Laplacian in spherical coordinates

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Laplacian in spherical coordinates refers to the three-dimensional Laplace operator

$\displaystyle \Delta =\nabla^2= \partial_x^2 + \partial_y^2 + \partial_z^2 $

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.

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Spherical system of coordinates

Certain confusion can be related with different notations, denoted with the same term "spherical coordinates".

In Quantum Mechanics, often, the following system is used:

$x= r \cos(\theta) \cos(\phi)$

$y= r \cos(\theta) \sin(\phi)$

$z= r \sin(\theta)$

In Geography and Astronomy, meaning of variable $\theta$ is different:

$x= r \sin(\theta) \cos(\phi)$

$y= r \sin(\theta) \sin(\phi)$

$z= r \cos(\theta)$

Below, the "quantum mechanical" notations are used. In other articles in TORI, the system of notations should be specified each time before to use the spherical coordinates. Various publications use different notations, including different sense the spherical coordinates, their names, and different order of arguments of functions dependent on these coordinates.

Laplacian

In spherical coordinates, the Laplacian can be written as follows:

$\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 $

where $s=\sin(\theta)$

Laplacian in spherical coordinates is often used for separation of variables in linear equations with central symmetry. In particular, it is useful for the schroedinger equation in the coordinate representation, while the potential has central symmetry. This separation is used in Atomic physics and molecular physics.

Separation of variables

For the single particle of mass $m$ in the 3-dimensional space with potential dependent only on $r$, the stationary schroedinger can be written as follows:

$\displaystyle \frac{-\hbar^2}{2m} \Delta \psi + (U( r ) -E) \psi =0$

Search for the solution $\psi$ in the following form:

$\psi=\psi(r,\theta,\phi)= R( r) \Theta(\theta) \Phi(\phi)$

The substitution gives:

$\displaystyle \frac{-\hbar^2}{2m} \frac{1}{r^2} \partial_r (r^2 R') \Theta \Phi + \frac{-\hbar^2}{2m} R \frac{1}{r^2 s} \partial_\theta (s \Theta' ) \Phi + \frac{-\hbar^2}{2m} R \Theta \frac{1}{r^2 s^2} \Phi'' + (U(r ) -E) R\Theta\Phi =0 $

Multiplication by $r^2/(R \Theta \Phi)$ gives

$\displaystyle \frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) + \frac{-\hbar^2}{2m} \frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta + \frac{-\hbar^2}{2m} \frac{1}{ s^2} \Phi''/\Phi =0 $

The first two terms do not depend on $\theta$ not on $\phi$, while the last two terms do not depend on $r$. Hence, sum of the first two terms should be constant, opposite to that of the sum of the last two terms. It is convenient to denote this constant

$\frac{-\hbar^2}{2m} L$

where $L$ is some constant. In such a way,

$\displaystyle \frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) =\frac{-\hbar^2}{2m} L $

$\displaystyle \frac{-\hbar^2}{2m} \frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta + \frac{-\hbar^2}{2m} \frac{1}{ s^2} \Phi''/\Phi = - \frac{-\hbar^2}{2m} L $

These two equations can be dented as "radial equation" and "orbital equation". The orbital equation can be simplified as follows:

$\displaystyle \frac{1}{ s} \partial_\theta (s \Theta' ) /\Theta + \frac{1}{ s^2} \Phi''/\Phi + L =0 $

Or, even simpler,

$\displaystyle s\partial_\theta (s \Theta' ) /\Theta + s^2 L =- \Phi''/\Phi $

The left hand side of this equation does not depend on $\phi$, and the right hand side does not depend on $\theta$. Hence, each of these prats should be constant, and we get the second constant of separation, let it be $m^2$:

$-\Phi''/\Phi=m^2$

$\displaystyle s\partial_\theta (s \Theta' ) /\Theta + s^2 L =m^2 $

The first of these equation has obvious solutions $\Phi(\phi)=\cos(m\phi)$, $\Phi(\phi)=\sin(m\phi)$, or any linear combination of these two solutions; continuity of the wave function implies that $m$ is integer.

