{‡}$	1973.922483399	78.9969211469	19.8996965019	5.5846390792	3.5000122149	3.499999999
	${\left(\Delta V_{2,0}\right)}^2$	0.00000182	0.00113739969	0.01815844553	0.2779838025	1.6246856738	1.624999999
$2s$	${\left(\Delta T_{2,0}\right)}^2$	0.00000182	0.00113739969	0.01815844553	0.2779838025	1.6246856738	1.624999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨TV⟩}_{2,0}$	0.00000182	0.00113739969	0.01815844553	0.2779838025	1.6246856738	1.624999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨VT⟩}_{2,0}$	0.00000182	0.00113739969	0.01815844553	0.2779838025	1.6246856738	1.624999999

Table 3. Table 3: ℰ n , ℓ , ( Δ ​ V n , ℓ ) 2 , ( Δ ​ T n , ℓ ) 2 , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ T ​ V ⟩ n , ℓ , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ V ​ T ⟩ n , ℓ subscript ℰ 𝑛 ℓ superscript Δ subscript 𝑉 𝑛 ℓ 2 superscript Δ subscript 𝑇 𝑛 ℓ 2 subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 ℓ \mathcal{E}_{n,\ell},\left(\Delta V_{n,\ell}\right)^{2},\left(\Delta T_{n,\ell}\right)^{2},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell} of 1 ​ s , 2 ​ s , 2 ​ p 1 𝑠 2 𝑠 2 𝑝 1s,2s,2p states in CHA at six selected r c subscript 𝑟 𝑐 r_{c} , namely, 0.1 , 0.2 , 0.5 , 1 , 5 , ∞ 0.1 0.2 0.5 1 5 0.1,0.2,0.5,1,5,\infty . See text for detail.

State	Property	$r_c=0.1$	$r_c=0.2$	$r_c=0.5$	$r_c=1$	$r_c=5$	$r_c=\infty$
	${ℰ}_{1,0}^{\P }$	468.993038659	111.069858836	14.7479700303	2.3739908660	$-$0.4964170065	$-$0.499999999
	${\left(\Delta V_{1,0}\right)}^2$	308.872889980	80.3808359891	14.5396201848	4.4909017616	1.0176222756	0.9999999999
$1s$	${\left(\Delta T_{1,0}\right)}^2$	308.872889980	80.3808359891	14.5396201848	4.4909017616	1.0176222756	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨TV⟩}_{1,0}$	308.872889980	80.3808359891	14.5396201848	4.4909017616	1.0176222756	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨VT⟩}_{1,0}$	308.872889980	80.3808359891	14.5396201848	4.4909017616	1.0176222756	0.9999999999
	${ℰ}_{n,l}^{†}$	1942.720354554	477.8516723922	72.6720391904	16.5702560934	0.1412542037	$-$0.1249999999
	${\left(\Delta V_{2,0}\right)}^2$	925.842896028	236.7351455444	40.5134596945	11.3096437104	0.8156705939	0.1874999999
$2s$	${\left(\Delta T_{2,0}\right)}^2$	925.842896028	236.7351455444	40.5134596945	11.3096437104	0.8156705939	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨TV⟩}_{2,0}$	925.842896028	236.7351455444	40.5134596945	11.3096437104	0.8156705939	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨VT⟩}_{2,0}$	925.842896028	236.7351455444	40.5134596945	11.3096437104	0.8156705939	0.1874999999
	${ℰ}_{2,1}^{‡}$	991.0075894411	243.10933211	36.6588758801	8.2231383161	0.0075939204	$-$0.124999999
	${\left(\Delta V_{2,1}\right)}^2$	47.98046148	12.14249373	2.01620344857	0.5370036884	0.0381647208	0.02083333333
$2p$	${\left(\Delta T_{2,1}\right)}^2$	47.98046148	12.14249373	2.01620344857	0.5370036884	0.0381647208	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨TV⟩}_{2,1}$	47.98046148	12.14249373	2.01620344857	0.5370036884	0.0381647208	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨VT⟩}_{2,1}$	47.98046148	12.14249373	2.01620344857	0.5370036884	0.0381647208	0.02083333333

Table 4. Table 4: ℰ n , ℓ , ( Δ ​ V n , ℓ ) 2 , ( Δ ​ T n , ℓ ) 2 , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ T ​ V ⟩ n , ℓ , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ V ​ T ⟩ n , ℓ subscript ℰ 𝑛 ℓ superscript Δ subscript 𝑉 𝑛 ℓ 2 superscript Δ subscript 𝑇 𝑛 ℓ 2 subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 ℓ \mathcal{E}_{n,\ell},\left(\Delta V_{n,\ell}\right)^{2},\left(\Delta T_{n,\ell}\right)^{2},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell} for 1 ​ s , 2 ​ s , 2 ​ p 1 𝑠 2 𝑠 2 𝑝 1s,~{}2s,~{}2p states in SCHA at five sets of ( r a , r b ) subscript 𝑟 𝑎 subscript 𝑟 𝑏 \left(r_{a},r_{b}\right) values. See text for detail.

State	Property	$r_a=0.1,r_b=0.5$	$r_a=0.2,r_b=1$	$r_a=0.5,r_b=2$	$r_a=1,r_b=5$	$r_a=2,r_b=8$
	${ℰ}_{1,0}$	27.27172629	5.92023765	1.34445210	$-$0.05806114	$-$0.07992493
	${\left(\Delta V_{1,0}\right)}^2$	1.01266084	0.25766775	0.04408223	0.011743288	0.0030965826
$1s$	${\left(\Delta T_{1,0}\right)}^2$	1.01266084	0.25766775	0.04408223	0.011743288	0.0030965826
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨TV⟩}_{1,0}$	1.01266084	0.25766777	0.04408222	0.011743282	0.0030965824
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨VT⟩}_{1,0}$	1.01266084	0.25766777	0.04408222	0.011743282	0.0030965824
	${ℰ}_{2,0}$	119.52182029	28.91900480	7.87809191	0.85031117	0.325553290
	${\left(\Delta V_{2,0}\right)}^2$	2.31747169	0.58308875	0.097543493	0.02432941	0.00630949665
$2s$	${\left(\Delta T_{2,0}\right)}^2$	2.31747169	0.58308875	0.097543493	0.02432941	0.00630949665
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨TV⟩}_{2,0}$	2.31747169	0.58308875	0.097543493	0.02432941	0.00630949665
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨VT⟩}_{2,0}$	2.31747169	0.58308875	0.097543493	0.02432941	0.00630949665
	${ℰ}_{2,1}$	40.49778250	9.26352721	2.09854297	0.088632364	-0.028352228
	${\left(\Delta V_{2,1}\right)}^2$	0.86315456	0.21982576	0.040223458	0.010141187	0.0028634216
$2p$	${\left(\Delta T_{2,1}\right)}^2$	0.86315456	0.21982576	0.040223458	0.010141187	0.0028634216
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨TV⟩}_{2,1}$	0.86315456	0.21982576	0.040223458	0.010141187	0.0028634216
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨VT⟩}_{2,1}$	0.86315456	0.21982576	0.040223458	0.010141187	0.0028634216

