This module presents the problem of determining the most likely position,
or orbital, of an electron around a hydrogen atom. The mechanics of
quantum particles are determined by a partial differential equation
known as Schrodinger's wave equation. This sets up an eigenvalue
problem that limits the possible energies of the electron. The
computational solution of Schrodinger's equation for hydrogen i
typically done via separation of variables, in which the radial solution
is separated from the angular solution, in effect reducing the problem
to 1 dimension and allowing for standard numerical integration techniques
to be applied.
The angular solution of Schrodinger's equation for the hydrogen
atom can be expressed in terms of the spherical harmonics. Once the
solution is obtained, a variety of computational techniques can be
used to visualize the orbitals for different energies.
The importance of matching the types of boundary conditions to the
problem will be demonstrated in the module, as well as limitations
of the numerical techniques used.