439: Variational Calculation on the Two-dimensional Hydrogen Atom

Normalized trial wave function:

$\psi ( \alpha , r) = \sqrt{ \frac{2}{ \pi}} \alpha e^{- \alpha r}$

$\int_{0}^{ \infty} \psi ( \alpha , r)^2 2 \pi r dr~~assume,~ \alpha > 0 \rightarrow 1$

Calculate electron kinetic energy:

$T ( \alpha ) = \int_{0}^{ \infty} \psi ( \alpha , r) \frac{-1}{2r} \frac{d}{dr} (r \frac{d}{dr} \psi ( \alpha , r)) 2 \pi r dr~~assume,~ \alpha >0 \rightarrow (-2) \alpha Z$

Calculate electron-nucleus potential energy:

$V_{NE} ( \alpha , Z) = \int_{0}^{ \infty} \psi ( \alpha , r) \frac{-Z}{r} \psi ( \alpha , r) 2 \pi r dr~assume,~ \alpha > 0 \rightarrow (-2) \alpha Z$

Calculate total electronic energy for the 2D H atom:

$$\alpha$$ = 1 $$\alpha$$ = Minimize (E, $$\alpha$$) $$\alpha$$ = 2 E( $$\alpha$$) = -2

Demonstrate that the virial theorem is satisfied:

$\frac{T ( \alpha )}{E( \alpha )} = -1~~~ \frac{T( \alpha )}{V_{NE} ( \alpha , 1)} = -0.5~~~ \frac{V_{NE} ( \alpha , 1)}{E( \alpha )} = 2$