Table 5. Table 5: ℰ n , ℓ , ( Δ ​ V n , ℓ ) 2 , ( Δ ​ T n , ℓ ) 2 , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ T ​ V ⟩ n , ℓ , ⟨ T ⟩ n , e ​ l ​ l ​ ⟨ V ⟩ n , ℓ − ⟨ V ​ T ⟩ n , ℓ subscript ℰ 𝑛 ℓ superscript Δ subscript 𝑉 𝑛 ℓ 2 superscript Δ subscript 𝑇 𝑛 ℓ 2 subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑛 𝑒 𝑙 𝑙 subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 ℓ \mathcal{E}_{n,\ell},\left(\Delta V_{n,\ell}\right)^{2},\left(\Delta T_{n,\ell}\right)^{2},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell},\langle T\rangle_{n,ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell} for 1 ​ s , 2 ​ s , 2 ​ p 1 𝑠 2 𝑠 2 𝑝 1s,2s,2p states in HICHA at six selected values of r c subscript 𝑟 𝑐 r_{c} , namely 0.1 , 0.2 , 0.5 , 1 , 5 , ∞ 0.1 0.2 0.5 1 5 0.1,0.2,0.5,1,5,\infty . See text for detail.

State	Property	$r_c=0.1$	$r_c=0.2$	$r_c=0.5$	$r_c=1$	$r_c=5$	$r_c=\infty$
	${ℰ}_{1,0}$	16.80524705	7.43767694	2.16863754	0.593771218¶	$-$0.404345971	$-$0.499999999
	${\left(\Delta V_{1,0}\right)}^2$	13.2294032	7.3539601	3.6335903	2.30437841	1.1794853	0.9999999999
$1s$	${\left(\Delta T_{1,0}\right)}^2$	13.2294032	7.3539601	3.6335903	2.30437841	1.1794853	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨TV⟩}_{1,0}$	13.2294032	7.3539601	3.6335903	2.30437841	1.1794853	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨VT⟩}_{1,0}$	13.2294032	7.3539601	3.6335903	2.30437841	1.1794853	0.9999999999
	${ℰ}_{2,0}$	45.89969929	22.186822249	8.25704419	3.771224646¶	0.434727738	$-$0.1249999999
	${\left(\Delta V_{2,0}\right)}^2$	18.4752785	9.8452409	4.4513850	2.5360027	0.78733209	0.1874999999
$2s$	${\left(\Delta T_{2,0}\right)}^2$	18.4752785	9.8452409	4.4513850	2.5360027	0.78733209	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨TV⟩}_{2,0}$	18.4752785	9.8452409	4.4513850	2.5360027	0.78733209	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨VT⟩}_{2,0}$	18.4752785	9.8452409	4.4513850	2.5360027	0.78733209	0.1874999999
	${ℰ}_{2,1}$	32.48998926	15.64056055	5.76850468	2.60273839¶	0.265263485	$-$0.124999999
	${\left(\Delta V_{2,1}\right)}^2$	1.5579056	0.8086356	0.3486783	0.1899865	0.05526280	0.02083333333
$2p$	${\left(\Delta T_{2,1}\right)}^2$	1.5579056	0.8086356	0.3486783	0.1899865	0.05526280	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨TV⟩}_{2,1}$	1.5579056	0.8086356	0.3486783	0.1899865	0.05526280	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨VT⟩}_{2,1}$	1.5579056	0.8086356	0.3486783	0.1899865	0.05526280	0.02083333333

Table 6. Table 6: ℰ n , ℓ , ( Δ ​ V n , ℓ ) 2 , ( Δ ​ T n , ℓ ) 2 , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ T ​ V ⟩ n , ℓ , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ − ⟨ V ​ T ⟩ n , ℓ subscript ℰ 𝑛 ℓ superscript Δ subscript 𝑉 𝑛 ℓ 2 superscript Δ subscript 𝑇 𝑛 ℓ 2 subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 ℓ \mathcal{E}_{n,\ell},\ \left(\Delta V_{n,\ell}\right)^{2},\left(\Delta T_{n,\ell}\right)^{2},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell} for 1 ​ s , 2 ​ s , 2 ​ p 1 𝑠 2 𝑠 2 𝑝 1s,2s,2p states in SPCHA at six selected sets of { V 0 , r c } subscript 𝑉 0 subscript 𝑟 𝑐 \{V_{0},r_{c}\} sets. See text for detail.

State	Property	$V_0=0$	$V_0=0$	$V_0=1$	$V_0=4$	$V_0=10$	$V_0=\infty$
		$r_c=5.77827$	$r_c=4.87924$	$r_c=5.72824$	$r_c=5.75669$	$r_c=5.49360$	$r_c=5.80119$
	${ℰ}_{1,0}^{†}$	$-$0.9998090	$-$0.9990142	$-$0.999186	$-$0.998703	$-$0.997682	$-$0.998302
	${\left(\Delta V_{1,0}\right)}^2$	1.000433	1.003194	1.00266	1.00447	1.00848	1.0047
$1s$	${\left(\Delta T_{1,0}\right)}^2$	1.000433	1.003194	1.00266	1.00447	1.00848	1.0047
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨TV⟩}_{1,0}$	1.000433	1.003194	1.00266	1.00447	1.00848	1.0047
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨VT⟩}_{1,0}$	1.000433	1.003194	1.00266	1.00447	1.00848	1.0047
	${ℰ}_{2,0}$	$-$0.1578690	$-$0.0909114	$-$0.035144	0.0128918	0.0818295	0.0434530
	${\left(\Delta V_{2,0}\right)}^2$	0.355830	0.412397	0.56205	0.64650	0.774448	0.60752
$2s$	${\left(\Delta T_{2,0}\right)}^2$	0.355830	0.412397	0.56205	0.64650	0.774448	0.60752
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}$ $-$ ${⟨TV⟩}_{2,0}$	0.355830	0.412397	0.56205	0.64650	0.774448	0.60752
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}$ $-$ ${⟨VT⟩}_{2,0}$	0.355830	0.412397	0.56205	0.64650	0.774448	0.60752
	${ℰ}_{2,1}$	$-$0.1996605	$-$0.1587620	$-$0.1406809	$-$0.1172265	$-$0.0832120	$-$0.1022024
	${\left(\Delta V_{2,1}\right)}^2$	0.032028	0.0413657	0.046701	0.0637238	0.094861	0.0316629
$2p$	${\left(\Delta T_{2,1}\right)}^2$	0.032028	0.0413657	0.046701	0.0637238	0.094861	0.0316629
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}$ $-$ ${⟨TV⟩}_{2,1}$	0.032028	0.0413657	0.046701	0.0637238	0.094861	0.0316629
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}$ $-$ ${⟨VT⟩}_{2,1}$	0.032028	0.0413657	0.046701	0.0637238	0.094861	0.0316629