Equation for $\Theta$ is called Asimutal equation; it can be rewritten in the following form:

$\displaystyle s\partial_\theta (s \Theta' ) + s^2 L \Theta - m^2 \Theta=0 $

where, as before, $\Theta=\Theta(\theta)$, and $s=\sin(\theta)$

Sometimes it is written also in the following form

$\displaystyle \frac{1}{s} \partial_\theta ( s \Theta' ) + \left( L - \frac{m^2}{s^2} \right) \Theta=0 $

Its solutions can be expressed in therm of the Legendre functions. 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 $\ell$.

Radial part

At $L=\ell(\ell\!+\!1)$, for $R$, the Radial equation for hydrogen atom:

$\displaystyle \frac{-\hbar^2}{2m} \partial_r (r^2 R')/R +r^2 (U(r ) -E) =\frac{-\hbar^2}{2m} L $

$\displaystyle \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 $

The solution depends on potential $U$. For applications in atomic physics, the most important is case $U(r )=-e^2/r$ where $e$ is electron charge. This quantity should not be confused with integer parameter $\ell$, nor with mathematical constant $\mathrm e=\exp(1)\approx 2.71828$ ; as usually, fundamental mathematical constants (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.

$\displaystyle \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 $

$\displaystyle \frac{1}{r^2} \partial_r (r^2 R') - \frac{\ell(\ell\!+\!1)}{r^2} R + \left( -\frac{2m} {-\hbar^2} \frac{e^2}{r} -\frac{2m} {-\hbar^2} E \right) R = 0 $

$\displaystyle \frac{1}{r^2} \partial_r (r^2 R') - \frac{\ell(\ell\!+\!1)}{r^2} R + \left( \frac{2m} {\hbar^2} \frac{e^2}{r} + \frac{2m} {\hbar^2} E \right) R = 0 $

Let $R(r )=F(r/a)$ where $a$ has sense of size of the atom. Then

$\displaystyle \frac{1}{x^2} \partial_r (x^2 F'(x)) - \frac{\ell(\ell\!+\!1)}{x^2} F(x) + \left( \frac{2m} {\hbar^2} \frac{e^2 a}{r} + \frac{2m a^2} {\hbar^2} E \right) F(x) = 0 $

The natural choice

$\displaystyle a=\frac{\hbar^2}{2 m e^2}$

leads to equation

$\displaystyle \frac{1}{x^2} \partial_x (x^2 F'(x)) - \frac{\ell(\ell\!+\!1)}{x^2} F(x) + \left( \frac{1}{x} + v \right) F(x) = 0 $

where $\displaystyle v=\frac{\hbar^2}{2 m e^4}E$


The solution can be expressed through the Laguerre Polynomial. In Mathematica, they are denoted with identifier LaguerreL [1][2]

$\displaystyle L_n(x)=\sum_{k=0}^n \frac{(-1)^k}{k!} \mathrm{Binomial}(n,k) \, x^k $

$\displaystyle L_n^k(x)=\sum_{k=0}^n (-1)^k \, \partial_x^k L_{n+k}(x) =\sum_{k=0}^n (-1)^k \frac{ (n\!+\!k)!}{(n\!-\!m)!\, (k\!+\!m)!\, m!} \, x^m $

References

  1. http://mathworld.wolfram.com/LaguerrePolynomial.html
  2. http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld--A Wolfram Web Resource.

https://en.wikipedia.org/wiki/Laplace_operator

http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html Hydrogen Schrodinger Equation.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html#c1

http://pskgu.ru/ebooks/tihonov.html А. Н. ТИХОНОВ, А. А. САМАРСКИЙ. УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ. Часть II. Сферические функции. § 1. Полиномы Лежандра. (In Russian)

http://www.nat.vu.nl/~wimu/EDUC/MNW-lect-2.pdf Wim Ubachs. Quantum Mechanics and the hydrogen atom. 2016.

Keywords

Atomic physics, Asimutal equation‎, Hydrogen wave function, Molecular physics, Laplacian, Legendre function, LegendreP, Quantum mechanics,