Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

Abstract

Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

COMMENTS ABOUT THE DIRICHLET BOUNDARY CONDITION

It is known that in spherical coordinates 3-dimensional wave function is represented as

$$\psi \left( r \right) = R\left( r \right)Y_{l}^{m}\left( {\theta ,\varphi } \right) = \frac{{u\left( r \right)}}{r}Y_{l}^{m}\left( {\theta ,\varphi } \right).$$

((A.1))

Correspondingly, after rewritten the Laplacian in terms of polar coordinates two form of radial equations are derived

$$\begin{gathered} - \frac{{{{\hbar }^{2}}}}{{2m}}\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} + \frac{2}{r}\frac{d}{{dr}} - \frac{{l\left( {l + 1} \right)}}{{{{r}^{2}}}}} \right] \\ \times \,\,R\left( r \right) + V\left( r \right)R\left( r \right) = ER\left( r \right), \\ \end{gathered} $$

((A.2))

and

$$ - \frac{{{{\hbar }^{2}}}}{{2m}}\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} - \frac{{l\left( {l + 1} \right)}}{{{{r}^{2}}}} - V\left( r \right)} \right]u\left( r \right) = Eu\left( r \right).$$

((A.3))

P.A. Dirac wrote [34]: “Our equations … strictly speaking are not correct, but the error is restricted by only one point $r = 0$. It is necessary perform a special investigation of solutions of wave equations, that are derived by using the polar coordinates, to be convince are they valid in the point $r = 0$ (p. 161)”.

Let us discuss briefly the essence of this problem. In the teaching books and scientific articles two methods were applied in the transition from (A.2) to (A.3):

(1) The substitution

$$R\left( r \right) = \frac{{u\left( r \right)}}{r}$$

((A.4))

into Eq. (A.2) or

(2) Replacement of the differential expression

$$\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} + \frac{2}{r}\frac{d}{{dr}}} \right] \to \frac{1}{r}\frac{{{{d}^{2}}}}{{d{{r}^{2}}}}r.$$

((A.5))

Now we demonstrate that in both cases the mistakes are made.

After the substitution (A.4) we obtain (only the change in Laplacian is displayed)

$$\begin{gathered} \frac{1}{r}\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} + \frac{2}{r}\frac{d}{{dr}}} \right]u\left( r \right) + u\left( r \right)\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} + \frac{2}{r}\frac{d}{{dr}}} \right]\left( {\frac{1}{r}} \right) \\ + \,\,2\frac{{du}}{{dr}}\frac{d}{{dr}}\left( {\frac{1}{r}} \right). \\ \end{gathered} $$

((A.6))

It is identity. Now the last term cancels the first derivative term in the parenthesis and there remains

$$\frac{1}{r}\frac{{{{d}^{2}}u}}{{d{{r}^{2}}}} + u\left[ {\frac{{{{d}^{2}}}}{{d{{r}^{2}}}} + \frac{2}{r}\frac{d}{{dr}}} \right]\left( {\frac{1}{r}} \right).$$

((A.7))

The last term is zero, if we calculate it naively. But really it is a delta function [11]. So we obtain for above expression

$$\frac{1}{r}\frac{{{{d}^{2}}u}}{{d{{r}^{2}}}} - 4\pi {{\delta }^{{\left( 3 \right)}}}\left( r \right).$$

((A.8))

Therefore the representation of the Laplacian operator in the form (A.5) is not valid everywhere. Results are different in the one point, $r = 0.$

If we take into account this fact, we obtain the correct form of equation for the reduced wave function (using polar coordinates also for the delta function)

$$\begin{gathered} r\left( { - \frac{{{{d}^{2}}u\left( r \right)}}{{d{{r}^{2}}}} + \frac{{l\left( {l + 1} \right)}}{{{{r}^{2}}}}} \right) + \delta \left( r \right)u\left( r \right) \\ - \,\,\frac{{2m}}{{{{\hbar }^{2}}}}\left( {E - V\left( r \right)} \right)ru\left( r \right) = 0. \\ \end{gathered} $$

((A.9))

We see that the additional term, containing the delta function, vanishes only when

$$u\left( 0 \right) = 0.$$

((A.10))

Only in this case we can return to the usual form of reduced equation. Therefore the usual radial equation arises only together with the condition (A.10), which coincides to the Dirichlet boundary condition. No other boundary conditions are permissible for the reduced wave function [34, 35].

Therefore, when you use the reduced Schrodinger equation it is necessary to impose the reduced wave function $u\left( r \right)$ by the Dirichlet boundary condition (A.10) both for regular as well as singular potentials.

Among the listed papers the 3-dimensional case is considered only in [8]. There are two examples for the Coulomb and oscillator potentials, studied by the reduced Schrodinger equation. In addition the Robin boundary condition $u{\kern 1pt} '\left( 0 \right) + \alpha u\left( 0 \right) = 0$ is used, which is not correct as follows from above consideration. Here$\alpha$ is a self-adjoint extension parameter. They wrote: ’’The caseof wave functions that vanish at the origin (the standard or the Dirichlet boundary condition for the hydrogen atom) is recovered when $\alpha \to - \infty$ and $u\left( 0 \right) \to 0,$ while the product $u\left( 0 \right)\alpha$ remains finite”. In $l = 0$ for Coulomb potential $V = - \frac{k}{r}$ they write the modified form of virial theorem

$$A - \left\langle {{{u}_{n}}} \right|\frac{\kappa }{r}\left| {{{u}_{n}}} \right\rangle = 2{{E}_{n}}.$$

((A.11))

Here $A$ stands for the extra contribution, $\Pi$ in our notation, they derived (regularized version)

$${{A}_{\varepsilon }} = - \frac{{{{\hbar }^{2}}}}{{2m}}{{\left| {{{u}_{n}}\left( 0 \right)} \right|}^{2}}\left( {\xi \ln \left| \xi \right|\varepsilon + \xi - \alpha + ...} \right).$$

((A.12))

In the limit $\alpha \to - \infty ,$$u\left( 0 \right) \to 0,$ it follows

$${{A}_{\varepsilon }} = - \frac{{{{\hbar }^{2}}}}{{2m}}{{\left| {{{u}_{n}}\left( 0 \right)} \right|}^{2}}\alpha = \frac{{{{\hbar }^{2}}}}{{2m}}u_{n}^{ * }\left( 0 \right){{u}_{n}}\left( 0 \right)\alpha .$$

((A.13))

As $u\left( 0 \right)\alpha$ remains finite and $u\left( 0 \right) \to 0,$ one obtains ${{A}_{\varepsilon }} = 0$ and according to (A.11), we return to the usual virial theorem $- \left\langle {{{u}_{n}}} \right|\frac{\kappa }{r}\left| {{{u}_{n}}} \right\rangle = 2{{E}_{n}},$ from (5.4) for $s = l = 0$. Here $k = {{e}^{2}}.$

The same correspondence happens in case of harmonic oscillator.

Therefore, our modified virial theorem with Dirichlet boundary condition for $l = 0$ states gives the same results, as extended radial Hamiltonian with the Robin boundary condition [8]. In our case the procedure of self-adjoint extension is not need.

We have modified the more general hypervirial theorem in the framework of Dirichlet boundary condition, therefore Eqs. (2.16), (2.17) are new.