ABSTRACT
This chapter discusses a new method for the quantization of arbitrary smooth spaces on the problem of the one-dimensional harmonic oscillator with metric. It shows that the classical potential if directly been translated into the quantum metric picture does not provide fully satisfying solutions. The chapter demonstrates that metrics of cos- or Gaussian shape provide the necessary properties for the creation of perfect ground state solutions being similar to the classical solution. Situations with multiple probability maxima can easily be constructed by the means of the introduction of several centers of excitation. As various states of excitation can be combined, asymmetric solutions also are possible. Asymmetry can occur even though the original “setup” or placement of centers of excitement was completely symmetric. This becomes the more pronounced the higher the levels of excitation respectively levels of “virtuality”. In the classical cases, the quantization is automatically coming from boundary conditions.
