What is the time-independent Schroedinger equation

Time-dependent Schrödinger equation

In order to obtain discrete energy values, one applies the quantum mechanical operator for the energy, the Hamilton operator, to the wave function. The application of the Hamilton operator to the wave function is the Schrödinger equation. Solutions for the time-independent Schrödinger equation only exist for discrete values ​​of energy, which are called "eigenvalues" of energy.

For example, the energy eigenvalues ​​for the harmonic oscillator is:

The lowest vibration levels of diatomic molecules can often be approximated to the harmonic oscillator with sufficient accuracy to determine the force constants of the molecular bonds.

The energy eigenvalues ​​are discrete for small energies, but they become continuous for large energies, since the system is then no longer in a bound state. With a more realistic potential (e.g. that of a diatomic molecule) the energy eigenvalues ​​move closer and closer together up to the dissociation energy. The energy levels take on continuous values ​​after dissociation, as with free particles.

More about the Schrödinger equation