Application to Physical Systems

Abstract

We apply representation theory to several real quantum systems. A real microscopic system has the symmetry of three-dimensional rotation as well as the classical macroscopic system. We firstly discuss this rotation symmetry on the function space $L^2(\mathbb {R}^3)$ by using the algebra $\mathop {{\mathfrak {su}}}\nolimits (2)$. Since this symmetry is related to the three-dimensional rotation, it is called orbital angular momentum. Then, we discuss the Schrödinger equation with central potential. As a typical example, we address the case of hydrogen-type atom and explain the degeneracy of Hamiltonian in this case by using $\mathop {\mathrm{SO}}\nolimits (4,\mathbb {R})$. We also extend the discussion of the rotation symmetry to the d-dimensional case. Then, we introduce the spin system and discuss an interaction between the spin system and the orbital angular momentum system. Also, we discuss quarks that constitutes elements of atoms based on several groups. Since a quark is a fermion and our discussion is limited to the cases with two or three particles, the discussion for quarks in this chapter is much simpler than that of the photonic system because we need to discuss unlimited number of bosons in the photonic system. Finally, we discuss uncertainty relation for wake packets on various spaces, i.e., not only on the set of real numbers $\mathbb {R}$ but also on the one-dimensional and three-dimensional spheres $S^1$ and $S^3$.