### 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.

The fundamental systems:

• Free particle. In the absence of any potential, the particle can move freely; its wave function is a sine (or cosine or complex exponential) wave.
• By confining the particle in an infinite well potential, a set of energy eigenvalues (distinct states) emerges. Each state is characterised by a wave function whose period is such that it is a standing wave within the width of the well.
• If there is a finite potential drop, the particle's wave function is not forced to zero at the boundaries of the well. Instead, the wave function decays exponentially outside the well, which allows the particle to tunnel into a neighbouring well. This is exploited in the scanning tunnelling microscope.
• When dealing with atoms, it is more appropriate to use a Coulomb potential because of the positive charge of the nucleus.
• An atom is a three-dimensional Coulomb well, where the potential along the three Cartesian coordinates is approximately the same, i.e. the atom is approximately spherical. Therefore, it is worthwhile to switch to spherical coordinates at this point.
• In a hydrogen-like atom, a nucleus and one electron share a Coulomb well. Instead of dealing with the two masses independently, the reduced mass of the two objects and their common centre of gravity is used when solving the Schrödinger equation of the hydrogen atom.
• More complex atoms are covered by including the inner electron shells together with the nucleus in the atom core. These electrons do nothing but shield the positive charge of the nucleus.
• When combining two atoms, we can take the wave functions calculated for each one of them and add the properties of the chemical bond by taking into account the vibrational and rotational motions of the molecule, e.g. in order to predict infrared and Raman spectra. The vibrations are covered by the harmonic oscillator...
• ...and the rotations by the rigid rotor.

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:

• a suitable potential
• a kinetic energy term
• to decide which reference frame is best suited to the geometry of the problem (e.g. spherical coordinates for spherical objects, axial coordinates for oblong objects such as chemical bonds)
• a trial wave function (a complex exponential is usually a good guess)
• a set of boundary conditions (the Schrödinger equation is a heterogeneous 2nd order partial differential equation with 3 independent variables; as such it needs up to 2x3=6 boundary conditions). These could be nodes of the wave function at a potential step or a continuity condition when an angular variable makes a full circle.

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:

• The wave function must be continuous.
• It can be normalised. (The particle must be somewhere, so the integral of the probability density over all space must equal 1.) That is to say, its complex square must be integrable.
• Its 1st and 2nd derivatives must also be continuous.
• It must have just a single value for each set of coordinates.

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.