Application of Modified Hypervirial and Ehrenfest Theorems and Several Their Consequences

Abstract

It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.

APPENDIX

Let consider various examples and find corresponding integrals:

(2) The Hulten potential:

$$V = - \frac{{{{V}_{0}}}}{{{{e}^{{\frac{r}{a}}}} - 1}}\,.$$

(A.1)

The solution of the Schrodinger equation in this case ($l = 0$) is [36]

$$u = rR = {{C}_{{0{{n}_{r}}}}}{{e}^{{ - \eta r}}}F\left( {\alpha ,\beta ,\gamma ;{{e}^{{ - \frac{r}{a}}}}} \right).$$

(A.2)

Here

$$\begin{gathered} \eta = \sqrt { - \frac{{2m{{E}_{{0,{{n}_{r}}}}}}}{{{{\hbar }^{2}}}}} ,\,\,\,\,\alpha = \varepsilon + \sqrt {{{\varepsilon }^{2}} + {{\lambda }^{2}}} , \\ \beta = \varepsilon - \sqrt {{{\varepsilon }^{2}} + {{\lambda }^{2}}} ,\,\,\,\,\varepsilon = \eta a,\,\,\,\,\lambda = {{\left( {\frac{{2m{{a}^{2}}{{V}_{0}}}}{{{{\hbar }^{2}}}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}. \\ \end{gathered} $$

(A.3)

$F$is a Hypergeometric function, ${{C}_{{0{{n}_{r}}}}}$—normalization constant, and energy levels are obtainable from the condition

$$F\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right) = 0\,.$$

(A.4)

For this potential Eq. (4.18) takes the form

$$\begin{gathered} \frac{{C_{1}^{2}{{\hbar }^{2}}}}{{2m}} = \left\langle {\frac{{dV}}{{dr}}} \right\rangle = \frac{{{{V}_{0}}}}{a}{{\left( {{{C}_{{0{{n}_{r}}}}}} \right)}^{2}} \\ \times \,\,\int\limits_0^\infty {{{e}^{{ - 2\eta r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr. \\ \end{gathered} $$

(A.5)

Behavior of (A.2) at the origin gives

$$\mathop {\lim }\limits_{r \to 0} u(r) = - {{C}_{{0{{n}_{r}}}}}F'\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\frac{r}{a}\,.$$

(A.6)

or

$${{C}_{1}} = - {{C}_{{0{{n}_{r}}}}}F{\kern 1pt} '\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\frac{1}{a}\,.$$

(A.7)

Then Eq. (4.18) gives

$$\begin{gathered} \int\limits_0^\infty {{{e}^{{ - 2\eta r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr \\ = - \frac{{a{{\hbar }^{2}}}}{{2m{{V}_{0}}{{a}^{2}}}}F{\kern 1pt} {{'}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\,, \\ \end{gathered} $$

(A.8)

or, accounting notations (A.3), we have

$$\begin{gathered} \int\limits_0^\infty {{{e}^{{ - 2\eta r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr \\ = - \frac{a}{{{{\lambda }^{2}}}}F{\kern 1pt} {{'}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\,. \\ \end{gathered} $$

(A.9)

But from (A.4) and

$$2\eta = \frac{{\alpha + \beta }}{a}\,,$$

(A.10)

and finally, we get

$$\begin{gathered} \int\limits_0^\infty {{{e}^{{ - \frac{{\alpha + \beta }}{a}r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr \\ = - \frac{a}{{{{\lambda }^{2}}}}{{{F'}}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\,. \\ \end{gathered} $$

(A.11)

Moreover, again from (A.3)

$${{\lambda }^{2}} = - \alpha \beta \,.$$

(A.12)

Therefore

$$\begin{gathered} \int\limits_0^\infty {{{e}^{{ - \frac{{\alpha + \beta }}{a}r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr \\ = \frac{a}{{\alpha \beta }}{{{F'}}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,1} \right)\,. \\ \end{gathered} $$

(A.13)

Now we can use the recurrence relation for derivatives of this function [37]

$$F{\kern 1pt} '\left( {a,b,c;z} \right) = \frac{{ab}}{c}F\left( {a + 1,b + 1,c + 1;z} \right)\,.$$

(A.14)