Table 7. Table 7: ℰ n , ℓ , ( Δ ​ V n , ℓ ) 2 , ( Δ ​ T n , ℓ ) 2 , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ subscript ℰ 𝑛 ℓ superscript Δ subscript 𝑉 𝑛 ℓ 2 superscript Δ subscript 𝑇 𝑛 ℓ 2 subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ \mathcal{E}_{n,\ell},\left(\Delta V_{n,\ell}\right)^{2},\left(\Delta T_{n,\ell}\right)^{2},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell} − - ⟨ T ​ V ⟩ n , ℓ , ⟨ T ⟩ n , ℓ ​ ⟨ V ⟩ n , ℓ subscript delimited-⟨⟩ 𝑇 𝑉 𝑛 ℓ subscript delimited-⟨⟩ 𝑇 𝑛 ℓ subscript delimited-⟨⟩ 𝑉 𝑛 ℓ \langle TV\rangle_{n,\ell},\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell} − - ⟨ V ​ T ⟩ n , ℓ subscript delimited-⟨⟩ 𝑉 𝑇 𝑛 ℓ \langle VT\rangle_{n,\ell} for 1 ​ s , 2 ​ s , 2 ​ p 1 𝑠 2 𝑠 2 𝑝 1s,2s,2p states in HPCHA at five selected r c subscript 𝑟 𝑐 r_{c} , namely, 0.1 , 0.2 , 0.5 , 1 , 5 0.1 0.2 0.5 1 5 0.1,0.2,0.5,1,5 , having U = 10 , w = 1000 formulae-sequence 𝑈 10 𝑤 1000 U=10,w=1000 . Last column indicates the values at r c = ∞ subscript 𝑟 𝑐 r_{c}=\infty and U = 0 𝑈 0 U=0 . See text for detail.

State	Property	$r_c=0.1$	$r_c=0.2$	$r_c=0.5$	$r_c=1$	$r_c=5$	$r_c=\infty ,U=0$
	${ℰ}_{1,0}^{\S }$	9.4871580	9.35868	5.25360	1.1528598	-0.4973688	$-$0.499999999
	${\left(\Delta V_{1,0}\right)}^2$	1.15378	2.4119	6.6390	3.25938	1.0133575	0.9999999999
$1s$	${\left(\Delta T_{1,0}\right)}^2$	1.15378	2.4119	6.6390	3.25938	1.0133575	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨TV⟩}_{1,0}$	1.15378	2.4119	6.6390	3.25938	1.0133575	0.9999999999
	${⟨T⟩}_{1,0}{⟨V⟩}_{1,0}-{⟨VT⟩}_{1,0}$	1.15378	2.4119	6.6390	3.25938	1.0133575	0.9999999999
	${ℰ}_{2,0}$	9.8734148	9.8593719	9.7728942	9.029792	0.10745905	$-$0.1249999999
	${\left(\Delta V_{2,0}\right)}^2$	0.20807	0.346874	0.153805	5.11808	0.7615043	0.1874999999
2s	${\left(\Delta T_{2,0}\right)}^2$	0.20807	0.346874	0.153805	5.11808	0.7615043	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨TV⟩}_{2,0}$	0.20807	0.346874	0.153805	5.11808	0.7615043	0.1874999999
	${⟨T⟩}_{2,0}{⟨V⟩}_{2,0}-{⟨VT⟩}_{2,0}$	0.20807	0.346874	0.153805	5.11808	0.7615043	0.1874999999
	${ℰ}_{2,1}$	9.8749992211	9.87497532482	9.869939026	4.980371	$-$0.011992	$-$0.124999999
	${\left(\Delta V_{2,1}\right)}^2$	0.02083685	0.0209243374	0.04006099	0.36608	0.03609	0.02083333333
2p	${\left(\Delta T_{2,1}\right)}^2$	0.02083685	0.0209243374	0.04006099	0.36608	0.03609	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨TV⟩}_{2,1}$	0.02083685	0.0209243374	0.04006099	0.36608	0.03609	0.02083333333
	${⟨T⟩}_{2,1}{⟨V⟩}_{2,1}-{⟨VT⟩}_{2,1}$	0.02083685	0.0209243374	0.04006099	0.36608	0.03609	0.02083333333

Equations

$$(\hat{T}+\hat{V}){\psi }_n(\tau )={\mathcal{E}}_n{\psi }_n(\tau ),$$

$$⟨{\hat{T}}^2{⟩}_n+⟨\hat{T}\hat{V}{⟩}_n={\mathcal{E}}_n⟨\hat{T}{⟩}_n.$$

$$⟨{\hat{T}}^2{⟩}_n-⟨\hat{T}{⟩}_n^2=⟨\hat{V}{⟩}_n⟨\hat{T}{⟩}_n-⟨\hat{T}\hat{V}{⟩}_n.$$

$$⟨{\hat{V}}^2{⟩}_n-⟨\hat{V}{⟩}_n^2=⟨\hat{T}{⟩}_n⟨\hat{V}{⟩}_n-⟨\hat{V}\hat{T}{⟩}_n$$

$$⟨{\hat{T}}^2{⟩}_n-⟨\hat{T}{⟩}_n^2$$

$$(\Delta {\hat{T}}_n{)}^2=(\Delta {\hat{V}}_n{)}^2$$

$$⟨{\hat{T}}^2{⟩}_n={\mathcal{E}}_n\left({\mathcal{E}}_n-2⟨\hat{V}{⟩}_n\right)+⟨{\hat{V}}^2{⟩}_n.$$

$$⟨{\varphi }_1|{\varphi }_1⟩⟨{\varphi }_2|{\varphi }_2⟩$$

$$(\Delta {\hat{T}}_n{)}^2(\Delta {\hat{V}}_n{)}^2$$

$$|\hat{T}-⟨\hat{T}{⟩}_n|{\psi }_n⟩=\alpha |\hat{V}-⟨\hat{V}{⟩}_n|{\psi }_n⟩,$$

$${\alpha }^2(\Delta {\hat{T}}_n{)}^2(\Delta {\hat{V}}_n{)}^2$$

$${\alpha }^2$$

$$(\Delta {\hat{T}}_n^2)(\Delta {\hat{V}}_n{)}^2=|⟨\hat{T}\hat{V}{⟩}_n-⟨\hat{T}{⟩}_n⟨\hat{V}{⟩}_n{|}^2$$

$$(\Delta {\hat{T}}_n{)}^2=(\Delta {\hat{V}}_n{)}^2=|⟨\hat{T}\hat{V}{⟩}_n-⟨\hat{T}{⟩}_n⟨\hat{V}{⟩}_n|$$

