Fundamental systems

Fig.: Water molecule with relevant fundamental systems.

The Schrödinger equation has been set up and solved for a number of fundamental systems. The purpose of the fundamental systems is to simplify solving the Schrödinger equation by focussing on only one particular property of a real system at a time. In many cases, good approximate solutions for real systems can be obtained by linear combination of a small number of relevant fundamental systems. For example, in a water molecule, we have two hydrogen atoms and another atom, which we can try to treat as hydrogen-like to start with. In addition, there are two bonds which can vibrate and the molecule can rotate around its two-fold symmetry axis.

Fig.: The relationships between the fundamental systems.

The fundamental systems:

The harmonic oscillator can be developed further to explain the vibrational states of solids (phonons), while the combination of many atoms with their Coulomb potentials leads to the electronic band structure observed in solids. Molecules, on the other hand, are best described by a combination of harmonic oscillators (one for each bond) and rigid rotors (one for each symmetry axis).

Solving the Schrödinger equation

To solve the Schrödinger equation for a particular system, we need:

The solutions are pairs of wave functions and their energy eigenvalues.

Properties of a wave function

The wave function of a particle is a complex function (in the sense of having real and imaginary parts). Its complex square (i.e. the product of the function and its complex conjugate) is a real and positive function which represents the probability density of finding the particle as a function of the spatial coordinates.

Therefore, only those solutions of the Schrödinger equation which satisfy the following conditions are physically sensible:

Next we want to find the wave functions of the hydrogen atom. To do this, we'll need the expression for the del operator in spherical coordinates.