And at the end we have

$$\begin{gathered} \int\limits_0^\infty {{{e}^{{ - \frac{{\alpha + \beta }}{a}r}}}{{e}^{{\frac{r}{a}}}}{{{\left( {{{e}^{{\frac{r}{a}}}} - 1} \right)}}^{2}}{{F}^{2}}\left( {\alpha ,\,\beta ,\,\gamma ;\,{{e}^{{ - \frac{r}{a}}}}} \right)} dr \\ = \frac{{\alpha \beta a}}{{{{\gamma }^{2}}}}{{F}^{2}}\left( {\alpha + 1,\,\beta + 1,\,\gamma + 1;\,1} \right)\,. \\ \end{gathered} $$

(A.15)

(3) The Morse potential:

$$V\left( r \right) = D\left( {{{e}^{{ - 2\alpha x}}} - 2{{e}^{{ - \alpha x}}}} \right);\,\,\,\,x = \frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}.$$

(A.16)

This potential has a wide application in Chemistry for studying two-atomic molecules.

The solution of the s-wave Schrodinger equation looks like [36]

$$u = rR = {{C}_{{0{{n}_{r}}}}}{{y}^{{\frac{\beta }{\alpha }}}}{{e}^{{ - \frac{y}{2}}}}F\left( {a,c,y} \right)\,.$$

(A.17)

Here $F$is a confluent hypergeometric function, and the following notations are used

$$\begin{gathered} y = \frac{{2\gamma }}{\alpha }{{e}^{{ - \alpha x}}};\,\,\,\,c = 2\frac{\beta }{\alpha } + 1\,;\,\,\,\,\,a = \frac{1}{2}c - \frac{\gamma }{\alpha }; \\ {{\beta }^{2}} = - \frac{{2mEr_{0}^{2}}}{{{{\hbar }^{2}}}} > 0,\,\,\,\,{{\gamma }^{2}} = \frac{{2mDr_{0}^{2}}}{{{{\hbar }^{2}}}}\,. \\ \end{gathered} $$

(A.18)

Eigenvalues equation is

$$u\left( 0 \right) = F\left( {a,c,{{y}_{0}}} \right) = 0;\,\,\,\,{{y}_{0}} = \frac{{2\gamma }}{\alpha }{{e}^{\alpha }}.$$

(A.19)

Proceeding in a similar way as above, one finds the following equality

$$\begin{gathered} \int\limits_0^\infty {{{F}^{2}}\left( {a,c,\frac{{2\gamma }}{\alpha }{{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)\left( {{{e}^{{ - 2\alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}} - {{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)dr} \\ = \frac{{{{r}_{0}}}}{{2{{\gamma }^{2}}\alpha }}{{y}_{0}}^{{\frac{{2\beta }}{\alpha } + 2}}{{e}^{{ - {{y}_{0}}}}}\frac{{{{a}^{2}}}}{{{{c}^{2}}}}{{\left[ {F\left( {a + 1,c + 1;{{y}_{0}}} \right)} \right]}^{2}}\,. \\ \end{gathered} $$

(A.20)

Consider now relation (4.42), which gives

$$\begin{gathered} D\left\langle {r\left( {{{e}^{{ - 2\alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}} - {{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)} \right\rangle \\ = 2D\left\langle {\left( {{{e}^{{ - 2\alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}} - {{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)} \right\rangle - 2E. \\ \end{gathered} $$

(A.21)

In explicit form this equation means, according to notations (A.18)

$$\begin{gathered} \int\limits_0^\infty {{{F}^{2}}\left( {a,c,\frac{{2\gamma }}{\alpha }{{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)\left( {{{e}^{{ - 2\alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}} - {{e}^{{ - \alpha \left( {\frac{{r - {{r}_{0}}}}{{{{r}_{0}}}}} \right)}}}} \right)rdr} \\ = \frac{{{{r}_{0}}}}{{{{\gamma }^{2}}\alpha }}{{y}_{0}}^{{c + 1}}{{e}^{{ - {{y}_{0}}}}}\frac{{{{a}^{2}}}}{{{{c}^{2}}}} \\ \times \,\,{{\left[ {F\left( {a + 1,c + 1;{{y}_{0}}} \right)} \right]}^{2}} + \frac{{{{\alpha }^{2}}{{{\left( {c - 1} \right)}}^{2}}}}{{4{{\gamma }^{2}}}}\,. \\ \end{gathered} $$

(A.22)