$$|\hat{T}-⟨\hat{T}{⟩}_n|{\psi }_n⟩$$

$$(\hat{T}+\hat{V})|{\psi }_n⟩$$

$$(\hat{T}-\hat{V})|{\psi }_n⟩=[⟨\hat{T}{⟩}_n-⟨\hat{T}{⟩}_n]|{\psi }_n⟩,$$

$$(\Delta {\hat{H}}_n{)}^2=(\Delta {\hat{T}}_n{)}^2+(\Delta {\hat{V}}_n{)}^2+[⟨\hat{T}\hat{V}{⟩}_n-⟨\hat{T}{⟩}_n⟨\hat{V}{⟩}_n]+[⟨\hat{V}\hat{T}{⟩}_n-⟨\hat{T}{⟩}_n⟨\hat{V}{⟩}_n]$$

$${\left(\Delta {\hat{H}}_n\right)}^2=0$$

$$\left[-\frac{1}{2}\frac{d^2}{d{r}^2}+\frac{\ell (\ell +1)}{2r^2}+v(r)+v_c(r)\right]{\psi }_{n,\ell }(r)={\mathcal{E}}_{n,\ell }{\psi }_{n,\ell }(r),$$

$$(\Delta {\hat{T}}_{n,\ell }{)}^2=⟨{\hat{T}}^2{⟩}_{n,\ell }-⟨\hat{T}{⟩}_{n,\ell }^2,$$

$$(\Delta {\hat{V}}_{n,\ell }{)}^2=⟨v(r{)}^2{⟩}_{n,\ell }+⟨v(r)v_c(r){⟩}_{n,\ell }+⟨v_c(r)v(r){⟩}_{n,\ell }+⟨v_c(r{)}^2{⟩}_{n,\ell }-⟨v(r){⟩}_{n,\ell }^2-⟨v_c(r){⟩}_{n,\ell }^2$$

$$-2⟨v(r){⟩}_{n,\ell }⟨v_c(r){⟩}_{n,\ell },$$

$$⟨\hat{T}{⟩}_{n,\ell }⟨\hat{V}{⟩}_{n,\ell }-⟨\hat{V}\hat{T}{⟩}_{n,\ell }=⟨\hat{T}{⟩}_{n,\ell }\left[⟨v(r){⟩}_{n,\ell }+⟨v_c(r){⟩}_{n,\ell }\right]-⟨\hat{T}v(r){⟩}_{n,\ell }-⟨\hat{T}{v}_c(r){⟩}_{n,\ell }$$

$$⟨\hat{T}{⟩}_{n,\ell }⟨\hat{V}{⟩}_{n,\ell }-⟨\hat{T}\hat{V}{⟩}_{n,\ell }=⟨\hat{T}{⟩}_{n,\ell }\left[⟨v(r){⟩}_{n,\ell }+⟨v_c(r){⟩}_{n,\ell }\right]-⟨v(r)\hat{T}{⟩}_{n,\ell }-⟨v_c(r)\hat{T}{⟩}_{n,\ell }.$$

$$-\frac{1}{2}\frac{d^2{\psi }_n}{d{x}^2}+4{\alpha }^2{x}^2{\psi }_n+v_c{\psi }_n={\mathcal{E}}_n{\psi }_n,$$

$${\psi }_e(x)=N_e{}_1{F}_1\left[\left(\frac{1}{4}-\frac{{\mathcal{E}}_n}{4\sqrt{2}\alpha }\right),\frac{1}{2},2\sqrt{2}\alpha x^2\right]e^{-\sqrt{2}\alpha x^2},$$

$${\psi }_o(x)=N_{o}x{}_1{F}_1\left[\left(\frac{3}{4}-\frac{{\mathcal{E}}_n}{4\sqrt{2}\alpha }\right),\frac{3}{2},2\sqrt{2}\alpha x^2\right]e^{-\sqrt{2}\alpha x^2}.$$

$$⟨\hat{T}\hat{V}{⟩}_n=⟨\hat{T}v(x){⟩}_n+⟨\hat{T}{v}_c{⟩}_n=⟨\hat{T}v(x){⟩}_n.$$

$$⟨\hat{T}{v}_c{⟩}_n$$

$$⟨\hat{V}\hat{T}{⟩}_n=⟨v(x)\hat{T}{⟩}_n+⟨v_c\hat{T}{⟩}_n=⟨v(x)\hat{T}{⟩}_n$$

$$⟨{\hat{V}}^2{⟩}_n=⟨(v(x{)}^2{⟩}_n+⟨v(x)v_c{⟩}_n+⟨v_{c}v(x){⟩}_n+⟨v_c^2{⟩}_n=⟨(v(x){)}^2{⟩}_n$$

$$⟨\hat{V}{⟩}_n=⟨v(x){⟩}_n+⟨v_c{⟩}_n=⟨v(x){⟩}_n.$$

$${\left(\Delta {\hat{T}}_n\right)}^2={\left(\Delta {\hat{V}}_n\right)}^2=⟨\hat{T}{⟩}_n⟨v(x){⟩}_n-⟨v(x)\hat{T}{⟩}_n=⟨\hat{T}{⟩}_n⟨v(x){⟩}_n-⟨\hat{T}v(x){⟩}_n.$$

Full text

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

Keywords: Virial theorem, soft confinement, hard confinement, homogeneous confinement, sufficient condition.

I introduction

Over the last twenty years confined quantum systems have emerged as a topic of considerable significance for physicists, chemists, biologists sabin2009 . Invention and advancement of contemporary experimental techniques have given the required insight about responses of matter under such constrained environments. Furthermore, recent progress in nano-science and nano-technology has inspired extensive research activity to explore and acquire more thorough, in-depth understanding. Nowadays, various physical, chemical processes are carried out in spatially confined environment. They have profound applications in diverse area of research, like condensed matter, semiconductor physics, astrophysics pang11 , nano-technology, quantum dot, wire and well sen2014electronic . In recent years, these models are also employed to interpret the trapping of atoms, molecules inside fullerene cage, zeolite cavity sabin2009 ; sen2014electronic ; sen12 etc.

A quantum particle under the influence of confinement displays many fascinating, distinctive changes in observable physical, chemical properties aquino16 ; yu17 . Usually, the Schrödinger equation (SE) can not be solved exactly; therefore, one has to take recourse to approximate methods. The perturbative approach leads to an asymptotic series fernandez82 , and standard linear variation method is fraught with the problem of proper boundary behavior, as familiar orthonormal basis sets do not vanish at finite boundaries. Thus linear combinations of such bases are explicitly inappropriate in representing their eigenstates. Recently for some central potentials (harmonic oscillator, H atom, pseudoharmonic oscillator, etc.) under hard confinement condition, such equation can be solved exactly. These eigenfunctions can then be used as appropriate orthonormal basis set in other confined systems ghosal16 .

