Thomas–Fermi equation

Introducing the transformation ${\displaystyle z=y/x}$ converts the equation to

${\displaystyle {\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dz}{dx}}\right)-z^{3/2}=0}$

This equation is similar to Lane–Emden equation with polytropic index ${\displaystyle 3/2}$ except the sign difference. The original equation is invariant under the transformation ${\displaystyle x\rightarrow cx,\ y\rightarrow c^{-3}y}$. Hence, the equation can be made equidimensional by introducing ${\displaystyle y=x^{-3}u}$ into the equation, leading to

${\displaystyle x^{2}{\frac {d^{2}u}{dx^{2}}}-6x{\frac {du}{dx}}+12u=u^{3/2}}$

so that the substitution ${\displaystyle u=e^{t}}$ reduces the equation to

${\displaystyle {\frac {d^{2}u}{dt^{2}}}-7{\frac {du}{dt}}+12u=u^{3/2}.}$

If ${\displaystyle w(u)={\frac {du}{dt}}}$ then the above equation becomes

${\displaystyle w{\frac {dw}{du}}-7w+12u=u^{3/2}.}$

But this first order equation has no known explicit solution, hence, the approach turns to either numerical or approximate methods.

The equation has a particular solution ${\displaystyle y_{p}(x)}$, which satisfies the boundary condition that ${\displaystyle y\rightarrow 0}$ as ${\displaystyle x\rightarrow \infty }$, but not the boundary condition y(0)=1. This particular solution is

${\displaystyle y_{p}(x)={\frac {144}{x^{3}}}.}$

Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.[7] If the transformation ${\displaystyle x=1/t,\ w=yt}$ is introduced, the equation becomes

${\displaystyle t^{4}{\frac {d^{2}w}{dt^{2}}}=w^{3/2},\quad w(0)=0,\ w(\infty )\sim t.}$

The particular solution in the transformed variable is then ${\displaystyle w_{p}(t)=144t^{4}}$. So one assumes a solution of the form ${\displaystyle w=w_{p}(1+\alpha t^{\lambda })}$ and if this is substituted in the above equation and the coefficients of ${\displaystyle \alpha }$ are equated, one obtains the value for ${\displaystyle \lambda }$, which is given by the roots of the equation ${\displaystyle \lambda ^{2}+7\lambda -6=0}$. The two roots are ${\displaystyle \lambda _{1}=0.772,\ \lambda _{2}=-7.772}$, where we need to take the positive root to avoid the singularity at the origin. This solution already satisfies the first boundary condition (${\displaystyle w(0)=0}$), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary ${\displaystyle n}$

${\displaystyle W=w_{p}(1+\beta t^{\lambda })^{n}=[144t^{3}(1+\beta t^{\lambda })^{n}]t.}$

The second boundary condition will be satisfied if ${\displaystyle 144t^{3}(1+\beta t^{\lambda })^{n}=144t^{3}\beta ^{n}t^{\lambda n}(1+\beta ^{-1}t^{-\lambda })^{n}\sim 1}$ as ${\displaystyle t\rightarrow \infty }$. This condition is satisfied if ${\displaystyle \lambda n+3=0,\ 144\beta ^{n}=1}$ and since ${\displaystyle \lambda _{1}\lambda _{2}=-6}$, Sommerfeld found the approximation as ${\displaystyle \lambda =\lambda _{1},\ n=-3/\lambda _{1}=\lambda _{2}/2}$. Therefore, the approximate solution is

${\displaystyle y(x)=y_{p}(x)\{1+[y_{p}(x)]^{\lambda _{1}/3}\}^{\lambda _{2}/2}.}$

This solution predicts the correct solution accurately for large ${\displaystyle x}$, but still fails near the origin.

Enrico Fermi[8] provided the solution for ${\displaystyle x\ll 1}$ and later extended by Edward B. Baker.[9] Hence for ${\displaystyle x\ll 1}$,

${\displaystyle {\begin{aligned}y(x)={}&1-Bx+{\frac {1}{3}}x^{3}-{\frac {2B}{15}}x^{4}+\cdots {}\\[6pt]&\cdots +x^{3/2}\left[{\frac {4}{3}}-{\frac {2B}{5}}x+{\frac {3B^{2}}{70}}x^{2}+\left({\frac {2}{27}}+{\frac {B^{3}}{252}}\right)x^{3}+\cdots \right]\end{aligned}}}$

where ${\displaystyle B\approx 1.588071}$.[10][11]

It has been reported by Salvatore Esposito[12] that the Italian physicist Ettore Majorana found in 1928 a semi-analytical series solution to the Thomas–Fermi equation for the neutral atom, which however remained unpublished until 2001. Using this approach it is possible to compute the constant B mentioned above to practically arbitrarily high accuracy; for example, its value to 100 digits is ${\displaystyle B=1.588071022611375312718684509423950109452746621674825616765677418166551961154309262332033970138428665}$.