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