Part I (Introduction).- Part II (Theory). - 2. Special orthogonal groups (Introduction; Abstract groups;Orthogonal group SO(n); Tensor representations of the SO(n); \Gamma matrix groups; Spinor representations of the SO(n); Concluding remarks).- 3. Rotational symmetry and Schrödinger equation in N-dimensional space (Introduction; Rotation operator; Orbital angular momentum operators; The linear momentum operators;Radial momentum operator; Spherical harmonic polynomials; Schrödinger equation for a two-body system; Concluding remarks).- 4. Dirac equation in higher dimensions (Introduction; Dirac equation in N+1 dimensions; The radial equation; Application to hydrogen atom; Concluding remarks).- 5. Klein-Gordon equation in higher dimensions (Introduction; The Radial equation; Application to hydrogen atom; Concluding remarks).- Part III (Application in Non-relativistic Quantum Mechanics).- 6. Harmonic oscillator (Introduction; Exact solutions of harmonic oscillator; Recurrence relations for the radioal function; Realization of dynamic group SU(1, 1); Generalization to pseudoharmonic ooscillator; Position and momentum information entropy; Conclusions).- 7. Coulomb potential (Introduction; Exact solution; Shift operators; Mapping between Coulumb and harmonic oscillator radial functions; Realization of dynamic of dynamic group SU (1, 1); Generalization to Kratzer potential; Concluding remarks).- 8. Wave function ansatz method (Introduction; Sextic potential; Singular one-fraction power potential; Mixture potential; Non-polynomial potential; Screened Coulomb potential; Morse potential; Conclusions).- 9. Levinson theorem for Schrödinger equation (Introduction; Scattering states and phase shifts; Bound states; Sturm--Liouville theorem; Levinson theorem; Discussions; Conclusions).- 10. Generalized hypervirial theorem for Schrödinger equation (Introduction; Generalized Blanchard's and Kramers' recurrence relations; Applications to central potentials; Conclusions).- 11. Exact quantization rule and Langer modification (Introduction; WKB approximation; Exact quantization rule; Application to trigonometric Rosen-Morse potential; Proper quantization rule; Illustrations of proper quantization rule; Langer modification in D dimensions; Calculations of logarithmic derivatives of wavefunction; Conclusions).- 12. Schrödinger equation with position-dependent mass (Introduction; Formalism; Applications to harmonic oscillator and Coulomb potential; Conclusions).- Part IV (Application in Relativistic Quantum Mechanics).- 13. Dirac equation with Coulomb potential (Introduction; Exact solutions of hydrogen-like atoms; Analysis of eigenvalues; Generalization to the Dirac equation with Coulomb potential plus scalar potential; Concluding remarks).- 14. Klein-Gordon equation with Coulomb potential (Introduction; Eigenfunctions and eigenvalues; Analysis of eigenvalues; Generalization: Klein-Gordon equation with Coulomb plus scalar potential; Comparison theorem; Conclusions).- 15. Levinson theorem for Dirac equation (Introduction; Generalization Sturm-Liouville theorem; Number of bound states; Relativistic Levinson theorem; Discussions; Friedel Theorem; Comparison theorem; Conclusions).- 16. Generalized hypervirial theorem for Dirac equation (Introduction; Relativistic recurrence relation; Diagonal case; Conclusions).- 17. Kaluza-Klein theory (Introduction; (4+D) -dimensional Kaluza-Klein theories; Paritcle spectrum of Kaluza-Klein theories for ferminions; Warped extra dimensions; Conclusions).- PART V (Conclusions and Outlooks).- 18. Conclusions and outlooks.- Appendices.- References.- Index.