In this paper, the n-th root of a matrix is defined, and the explicit form of n-th root of an Hermitian matrix is given. A new method for diagonalizing quadratic Hamiltonians is proposed. Also, a class of quantum operators is induced by the linear transformation in configuration space, and its unitary properties and transformation behavior are studied. Our new method based on n-th root of matrices can develop the mathematical methods of quantum mechanics and quantum optics, and can also be applied to engineering, quantum optics and quantum fields states with squeezing properties, as well as the binomial field states.
On Physical Principles of Quantum Mechanics