The Schrödinger equation:

The Schrödinger equation cannot be derived from the principles of classical physics. Its validity is seen from its success in describing the motion of microscopic particles. The Schrödinger equation is the equation of motion of the wavefunction. A wavefunction is a mathematical expression describing the amplitude of the wave at any given location in space and time. In our case the amplitude is a probability amplitude since the absolute square of the wavefunction is interpreted as the probability to find the microscopic particle at a given set of coordinates in space:

The wavefunction of a "free particle" can be described as:

The time-dependent Schrödinger equation:

The time-dependent form of Schrödinger's equation describes the time-evolution of the wavefunction of a particle in a given potential V:

with

This form has to be used when the potential V in which the particle is moving is time-dependent. It applies when the transition between different eigenstates of the time-independent Schrödinger equation are considered, for example when optical transitions in an atom have to be calculated, or in scattering experiments.

The time-independent Schrödinger equation:

The time-independent form of Schrödinger's equation is applied in those cases where the potential V is a constant in time.

It is used in the calculation of the energy states of atoms and molecules (i.e. electronic, vibrational, rotational, etc. states)

The free particle:

The case of a free particle is given when no forces act on the particle. This is the case when the potential V is a constant with respect to the time and space coordinates. For simplicity we can set V = 0. The time-independent Schrödinger equation in a one-dimensional case is:

The mathematical solution of this second order differential equation is:

with A and B being integration constants that depend on the boundary conditions of the problem. The term with amplitude A describes a free particle moving in positive x-direction, the one with amplitude B a particle moving in negative x-direction. If we insert equation (7) into eq. (6) we obtain the energy of the free particle as:

Using a slightly different version of de Broglie's equation

and substituting its expression for p in eq. (8) we obtain the classical expression for the energy of a free particle:

Operators and observables:

The time-independent Schrödinger equation has the form of a so-called eigenvalue equation. The general form of such an equation is:

where f is the eigenfunction, is the eigenvalue, and the operator (usually written with a hat).

Each operator usually has an infinite set of eigenfunctions. Each eigenfunction is associated with a specific eigenvalue. Sometimes the same eigenvalue is associated with more than one eigenfunction. In those degenerate cases there are n-1 degrees of freedom for writing the eigenfunctions where n is the degeneracy.

The eigenfunction describes the state of the particle. The operator is associated with an experimental observable which can be measured. If the system is in a pure state, i.e. one of the eigenfunctions of the operator , the measurement is exact and returns the eigenvalue in every case as the measured value.

If the particle is not in a pure state, i.e. not described by an eigenfunction of , it may always be described by a linear combination of eigenfunctions

A measurement of the observable will in this case return one of the eigenvalues with a probability of , i.e. the absolute square of the linear coefficient of the eigenfunction associated with it. If one does infinitely many measurements of the same observable all the eigenvalues will be sampled according to their statistical weights . The experimental result can then be described by the expectation value which is defined as the weighted average of the individual measurements:

In the time-independent Schrödinger equation the operator is the so-called Hamilton-operator:

It is the operator associated with the energy of the system. Its eigenvalues are the energy terms which are available to the particular system under study. Its observable is the total energy E with its expectation value defined as:

It is also possible to define the operators of space , linear momentum , kinetic energy , etc.

The general rule to obtain a quantum-mechanical operator of an experimental observable is to express the classical form of the observable in terms of space and momentum coordinates and substitute them with their operators. Care has to be taken not to lightly reverse the order in which operators are written. The commutative rule is not generally valid for operators. In those cases where it is valid it takes on a special meaning:

The commutator of two quantum-mechanical operators and , is defined as:

If it is zero, the two operators commute. This means that their respective observables, and , can be measured precisely at the same time. This is because the eigenfunctions of one operator are automatically eigenfunctions of the second operator. If the system is in a pure state described by one of these eigenfunctions the application of the operators will return sharp, well-defined eigenvalues in both cases, i.e. allow for a precise measurement of these observables.

where and , are so-called conjugate coordinates.

Examples are:

Space and linear momentum:

Energy and time:

Angle and angular momentum:

The delta in these expression has to be interpreted as the rms deviation of an infinite set of measurements on the system.