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<p><b>New page</b></p><div>[[Laplacian in spherical coordinates]] refers to the three-dimensional [[Laplace operator]]<br />
<br />
$\displaystyle \Delta =\nabla^2= \partial_x^2 + \partial_y^2 + \partial_z^2 $<br />
<br />
where $x$, $y$, $z$ are cartesian coordinates, and $\partial_n$ differentiates with respect to $n$th coordinate, where $n$ = "$x$", "$y$" or "$z$", and the Schroediner equation for the hydrogen atom.<br />
<br />
==Spherical system of coordinates==<br />
Certain confusion can be related with different notations, denoted with the same term "spherical coordinates".<br />
<!-- after the change to the [[spheric system of coordinates]] $r$, $\theta$, $\phi$ such that <br />
!--><br />
<br />
In Quantum Mechanics, often, the following system is used:<br />
<br />
$x= r \cos(\theta) \cos(\phi)$<br />
<br />
$y= r \cos(\theta) \sin(\phi)$<br />
<br />
$z= r \sin(\theta)$<br />
<br />
In Geography and Astronomy, meaning of variable $\theta$ is different:<br />
<br />
$x= r \sin(\theta) \cos(\phi)$<br />
<br />
$y= r \sin(\theta) \sin(\phi)$<br />
<br />
$z= r \cos(\theta)$<br />
<br />
Below, the "quantum mechanical" notations are used.<br />
In other articles in TORI, the system of notations should be specified each time before to use the spherical coordinates.<br />
Various publications use different notations, including different sense the spherical coordinates, their names, and different order of arguments of functions dependent on these coordinates.<br />
<br />
==Laplacian==<br />
<br />
In spherical coordinates, the Laplacian can be written as follows:<br />
<br />
$\displaystyle \Delta = \frac{1}{r^2} \partial_r (r^2 \partial_r) + \frac{1}{r^2 s} \partial_\theta (s \partial_\theta) + \frac{1}{r^2 s^2} \partial_\phi^2 $<br />
<br />
where $s=\sin(\theta)$<br />
<br />
[[Laplacian in spherical coordinates]] is often used for separation of variables in linear equations with central symmetry.<br />
In particular, it is useful for the [[schroedinger equation]] in the coordinate representation, while the potential has central symmetry.<br />
This separation is used in [[Atomic physics]] and [[molecular physics]].<br />
<br />
==Separation of variables==<br />
<br />
For the single particle of mass $m$ in the 3-dimensional space with potential dependent only on $r$, the [[stationary schroedinger]]<br />
can be written as follows:<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m}<br />
\Delta \psi + (U( r ) -E) \psi =0$<br />
<br />
Search for the solution $\psi$ in the following form:<br />
<br />
$\psi=\psi(r,\theta,\phi)= R( r) \Theta(\theta) \Phi(\phi)$<br />
<br />
The substitution gives:<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \frac{1}{r^2} \partial_r (r^2 R') \Theta \Phi + <br />
\frac{-\hbar^2}{2m} R \frac{1}{r^2 s} \partial_\theta (s \Theta' ) \Phi +<br />
\frac{-\hbar^2}{2m} R \Theta \frac{1}{r^2 s^2} \Phi''<br />
+ (U(r ) -E) R\Theta\Phi =0<br />
$<br />
<br />
Multiplication by $r^2/(R \Theta \Phi)$ gives<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) +<br />
\frac{-\hbar^2}{2m} \frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta +<br />
\frac{-\hbar^2}{2m} \frac{1}{ s^2} \Phi''/\Phi<br />
=0<br />
$<br />
<br />
The first two terms do not depend on $\theta$ not on $\phi$, while the last two terms do not depend on $r$.<br />
Hence, sum of the first two terms should be constant, opposite to that of the sum of the last two terms.<br />
It is convenient to denote this constant <br />
<br />
$\frac{-\hbar^2}{2m} L$<br />
<br />
where $L$ is some constant. In such a way,<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) =\frac{-\hbar^2}{2m} L<br />
$<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta +<br />
\frac{-\hbar^2}{2m} \frac{1}{ s^2} \Phi''/\Phi<br />
= - \frac{-\hbar^2}{2m} L<br />
$<br />
<br />
These two equations can be dented as "radial equation" and "orbital equation". <br />
The orbital equation can be simplified as follows:<br />
<br />
$\displaystyle<br />
\frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta +<br />
\frac{1}{ s^2} \Phi''/\Phi<br />
+ L =0<br />
$<br />
<br />
Or, even simpler,<br />
<br />
$\displaystyle<br />
s\partial_\theta (s \Theta' ) /\Theta + s^2 L =- \Phi''/\Phi<br />
$<br />
<br />
The left hand side of this equation does not depend on $\phi$, and the right hand side does not depend on $\theta$.<br />
Hence, each of these prats should be constant, and we get the second constant of separation, let it be $m^2$:<br />
<br />
$-\Phi''/\Phi=m^2$<br />
<br />
$\displaystyle s\partial_\theta (s \Theta' ) /\Theta + s^2 L =m^2<br />
$<br />
<br />
The first of these equation has obvious solutions <br />
$\Phi(\phi)=\cos(m\phi)$, <br />
$\Phi(\phi)=\sin(m\phi)$, <br />
