130_notes.dvi

1D Molecule

-4 -2 0 2 4

x

u(x)

V(x)

u0(x)

u1(x)

Two Hydrogen atoms bind together to form a molecule with a separation of 0.74 Angstroms, just
larger than the Bohr radius of 0.53 Angstroms. The binding energy (for the two electrons) is about 4.5
eV. If we approximate the Coulomb potential by with a delta function, settingaV 0 = (0.53)(2)(13.6)
eV Angstroms, our very naive calculation would give 1.48 eV for one electron, which is at least the
right order of magnitude.

Theodd parity solutionhas an energy that satisfies the equation

2 maV 0

κ ̄h^2

= 1 + coth(κd).

2 maV 0

κ ̄h^2

> 2

κ <

maV 0

̄h^2

This energy is larger than for one delta function. This state would becalledanti-bonding.

9.5 The Delta Function Model of a Crystal*

The Kronig-Penny model of a solid crystal contains aninfinite array of repulsive delta func-
tions.

V(x) =aV 0

∑∞

n=−∞

δ(x−na)