In 1937, the first model for confined quantum system, a H atom trapped inside an impenetrable barrier was proposed to understand its behavior under extreme pressure michels37 . With time this was found to be somehow restrictive for practical purposes, leading to the development of so-called penetrable barriers. For sake of convenience, it may be appropriate to categorize different confining potentials, following katriel12 , in two broad classes, namely (i) a penetrable potential which is finitely bounded from above, whereas in an impenetrable case, it rises to infinity at large $r$ (ii) A continuous potential will be termed as smooth while a sharp potential possesses discontinuity. In case of impenetrable, sharp condition, a potential is modified by the addition of a term that disappear up to a certain distance from origin, rising to infinity thereafter. Such potentials are defined as $V=V(r)$ at $0\leq r\leq r_{c}$ and $V=\infty$ at $r>r_{c}$ ($r_{c}$ implies confinement radius). In this situation, the Dirichlet boundary condition $R_{n,\ell}(0)=R_{n,\ell}(r_{c})=0$ is obeyed sabin2009 . On the other hand, an impenetrable, smooth potential is defined as $V=V(r)+V_{c}(r)$, where $V_{c}(r)$ is the confining potential that becomes infinity at $r\rightarrow\infty$, and remain continuous otherwise wilson94 ; patil09 . Similarly, for penetrable, sharp case the potential has the form: $V=V(r)$ at $0\leq r\leq r_{c}$ and $V=V_{c}(r)$ at $r>r_{c}$, where, $V_{c}(r)$ is the confining potential koo79 . Finally, in the penetrable, smooth case it becomes, $V=V(r)+V_{c}(r)$ adamowski00 . In recent years, various models were proposed and investigated by many authors sen2014electronic ; aquino13 ; sen05 ; randazzo16 ; cabrera16 , especially in the context of H atom, maintaining these confinement conditions, revealing numerous striking features sabin2009 ; sen2014electronic ; stevanocic10 ; montgomery12 ; cabrera13 .

Extensive theoretical calculations have been made in case of confined harmonic oscillator (CHO) (1D, 2D, 3D, $d$ dimension) coll17 ; aquino97 ; campoy2002 ; montgomery07 ; roy14 ; ghosal16 and confined hydrogen atom (CHA) inside an impenetrable cavity coll17 ; goldman92 ; aquino95 ; garza98 ; laughlin02 ; burrows06 ; aquino07cha ; baye08 ; ciftci09 ; sen2014electronic ; roy15a ; centeno17 . They offer many extraordinary features, especially relating to simultaneous, incidental, inter-dimensional degeneracy montgomery07 in their energy spectra. Effect of contraction on ground and excited energy states, as well as other properties like hyperfine splitting constant, dipole shielding factor, nuclear magnetic screening constant, static and dynamic polarizability, etc., were explored sabin2009 ; sen2014electronic ; sen12 . A wide range of attractive analytical and numerical approaches including perturbation theory, Padé approximation, WKB method, Hypervirial theorem, power-series solution, supersymmetric quantum mechanics, Lie algebra, Lagrange-mesh, asymptotic iteration, generalized pseudo-spectral method, etc., were invoked to solve the relevant eigenvalue problem goldman92 ; aquino95 ; garza98 ; laughlin02 ; burrows06 ; aquino07cha ; baye08 ; ciftci09 ; roy15a . Exact solutions burrows06 of CHA are expressible in terms of Kummer M-function (confluent hypergeometric).

In quantum mechanics, stationary states of a bound system satisfy the virial theorem (VT). In fact, it is a necessary condition for a quantum stationary state to follow lowdin59 . Historically the quantum mechanical VT was derived from analogy with classical counterpart; for a non-relativistic Hamiltonian, it offers a relation between expectation values of kinetic energy and directional derivatives of potential energy. In this regard, it is important to point out that a variationally optimized wave function also obeys the VT. Hence, it becomes a necessary condition for an exact wave function to obey. On the contrary, obeying this relation will not ensure that the state to be exact. After some controversy, it is now generally accepted that the standard form of VT is not obeyed in enclosed condition; rather a modified form is invoked. Several attempts were made to find an appropriate form of VT in such systems cotrell51 ; brown58 ; fernandez82 . Previously, some semi-classical strategies based on Wilson-Sommerfeld rule and uncertainty principle were also adopted to construct VT in such systems mukhopadhyay05 . In recent years, standard form of VT and Hellmann-Feynman theorem (HFT) were combined to design new virial-like expression for penetrable and impenetrable CHA katriel12 , however, the mathematical forms of the expressions change from system to system. Importantly, all these relations can only serve as necessary condition for an exact state to obey. In this endeavour our aim is to design a uniform virial-like expression for both free and confined conditions using time-independent SE, the Hyper-virial theorem (HVT) hirsh60 , along with mean square values and expectation values of potential and kinetic energy operators. Apart from that, a new relation involving SE and HVT has been derived, which can serve as a sufficient condition (only true for exact states) for a bound stationary state to obey. In this scenario, detailed derivation of these relations are given in Sec. II. Next we proceed to verify the utility and applicability of these relations in the context of several representative confined systems. We begin Sec. III with the oldest, primitive model of hard confinement, where the potential is trapped inside an infinite wall satisfying the Dirichlet boundary condition. In this category, at first, we discuss the typical and most prolific cases of CHO (1D, 3D) as well as a CHA. Later, this is extended to the so-called shell-confined H-atom (SCHA), in order to understand the role of nodal structure in confined condition. This can be potentially treated as a confined off-centre model, needed to probe quantum wells/dots. With time, a new model for off-centre quantum dot structures was also adopted, but within the frame work of Newmann boundary condition, a prominent examples being the trapping of H atom inside a homogeneous, impenetrable cavity (HICHA). It my be noted that, at $r_{c}\rightarrow 0$ this behaves similar to CHA, while at $r_{c}\rightarrow\infty$ it resembles a free H-atom (FHA). In order to make these artificial atomic models more realistic, a finite wall was placed at a certain $r_{c}$; this has been widely used to study the properties of encapsulated atoms within fullerene cage and zeolite cavity. As an approximation to this, we explore the case of a H atom inside an inhomogeneous, penetrable spherical cavity (SPCHA). Apart from that, to incorporate the interaction of particle with the environment homogeneous, penetrable confinement model was proposed–for this we consider an H atom under similar condition (HPCHA). This will help us about the advantages of presently derived relations in the pursuit of confined quantum systems. Section IV makes a few concluding remarks.

II Theoretical Formalism

The time-independent non-relativistic SE for a system may simply be written as,

[TABLE]

where, $\hat{T},\hat{V}$ are usual kinetic and potential energy operators, while $\tau$ is a generalized variable. After some straightforward algebra (multiplying both sides by $\hat{T}$, integrating over whole space and rearranging), one gets,

[TABLE]

Now, replacing $\mathcal{E}_{n}=\langle\hat{T}\rangle_{n}+\langle\hat{V}\rangle_{n}$ in Eq. (2) produces,

[TABLE]

A similar consideration using $\hat{V}$ leads to the following equation,

[TABLE]

From hypervirial theorem, it can be proved that, $\langle\hat{T}\hat{V}\rangle_{n}=\langle\hat{V}\hat{T}\rangle_{n}$. Hence, from Eqs. (3)-(4), one obtains,

[TABLE]

This relation suggests that, the magnitude of error incurred in $\langle\hat{T}\rangle_{n}$ and $\langle\hat{V}\rangle_{n}$ are equal. Now, one can easily interpret the fact that, $\mathcal{E}_{n}$ is a sum of two average quantities but still provides exact result. It is due to the cancellation of errors between $\langle\hat{T}\rangle_{n}$ and $\langle\hat{V}\rangle_{n}$.

