Quantum Mechanics on S 1 Based on Dirac Formalism

We deal with quantum mechanics on S 1 (embedded to R 2 ) based on Dirac formalism. To conclude that the dynamical system is a quantum mechanical system we have to show selfadjointness of the dynamical variables. First we show selfadjointness of the momentum operators and then study their spectra. We find that the continuous spectrum of each momentum operator coincides with the set of all real numbers. Second we discuss selfadjointness of the Hamiltonian for a free quantum mechanical particle moving on S 1 together with its spectrum. We see that the spectrum of the Hamiltonian consists of eigenvalues only. Finally we apply these results to the Cauchy problem for the Schrödinger equation for a free quantum mechanical particle moving on S 1 .

Keywords:

• Selfadjointness
• Spectrum
• Quantum Mechanics On The 1-sphere
• Dirac Formalism