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Chemistry LibreTexts

415: Numerical Solutions for the Hydrogen Atom Radial Equation

  • Page ID
    137744
  • Reduced mass: \( \mu\) = 1

    Angular momentum: L = 0

    Integration limit: rmax = 18

    Nuclear charge: Z = 1

    Solve Schrödinger's equation numerically. Use Mathcad's ODE solve block:

    Given

    \[ \frac{-1}{2 \mu} \frac{d^2}{dr^2} \psi (r) - \frac{1}{r \mu} \frac{d}{dr} \psi (r) + \bigg[ \frac{L( L + 1)}{2 \mu r^2} + \frac{1}{2} kr^2 \bigg] \psi (r) = E \psi (r)~~~ \psi (.0001) = .1~~~ \psi '(.0001) = 0.1\]

    \[ \psi = Odesolve (r, r_{max})\]

    Normalize wave function:

    \[ \psi (r) = \left( \int_{0}^{r_{max}} \psi (r)^2 4 \pi r^2 dr \right) ^{ \frac{-1}{2}} \psi (r)\]

    Energy guess:

    E = -.125 r = 0, .001 .. rmax

    Screen Shot 2019-02-19 at 11.44.13 AM.png

    Calculate average position:

    \[ \int_{0}^{r_{max}} \psi (r) r \psi (r) 4 \pi r^2 dr = 5.997\]

    Calculate kinetic energy:

    \[ \int_{0}^{r_{max}} \psi (r) \bigg[ \frac{-1}{2 \mu} \frac{d^2}{dr^2} \psi (r) - \frac{1}{r \mu} \frac{d}{dr} \psi (r) + \bigg [ \frac{L(L + 1)}{2 \mu r^2} \bigg] \psi (r) \bigg] 4 \pi r^2 dr = 0.125\]

    Calculate potential energy:

    \[ \int_{0}^{r_{max}} \psi (r) \frac{-Z}{r} \psi (r) 4 \pi r^2 dr = -0.25\]