(4) Wood–Saxon potential:

$$V\left( r \right) = - \frac{{{{V}_{0}}}}{{1 + {{e}^{{\frac{{r - R}}{a}}}}}};\,\,\,\,a \ll R$$

(A.23)

It is applied for description of neutron-nucleus interaction. For s-wave Schrodinger equation the solution is [36]

$$\begin{gathered} u = rR = {{C}_{{0{{n}_{r}}}}}{{y}^{\nu }}{{\left( {1 - y} \right)}^{\mu }} \\ \times \,\,F\left( {\mu + \nu ,\mu + \nu + 1,\,2\nu + 1;\,y} \right)\,. \\ \end{gathered} $$

(A.24)

Here $F$ is a hypergeometric function and

$$\begin{gathered} y = \frac{1}{{1 + {{e}^{{\frac{{r - R}}{a}}}}}};\,\,\,\,\nu = \beta ;\,\,\,\,\,{{\mu }^{2}} = {{\beta }^{2}} - {{\gamma }^{2}}; \\ {{\beta }^{2}} = - \frac{{2mE{{a}^{2}}}}{{{{\hbar }^{2}}}} > 0,\,\,\,\,{{\gamma }^{2}} = \frac{{2m{{V}_{0}}{{a}^{2}}}}{{{{\hbar }^{2}}}}\,. \\ \end{gathered} $$

(A.25)

Spectrum is derived by the condition

$$\begin{gathered} u\left( 0 \right) = F\left( {\mu + \nu ,\mu + \nu + 1,2\nu + 1;{{y}_{0}}} \right) = 0; \\ {{y}_{0}} = \frac{1}{{1 + {{e}^{{ - \frac{R}{a}}}}}}. \\ \end{gathered} $$

(A.26)

It is easy to show that

$$\begin{gathered} \mathop {\lim }\limits_{r \to 0} u(r) = - {{C}_{{0{{n}_{r}}}}}{{y}_{0}}^{{\nu + 1}}{{\left( {1 - {{y}_{0}}} \right)}^{\mu }} \\ \times \,\,F{\kern 1pt} '\left( {\mu + \nu ,\mu + \nu + 1,2\nu + 1;{{y}_{0}}} \right)\frac{r}{a}\frac{{{{e}^{{ - \frac{R}{a}}}}}}{{1 + {{e}^{{ - \frac{R}{a}}}}}}. \\ \end{gathered} $$

(A.27)

So

$$\begin{gathered} {{C}_{1}} = - {{C}_{{0{{n}_{r}}}}}{{y}_{0}}^{{\nu + 1}}{{\left( {1 - {{y}_{0}}} \right)}^{\mu }} \\ \times \,\,F{\kern 1pt} '\left( {\mu + \nu ,\mu + \nu + 1,2\nu + 1;{{y}_{0}}} \right)\frac{1}{a}\frac{{{{e}^{{ - \frac{R}{a}}}}}}{{1 + {{e}^{{ - \frac{R}{a}}}}}}. \\ \end{gathered} $$

(A.28)

Analogous consideration, as above, gives the following relation

$$\begin{gathered} \int\limits_0^\infty {{{F}^{2}}\left( {\mu + \nu ,\mu + \nu + 1,2\nu + 1;\frac{1}{{1 + {{e}^{{\frac{{r - R}}{a}}}}}}} \right)} \\ \times \,\,\frac{{{{e}^{{\frac{{r - R}}{a}}}}}}{{{{{\left[ {1 + {{e}^{{\frac{{r - R}}{a}}}}} \right]}}^{2}}}}dr = - \frac{{{{a}^{3}}}}{{{{\gamma }^{2}}}}\left[ {y_{0}^{{\nu + 1}}{{{\left( {1 - {{y}_{0}}} \right)}}^{\mu }}} \right. \\ \times \,\,{{\left. {F{\kern 1pt} '\left( {\mu + \nu ,\,\mu + \nu + 1,\,2\nu + 1;\,{{y}_{0}}} \right)\frac{1}{a}\frac{{{{e}^{{ - \frac{R}{a}}}}}}{{1 + {{e}^{{ - \frac{R}{a}}}}}}} \right]}^{2}}\,. \\ \end{gathered} $$

(A.29)

We do not find these integrals (A.15), (A.20), (A.22) and (A.29) in accessible to us Tables and Handbooks.