Self-adjoint Hamilton Operators
The time evolution of a classical mechanical system is governed by the Hamilton function. Similarly, the Hamilton operator determines the time evolution of a quantum mechanical system and this operator provides information about the total energy of the system in specific states. In both cases it is important that the Hamilton operator is self-adjoint in the Hilbert space of the quantum mechanical system. Thus we are faced with the mathematical task of constructing a self-adjoint Hamilton operator out of a given classical Hamilton function. The Hamilton function is the sum of the kinetic and the potential energy. For the construction of the Hamilton operator this typically means that we have to add to unbounded self-adjoint operators.
KeywordsHilbert Space Hamilton Function Momentum Operator Symmetric Operator Potential Operator
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