Schroedinger Equation for Hydrogen Atom

The Schroedinger equation is:

\begin{eqnarray} (-\frac{\hbar^2}{2m}\nabla^2-\frac{Z e^2}{4\pi\varepsilon_0 r})\psi(\vec{r})=E \psi(\vec{r}) \end{eqnarray}

using ansatz:

$\psi(\vec{r}) = Y_{lm}(\hat{r})\; u(r)/r$

and introducing dimensionless variables:

\begin{eqnarray} x = \frac{r}{r_B}\\ \varepsilon = \frac{E}{E_0} \end{eqnarray}

where \begin{eqnarray} && r_B = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2} \approx 0.529 A\\ && E_0 = \frac{\hbar^2}{2 m r_B^2} == Ry \approx 13.6 eV \end{eqnarray}

we get the differential equation

\begin{eqnarray} u''(x)- \left(\frac{l(l+1)}{x^2}-\frac{2Z}{x}-\varepsilon\right)u(x)=0 \end{eqnarray}

Next we rewrite into the system of first order equations:

\begin{eqnarray} y = \left(u(x),u'(x)\right)\\ \frac{dy}{dx} = \left(u'(x),u''(x)\right) \end{eqnarray}

with boundary conditions \begin{eqnarray} &&u(0) = 0 \rightarrow \psi(0)<\infty\\ &&u(\infty)=0 \rightarrow \int |\psi(r)|^2 r^2 dr \propto \int u^2(r)dr < \infty \end{eqnarray}

Because boundary conditions are given at the two ends, we need so-called shooting method

Shooting algorithm:

Suppose the two boundary condistions are given at $a$ and $b$, i.e., $u(a)=u(b)=0$. Then

Some remarks

The implementation will follow these steps

  1. call SciPy routine 
    integrate.odeint
    to integrate the one-electron Schroedinger equation. Note that the distance is measured in units of bohr radius and energy units is Rydberg ($1 Ry = 13.6058...eV$)

  2. The boundary conditions are $u(0)=0$ and $u(\infty)=0$. Use shooting method to obtain wave functions:
    1. Use logarithmic mesh of radial points for integration. Start integrating from a large distance ($R_{max} \sim 100$). At $R_{max}$ choose $u=0$ and some nonzero (not too large) derivative.
    2. Integrate the Schroedinger equation down to $r=0$. If your choice for the energy $\varepsilon$ corresponds to the bound state, the wave function at $u(r=0)$ will be zero.

  3. Start searching for the first bound state at sufficiently negative energy (for example $\sim -1.2 Z^2$) and increase energy in sufficiently small steps to bracket all necessary bound states. Ones the wave function at $r=0$ changes sign, use root finding routine, for example 
    optimize.brentq,
    to compute zero to very high precision. Store the index and the energy of the bound state for further processing.
  4. Ones bound state energies are found, recompute $u(r)$ for all bound states. Normalize $u(r)$ and plot them.

  5. Compute electron density for various atoms (for example He, Li, ..) neglecting Coulomb repulsion:

    Populate first $Z$ lowest laying electron states and compute $\rho = \sum_{lm\in occupied} u_{lm}^2(r)/(4\pi r^2)$. Each state with quantum number $l$ can take $2(2l+1)$ electrons. Be carefull, if atom is not one of the Nobel gases (He, Ne, ...) the last orbital is only partially filled.

    </ol>

Recall:

\begin{eqnarray} && y = (u(r), u'(r) )\\ && dy/dr = (u'(r), u''(r)) \end{eqnarray}\begin{eqnarray} u''(r)= \left(\frac{l(l+1)}{r^2}-\frac{2Z}{r}-\varepsilon\right)u(r) \end{eqnarray}

First we try linear mesh and forward integration. It is supposed to be unstable. We know the ground state has energy $E_0=-1 Ry$ and we should get $1s$ state with integrating Scroedinger equation.

Recal Euler's method and Runge Kutta method

Indeed the integration is unstable, and needs to be done in opposite direction. Let's try from large R.

Clearly the integration from infinity is stable, and we will use it here.

Logarithmic mesh is better suited for higher excited states, as they extend far away.

Lets create a subroutine of what we learned up to now:

Resulting charge density for a Ni-like Hydrogen atom

Numerov algorithm

The general purpose integration routine is not the best method for solving the Schroedinger equation, which does not have first derivative terms.

Numerov algorithm is better fit for such equations, and its algorithm is summarized below.

The second order linear differential equation (DE) of the form

\begin{eqnarray} x''(t) = f(t) x(t) + u(t) \end{eqnarray}

is a target of Numerov algorithm.

Due to a special structure of the DE, the fourth order error cancels and leads to sixth order algorithm using second order integration scheme.

If we expand x(t) to some higher power and take into account the time reversal symmetry of the equation, all odd term cancel

\begin{eqnarray} x(h) = x(0)+h x'(0)+\frac{1}{2}h^2 x''(0)+\frac{1}{3!}h^3 x^{(3)}(0)+\frac{1}{4!}h^4 x^{(4)}(0)+\frac{1}{5!}h^5 x^{(5)}(0)+...\\ x(-h) = x(0)-h x'(0)+\frac{1}{2}h^2 x''(0)-\frac{1}{3!}h^3 x^{(3)}(0)+\frac{1}{4!}h^4 x^{(4)}(0)-\frac{1}{5!}h^5 x^{(5)}(0)+... \end{eqnarray}

hence

\begin{eqnarray} x(h)+x(-h) = 2x(0)+h^2 (f(0)x(0)+u(0))+\frac{2}{4!}h^4 x^{(4)}(0)+O(h^6) \end{eqnarray}

If we are happy with $O(h^4)$ algorithm, we can neglect $x^{(4)}$ term and get the following recursion relation

\begin{equation} x_{i+1}-2 x_i + x_{i-1} = h^2 (f_i x_i+u_i) . \end{equation}

But we know from the differential equation that

\begin{equation} x^{(4)} = \frac{d^2 x''(t)}{dt^2} = \frac{d^2}{dt^2}(f(t) x(t)+u(t)) \end{equation}

which can be approximated by

\begin{equation} x^{(4)}\sim \frac{f_{i+1}x_{i+1}+u_{i+1}-2 f_i x_i -2 u_i+ f_{i-1}x_{i-1}+u_{i-1}}{h^2} \end{equation}

Inserting the fourth order derivative into the above recursive equation (forth equation in his chapter), we get

\begin{equation} x_{i+1}-2 x_i+x_{i-1}=h^2(f_i x_i+u_i)+\frac{h^2}{12}(f_{i+1}x_{i+1}+u_{i+1}-2 f_i x_i -2 u_i+ f_{i-1}x_{i-1}+u_{i-1}) \end{equation}

If we switch to a new variable $w_i=x_i(1-\frac{h^2}{12} f_i)-\frac{h^2}{12}u_i$ we are left with the following equation

\begin{equation} w_{i+1} -2 w_i + w_{i-1} = h^2 (f_i x_i + u_i)+O(h^6) \end{equation}

The variable $x$ needs to be recomputed at each step with $x_i=(w_i+\frac{h^2}{12}u_i)/(1-\frac{h^2}{12}f_i)$.

Put it all together is the homework