Interestingly, using the condition $\langle\hat{T}\hat{V}\rangle_{n}=\langle\hat{V}\hat{T}\rangle_{n}$ and exploiting Eqs.  (3) and (4), one can reach the expression,

[TABLE]

Thus, instead of performing the fourth order derivative of $\psi_{n}(\tau)$, one can alternatively evaluate $\langle\hat{T}^{2}\rangle_{n}$ from a knowledge of $\mathcal{E}_{n}$, $\langle\hat{V}\rangle_{n}$ and $\langle\hat{V}^{2}\rangle_{n}$.

Now we wish to verify whether Eq. (5) is true for eigenstates only or not. Let us consider two functions having forms $\phi_{1}=|\hat{T}-\langle\hat{T}\rangle_{n}|\psi_{n}\rangle$ and $\phi_{2}=|\hat{V}-\langle\hat{V}\rangle_{n}|\psi_{n}\rangle$. Making use of Schwartz inequality, it is possible to write,

[TABLE]

This inequality becomes equality when $\phi_{1}$ and $\phi_{2}$ are linearly dependent. That implies,

[TABLE]

where $\alpha$ is a number. Putting this back in the inequality and doing some algebraic rearrangement, we get,

[TABLE]

Choice of $\alpha^{2}=1$ yields the following expression,

[TABLE]

Here $\alpha^{2}=1$. Now, left multiplying Eq. (8) by $\langle\psi_{n}|(T-\langle T\rangle_{n})|$, followed by integration over whole space and rearrangement leads to,

[TABLE]

Equation (11) is valid for two values of $\alpha$, namely, 1 or $-1$. When $\alpha=-1$,

[TABLE]

Which is nothing but Schrödinger Equation: $\hat{H}|\psi_{n}\rangle=\mathcal{E}_{n}|\psi_{n}\rangle$. Whereas $\alpha=1$ gives,

[TABLE]

which does not concern us here.

This above discussion suggests that, Eq. (11) is a necessary condition for a stationary state to obey and Eq. (5) is a special case of it. Now, to verify the suitability of Eq. (5), it is useful to study $\langle\hat{H}^{2}\rangle_{n}-\langle\hat{H}\rangle^{2}_{n}=\left(\Delta\hat{H}_{n}\right)^{2}$, because, only for eigenstates it is zero. Thus,

[TABLE]

Now, putting the condition of Eq. (5) in Eq. (14) one can obtain,

[TABLE]

This clearly states that, Eq. (5) is a sufficient condition for an eigenfunction to obey. Hence, once this relation is obeyed by $\psi_{n}$, it is an eigenfunction of that particular $\hat{H}$. But $(\Delta\hat{T}_{n})^{2}=(\Delta\hat{V}_{n})^{2}$ is a necessary condition for a quantum system to obey, which is actually a virial-like expression. Now, it will be interesting to examine the applicability of Eq. (5) in the context of confined quantum systems.

For our current purpose at hand, without loss of generality, our relevant radial SE under the influence of confinement is,

[TABLE]

where $v(r)$ signifies the unperturbed effective potential (for example, in a many-electron system that may include effective electron-nuclear attraction and electron-electron repulsion), and our desired confinement inside a spherical cage is accomplished by invoking the potential $v_{c}(r)$, with $\hat{V}=v(r)+v_{c}(r)$. Thus in a confinement scenario, validity of Eq. (5) can be checked by deriving the expressions of $\langle\hat{T}\hat{V}\rangle_{n,\ell}$, $\langle\hat{V}\hat{T}\rangle_{n,\ell}$, $\langle\hat{V}^{2}\rangle_{n,\ell}$ and $\langle\hat{V}\rangle_{n,\ell}$ (other integrals remain unchanged). Towards this end, Eq. (5) may be modified as follows:

[TABLE]

[TABLE]

[TABLE]

In what follows, we will analyze the above-mentioned criteria for a number of important confining potentials, as mentioned in the introduction section, viz., (i) CHO in 1D and 3D (ii) A H atom encapsulated in five different confining environments, namely, CHA, SCHA, HICHA, SPCHA and HPCHA. This will offer us the opportunity to understand the effect of boundary condition on derived relations. It may be recalled that, out of these seven different potentials, 1DCHO, 3DCHO and CHA are exactly solvable. However, it is instructive to note that, in order to construct the exact wave function for a specific state, one needs to supply energy eigenvalue, which is calculated using imaginary-time propagation roy15 and generalized pseudo-spectral roy04 ; sen06 ; roy13 ; roy14 ; roy15a method respectively for 1D and 3D problems. Except CHA, in all the remaining confining H atom cases, we have employed numerically calculated wave functions and energies through GPS scheme. Now, we can use relation in Eq. (5) to inspect the goodness of numerical wave function.

III Results and Discussion

We shall now discuss the results under four broad category of confinement conditions viz., (i) impenetrable, sharp (ii) impenetrable, smooth (iii) penetrable, sharp (iv) penetrable, smooth, sequentially.

III.1 Impenetrable, sharp confinement

In this condition, the desired confinement effect on $v(r)$ is imposed by invoking the following form of potential: $v_{c}(r)=+\infty$ for $r>r_{c}$, and 0 for $r\leq r_{c}$, where $r_{c}$ signifies radius of box. In such situation, Eq. (16) needs to be solved under Dirichlet boundary condition, $\psi_{n_{r},l}(0)=\psi_{n_{r},l}(r_{c})=0$. Four systems will be included, namely, 1DCHO, 3DCHO, CHA and SCHA, which are taken up one by one.

III.1.1 1DCHO

The single-particle time-independent non-relativistic SE in 1D is ($\alpha$ is force constant):

[TABLE]

where, the confining potential is defined as, $v_{c}=0$ for $x<|x_{c}|$ and $v_{c}=\infty$ for $x\geq|x_{c}|$. Here, $x_{c}$ signifies confinement length. Note that we consider only the symmetric case; while asymmetric confinement can also be worked out (omitted here). Equation (20) can be solved exactly using the boundary condition $\psi_{n}(-x_{c})=\psi_{n}(x_{c})=0$, to produce the following analytical closed forms for odd and even states ($\alpha=\sqrt{\frac{1}{8}}$, for sake of convenience),

[TABLE]

In this equation, $N_{e},N_{o}$ represent normalization constant for even and odd states respectively, $\mathcal{E}_{n}$, the energy of respective eigenstates has been calculated accurately by an imaginary-time evolution method roy15 , while ${}_{1}F_{1}\left[a,b,x\right]$ denotes the confluent hypergeometric function. Now, the expectation values will take following forms:

[TABLE]

One can make use of the property of Reimann integral to write,

[TABLE]