or any linear combination of these two solutions; continuity of the wave function implies that $m$ is integer.<br />
<br />
Equation for $\Theta$ is called [[Asimutal equation]]; it can be rewritten in the following form:<br />
<br />
$\displaystyle s\partial_\theta (s \Theta' ) + s^2 L \Theta - m^2 \Theta=0 $<br />
<br />
where, as before, $\Theta=\Theta(\theta)$, and $s=\sin(\theta)$<br />
<br />
Sometimes it is written also in the following form<br />
<br />
$\displaystyle \frac{1}{s} \partial_\theta ( s \Theta' ) + \left( L - \frac{m^2}{s^2} \right) \Theta=0 $<br />
<br />
Its solutions can be expressed in therm of the [[Legendre function]]s. For existence of the regular (periodic) solution, parameter $L$ should have value from certain discrete set of numbers, namely, $L=\ell(\ell\!+\!1)$ for some non–negative integer $\ell$.<br />
<br />
==Radial part==<br />
At $L=\ell(\ell\!+\!1)$, for $R$, the [[Radial equation for hydrogen atom]]:<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) =\frac{-\hbar^2}{2m} L<br />
$<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \frac{1}{r^2} \partial_r (r^2 R') - \frac{-\hbar^2}{2m} \frac{\ell(\ell\!+\!1)}{r^2} R+ (U(r ) -E) R = 0<br />
$<br />
<br />
The solution depends on potential $U$. For applications in atomic physics, the most important is case <br />
$U(r )=-e^2/r$ where $e$ is electron charge. <br />
This quantity should not be confused with integer parameter $\ell$, nor with mathematical constant <br />
$\mathrm e=\exp(1)\approx 2.71828$ ; as usually, fundamental mathematical constants <br />
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, $\mathrm e$, $\mathrm i$) are typed with Roman font; other parameters, even physical constants, should appear with Italics.<br />
<br />
$\displaystyle<br />
\frac{-\hbar^2}{2m} \frac{1}{r^2} \partial_r (r^2 R') - \frac{-\hbar^2}{2m} \frac{\ell(\ell\!+\!1)}{r^2} R+ \left(-\frac{e^2}{r} -E \right) R = 0<br />
$<br />
<br />
$\displaystyle<br />
\frac{1}{r^2} \partial_r (r^2 R') <br />
- \frac{\ell(\ell\!+\!1)}{r^2} R <br />
+ \left( -\frac{2m} {-\hbar^2} \frac{e^2}{r} -\frac{2m} {-\hbar^2} E \right) R = 0<br />
$<br />
<br />
$\displaystyle<br />
\frac{1}{r^2} \partial_r (r^2 R') <br />
- \frac{\ell(\ell\!+\!1)}{r^2} R <br />
+ \left( \frac{2m} {\hbar^2} \frac{e^2}{r} + \frac{2m} {\hbar^2} E \right) R = 0<br />
$<br />
<br />
Let $R(r )=F(r/a)$<br />
where $a$ has sense of size of the atom. Then<br />
<br />
$\displaystyle<br />
\frac{1}{x^2} \partial_r (x^2 F'(x)) <br />
- \frac{\ell(\ell\!+\!1)}{x^2} F(x) <br />
+ \left( \frac{2m} {\hbar^2} \frac{e^2 a}{r} + \frac{2m a^2} {\hbar^2} E \right) F(x) = 0<br />
$<br />
<br />
The natural choice <br />
<br />
$\displaystyle a=\frac{\hbar^2}{2 m e^2}$<br />
<br />
leads to equation<br />
<br />
$\displaystyle<br />
\frac{1}{x^2} \partial_x (x^2 F'(x)) <br />
- \frac{\ell(\ell\!+\!1)}{x^2} F(x) <br />
+ \left( \frac{1}{x} + v \right) F(x) = 0<br />
$<br />
<br />
where $\displaystyle v=\frac{\hbar^2}{2 m e^4}E$<br />
<br />
<br />
The solution can be expressed through the [[Laguerre Polynomial]].<br />
In Mathematica, they are denoted with identifier [[LaguerreL]]<br />
<ref><br />
http://mathworld.wolfram.com/LaguerrePolynomial.html<br />
</ref><ref><br />
http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html<br />
Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld--A Wolfram Web Resource.<br />
</ref><br />
<br />
$\displaystyle<br />
L_n(x)=\sum_{k=0}^n \frac{(-1)^k}{k!} \mathrm{Binomial}(n,k) \, x^k $<br />
<br />
$\displaystyle<br />
L_n^k(x)=\sum_{k=0}^n (-1)^k \, \partial_x^k L_{n+k}(x)<br />
=\sum_{k=0}^n (-1)^k \frac{ (n\!+\!k)!}{(n\!-\!m)!\, (k\!+\!m)!\, m!} \, x^m<br />
$<br />
<br />
==References==<br />
<references/><br />
<br />
https://en.wikipedia.org/wiki/Laplace_operator<br />
<br />
http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html<br />
<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html <br />
Hydrogen Schrodinger Equation.<br />
<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html#c1<br />
<br />
http://pskgu.ru/ebooks/tihonov.html<br />
А. Н. ТИХОНОВ, А. А. САМАРСКИЙ. УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ. Часть II. Сферические функции. § 1. Полиномы Лежандра. (In Russian)<br />
<br />
http://www.nat.vu.nl/~wimu/EDUC/MNW-lect-2.pdf<br />
Wim Ubachs. Quantum Mechanics and the hydrogen atom. 2016.<br />
<br />
==Keywords==<br />
<br />
[[Atomic physics]],<br />
[[Asimutal equation]],<br />
[[Hydrogen wave function]],<br />
[[Molecular physics]],<br />
[[Laplacian]],<br />
[[Legendre function]],<br />
[[LegendreP]],<br />
[[Quantum mechanics]],<br />
<br />
[[Category:Atomic physics]]<br />
[[Category:Differential operator]]<br />
[[Category:English]]<br />
[[Category:Legendre function]]<br />
[[Category:LegendreP]]<br />
[[Category:Molecular physics]]<br />
[[Category:Quantum mechanics]]</div>Maintenance script