Quantum Mechanics from a Linear Algebra Viewpoint

Yao Jiu-min

Elementary theory of representation is introduced firstly,including primary meaning,expressions of a state and an operator in various representations,as well as the transformation from one representation to another.Then,representation and its transformation can be understood from its own instinct physical viewpoint through analogying with the counterpart in geometric coordinates and linear algebra mathematically,

Günter Ludyk

Shuji Watanabe

First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on $\mathbb{C}^{2^{2M}}$, where $M$ is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.

Albert S. Schwarz

A. Mangoni