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  The equation

The equation introduced by the austrian physicist Erwin Schrödinger in 1926 is the fundamental equation to describe space and time dependency of a nonrelativistic quantum mechanic state. The importance of the equations for quantum mechanincs is comparable to the importance of Newton's second law for classical mechanics.

The equation describes the state of a single particle (electron or atom) in an extern potential. If we describe the particle by use of the wave function , it has to fulfill the Schrödinger equation

where describes the potential, the elementary charge and the electron mass.

is interpreted as the probability that the electron is in space x at time t.

With that the Schrödinger equation keeps to important properties:

  • Probability conservation

          The probabilty for the electron beiing in the complete space equals 1 at any time.

  • Energy conservation

         The complete energy of the system is equal at any time.

Goal for the further numerical treatment is to keep these properties even for the numerical approximations.