Explanation of covalent bond from physics point of view?

Covalent bonding is a quantum-mechanical phenomenon that can't be explained classically. The basic idea can be demonstrated in the case of a hydrogen molecule if you think of the electrons as two particles in a box. By joining the two hydrogen atoms together into a molecule, you make the "box" twice as long. Let's pretend that this is a rectangular box. Lengthening the cubical box into a $2\times 1 \times 1$ box makes the wavelength along the long axis twice as big. By the de Broglie relation $p=h/\lambda$, this reduces the momentum in that direction by a factor of $2$, which makes the associated kinetic energy $1/4$ as much. The lowered energy makes the system more stable. To pull it apart, you would need to add energy.