The first and third integrals turn out as zero because $\psi_{n}(x)=0$ when $x\geq|x_{c}|$, whereas the second integral becomes zero as, $v_{c}=0$ inside the box. Similarly,

[TABLE]

[TABLE]

[TABLE]

Thus, for a 1DCHO, with the help of above equations, Eq. (5) may be recast as,

[TABLE]

Thus it is evident from Eq. (27) that, $v_{c}$ has no contribution in the desired expectation values. Hence the only difference between the free and enclosed system is that, in the latter, the boundary has been reduced to a finite region from infinity. Numerical values of $\mathcal{E}_{n}$, $(\Delta\hat{T}_{n})^{2}$, $(\Delta\hat{V}_{n})^{2}$, $\langle T\rangle_{n}\langle V\rangle_{n}-\langle TV\rangle_{n}$ and $\langle T\rangle_{n}\langle V\rangle_{n}-\langle VT\rangle_{n}$ are produced in Table I for $n=0,~{}1$ states of 1DCHO at six selected $x_{c}$ values, namely $0.1,~{}0.5,~{}1,~{}3,~{}5,~{}\infty$, that cover a large region of confinement. In all these six $x_{c}$, $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ remain in excellent agreement with available literature results as compared in roy15 , and hence not repeated here. However, no direct reference could be found for the expectation values to tally. It is easily noticed that, in both free and confined condition, Eq. (5) is obeyed, as all the expectation values offer identical results, which validates the the applicability of our newly designed theorem in case of 1D CHO. Additionally, with increase in $x_{c}$, both $\Delta\hat{T}^{2}$ and $\Delta\hat{V}^{2}$ increase, which presumably occurs as the wave function delocalizes with $x_{c}$. Consequently, the difference between mean square and average values of $\hat{T},~{}\hat{V}$ tends to grow.

III.1.2 3DCHO

The isotropic harmonic oscillator has the form, $v(r)=\frac{1}{2}\omega r^{2}$, where $\omega$ signifies the oscillation frequency. The exact generalized radial wave function of a 3DCHO is mathematically expressed as montgomery07 ,

[TABLE]

Here $N_{n_{r},\ell}$ represents the normalization constant, $\mathcal{E}_{n_{r},\ell}$ corresponds to the energy of a given state characterized by quantum numbers $n_{r},\ell$. Note that, the levels are designated by $n_{r}+1$ and $\ell$ values, such that $n_{r}=\ell=0$ signifies $1s$ state. The radial quantum number $n_{r}$ relates to $n$ as $n=2n_{r}+\ell$.

The relevant expectation values will now take following forms,

[TABLE]

This occurs because $\langle\hat{T}v_{c}(r)\rangle_{n_{r},\ell}=0$, due to the wave function vanishing when $r\geq r_{c}$. A similar argument ($\langle v_{c}(r)\hat{T}\rangle_{n_{r},\ell}=0$) leads to the conclusion that,

[TABLE]

Then since $\langle v(r)v_{c}(r)\rangle_{n_{r},\ell}=\langle v_{c}(r)v(r)\rangle_{n_{r},\ell}=\langle v_{c}(r)^{2}\rangle_{n_{r},\ell}=0$, we can write,

[TABLE]

And finally, one can derive (since $\langle v_{c}(r)\rangle_{n_{r},\ell}=0$),

[TABLE]

Thus, for a 3DCHO, Eq. (5) remains unchanged,

[TABLE]

Thus we observe that, similar to 1DCHO, here also the perturbing (confining) potential makes no contribution on desired expectation values; only the boundary in confined system gets shifted to $r_{c}$, from $\infty$ of the corresponding free counterpart. It clearly indicates the validity of Eq. (5) in a 3DCHO. As an illustration, Table II imprints numerically calculated values of appropriate expectation values, for three low-lying ($1s,~{}1p,~{}2s$) states at six selected $r_{c}$’s, namely, $0.1,~{}0.5,~{}1,~{}2,~{}5,~{}\infty$. This again establishes the utility of Eq. (5) for such potential in both confined and free system, as evident from identical values of these quantities at all $r_{c}$’s–last column signifying the corresponding free system. Accurate energy values are quoted from GPS results roy14 . No literature results are available for average values considered here. Like the 1D case, here also $(\Delta\hat{T}_{n_{r},\ell})^{2},~{}(\Delta\hat{V}_{n_{r},\ell})^{2}$ increase with $r_{c}$.

III.1.3 CHA

We begin with the exact wave function for CHA, which assumes the following form burrows06 ,

[TABLE]

with $N_{n,\ell}$ denoting normalization constant, $\mathcal{E}_{n,\ell}$ corresponding to energy of a state represented by $n,\ell$ quantum numbers. The pertinent expectation values can be simplified as,

[TABLE]

In this instance, $\langle\hat{T}v_{c}(r)\rangle_{n,\ell}=0$, as the wave function vanishes for $r\geq r_{c}$. Use of same argument, along with the fact that $\langle v_{c}(r)\hat{T}\rangle_{n,\ell}=0$, leads to the following,

[TABLE]

Now since $\langle v(r)v_{c}(r)\rangle_{n,\ell}=\langle v_{c}(r)v(r)\rangle_{n,\ell}=\langle v_{c}(r)^{2}\rangle_{n,\ell}=0$, one can write,

[TABLE]

Again because $\langle v_{c}(r)\rangle_{n,\ell}=0$, it follows that,

[TABLE]

Thus, like the previous two systems, for CHA also, Eq. (5) remains unchanged, i.e.,

[TABLE]

This equation implies that, CHA satisfies the results given in Eq. (5); as before, $v_{c}$ has no impact on it. It has only introduced the boundary in a finite range. Table III demonstrates sample values of $\mathcal{E}_{n,\ell}$, $(\Delta\hat{T}_{n,\ell})^{2}$, $(\Delta\hat{V}_{n,\ell})^{2}$, $\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell}$ and $\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell}$ for same low-lying ($1s,~{}2s,~{}2p$) states of previous table, in CHA at same six selected $r_{c}$ values, namely $0.1,~{}0.2,~{}0.5,~{}1,~{}5,\infty$. For sake of completeness, accurate values of $\mathcal{E}_{n,\ell}$ are reproduced from roy15a . Once again, no literature results could be found to compare the numerically calculated expectation values. In both free and confining conditions, these results complement the conclusion of Eq. (5). In the passing, it is interesting to note that both $(\Delta\hat{T}_{n,\ell})^{2},(\Delta\hat{V}_{n,\ell})^{2}$ decrease with rise in $r_{c}$.

III.1.4 SCHA

In this case, the desired confinement is accomplished by introducing the following form of potential: $v_{c}=\infty,$ when $0<r\leq r_{a}$, $r\geq r_{b}$ and $v_{c}=0$ when $r_{a}<r<r_{b}$, where $r_{a},~{}r_{b}$ signify the inner and outer radius respectively. Expectation values of such potential can then be worked out as below,

[TABLE]

which, upon application of the property of Reimann integral provide,

[TABLE]

The first and third integrals contributes zero as wave function vanishes in these two regions. On the contrary, at $r_{a}<r<r_{b}$ region $v_{c}=0$; thus the second integral disappears. Same argument can be used to write,

[TABLE]

The second equality hold because $\langle v_{c}(r)\hat{T}\rangle_{n,\ell}=0$. Likewise, $\langle\hat{V}^{2}\rangle_{n,\ell}$ may be expressed as,

[TABLE]

since $\langle v(r)v_{c}(r)\rangle_{n,\ell}=\langle v_{c}(r)v(r)\rangle_{n,\ell}=\langle v_{c}(r)^{2}\rangle_{n,\ell}=0$. Next, utilizing $\langle v_{c}(r)\rangle_{n,ell}=0$, we get,

[TABLE]

Collecting all these fact, we can write the final expressions for SCHA,

[TABLE]

Equation (45) explains that, similar to three previous confined cases, SCHA satisfies the results given in Eq. (5). As before, the role of $v_{c}$ is to incorporated the effect of boundary on the wave function. As mentioned earlier, closed form analytical solutions are unavailable in this case as yet; we have employed the GPS method to extract eigenvalues and eigenfunctions of a definite state. Table IV produces the calculated values of various quantities for ground and two excited states ($1s,~{}2s,~{}2p$) of SCHA at five chosen sets of $r_{a},r_{b}$ values. The equality of four quantities at all shells once again justifies the validity of relations derived in Eq. (5). No literature is available to compare the computed expectation values.

III.2 Impenetrable, smooth/homogeneous confinement

One such potential, $v(r)=-\frac{1}{r}+\frac{1}{2}\omega r^{2}$ was first proposed in patil04 , to mimic the quantum-dot structure. Later, in 2012, this was modified into a generalized form katriel12 , v(r)=-\frac{1}{r}+\big{(}\frac{r}{r_{c}}\big{)}^{k} \big{(}k>1 and real; \frac{1}{2}\omega=(\frac{1}{r_{c}})^{k}\big{)}. At a fixed $r_{c}$, the perturbing potential takes following form,

[TABLE]

The required expectation values for this potential will then be given by,

[TABLE]

and

[TABLE]

Ultimately, we get the virial expression from Eq. (5) in following form,

[TABLE]

One striking difference from the previous impenetrable, sharp potentials is that, here the perturbing potential contributes in to the final form of expression. Now for the illustration, we choose $k=2$. In this scenario (finite positive $k$), at very small $r_{c}$, the potential blows up sharply, at $r_{c}\rightarrow\infty$ it behaves as free system, and at other definite $r_{c}$, it rises with $r$. Table V offers sample results for $\mathcal{E}_{n,\ell}$ and related quantities of Eqs. (5), for $1s,2s,2p$ states of HICHA at six selected $r_{c}$, namely $0.1,0.2,0.5,1,5,\infty$. Energies for these states, at $r_{c}=1$ could be compared with the known literature values patil04 , which shows reasonable agreement. The other computed quantities could not be compared due to the lack of reference values. Clearly, similar to the previous cases, these results also establish the applicability of our newly proposed virial-like expressions in HICHA.

III.3 Penetrable, sharp confinement

In this context, we have chosen the potential having following form,

[TABLE]

where $V_{0}$ is a positive constant. It was first introduced by koo79 in 1979. The expectation values in this case, are given by following expressions,

[TABLE]

where the property of Reimann integral has been used. Now,

[TABLE]

After some algebra, we eventually obtain the following expressions,

[TABLE]

Thus, analogous to HICHA, here also the perturbing term $V_{0}$ contributes to the expectation values. Table VI presents results of $\mathcal{E}_{n,\ell}$, along with the respective expectation values for $1s,2s,2p$ states of SPCHA at six selected sets of $\{V_{0},r_{c}\}$ values. Very few literature results are available, except the ground-state energy, which are duly quoted; our results display nice agreement. these results promote the validity of this virial like expression for SPCHA.

III.4 Penetrable, smooth/homogeneous confinement

One example of such potential is $v(r)=-\frac{1}{r}+v_{p,h}(r)$, where $v_{p,h}(r)=\frac{U_{0}}{e^{w\left(1-\frac{r}{r_{c}}\right)}+1}$, $U_{0},w$ both are positive and real. Its importance and utility has been discussed in aquino13 in the context of explaining the interactions present in artificial atoms. The relevant expressions can be derived as follows,

[TABLE]

and

[TABLE]

Eventually we arrive at the following expression after some algebra,

[TABLE]

Thus we notice that, similar to HICHA and SPCHA, here also the perturbing term $v_{p,h}(r)$ remains in the final expression.

In order to explain the result for HPCHA, we have taken $w=1000$ and $U_{0}=10$ as the potential parameters. Table VII reports the calculation of $\mathcal{E}_{n,\ell}$, $\left(\Delta\hat{T}_{n,\ell}\right)^{2}$, $\left(\Delta\hat{V}_{n,\ell}\right)^{2}$, $\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle TV\rangle_{n,\ell}$ and $\langle T\rangle_{n,\ell}\langle V\rangle_{n,\ell}-\langle VT\rangle_{n,\ell}$ for $1s,2s,2p$ states at five selected $r_{c}$, namely $0.1,0.2,0.5,1,5$. Apart form that, the last column clearly suggests at $r_{c}\rightarrow\infty$ and $U\rightarrow 0$ this system merges to FHA. These results, like the previous cases, demonstrate that relation (5) is valid for HPCHA. Ground-state energies at all these $r_{c}$’s are compared with the available literature results. No further comparison could be made due to lack of data.

IV Future and Outlook

A new virial-like relation ($(\Delta\hat{T}_{n})^{2}=(\Delta\hat{V}_{n})^{2}$) has been proposed for free, and confined quantum systems, by invoking SE and HVT. This can be used as an essential condition for an eigenstate to obey. Besides this, Eq. (5) in its complete form has been proven to be a sufficient condition for these bound, stationary states to obey. Generalized expressions have been derived for impenetrable, penetrable, and shell-confined quantum systems along with the sharp and smooth situations. The change in boundary condition does not influence the form of these relations. Their applicability has been tested and verified by doing pilot calculations on quantum harmonic oscillator and H atom–a total of seven different confining potentials, as well as the respective free systems. In all cases these conditions are found to be obeyed. In impenetrable and sharp (hard) confinement condition the perturbing term is not contributing in the final expression. But in impenetrable-smooth, penetrable-sharp, and penetrable-smooth cases it participates in the eventual form. There are several open questions that may lead to important conclusions, and require further scrutiny, such as, use of these sufficient conditions in the context of determining optimized wave function for various quantum systems, in both ground and excited states. Importantly, one can perform unconstrained optimization (without employing the orthogonality criteria) of trial states by adopting this condition. A parallel inspection on many-electron systems would be highly desirable.

V Acknowledgement

Financial support from DST SERB, New Delhi, India (sanction order: EMR/2014/000838) is gratefully acknowledged.