Unified Description of Matrix Mechanics and Wave Mechanics on Hydrogen Atom
Unified Description of Matrix Mechanics and Wave Mechanics on Hydrogen Atom
arXiv:1201.0136v5 [math-ph] 25 Mar 2013
Yongqin Wang1 , Lifeng Kang2
1
Department of Physics, Nanjing University, Nanjing 210008, China e-mail:yhwnju@hotmail.com
2
Faculty of Science, National University of Singapore, Singapore 117543
Abstract A new mathematical method is established to represent the operator, wave functions and square matrix in the same representation. We can obtain the specific square matrices corresponding to the angular momentum and RungeLenz vector operators with invoking assistance from the operator relations and the orthonormal wave functions. Furthermore, the first-order differential equations will be given to deduce the specific wave functions without using the solution of the second order Schr¨dinger equation. As a result, we will o unify the descriptions of the matrix mechanics and the wave mechanics on hydrogen atom. By using matrix transformations, we will also deduce the specific matrix representations of the operators in the SO(4,2) group. Keywords: Matrix Operator Wave function Hydrogen atom Quantum mechanics 1. Introduction In 1925, based on Niels Bohr’s correspondence principle, Werner Heisenberg represented the spatial coordinate q and the momentum p by the following form [1] q = [q(nm)e2πiν(nm)t ], p = [p(nm)e2πiν(nm)t ] (1)
Preprint submitted to Elsevier
March 26, 2013
They abandoned the representation (1) in favor of the shorter notation q = q(nm) p = p(nm)
Max Born and Pascual Jordan then wrote q substituted for q(nm) as a matrix [2] 0 q(01) 0 0 0 ··· q(10) 0 q(12) 0 0 ··· 0 q(21) 0 q(23) 0 · · · ··· ··· ··· ··· ··· ···
The founders of matrix mechanics tried to describe the mechanics quantum by the square matrix. They had not been successful because the source of the matrix could not be explained. In modern quantum mechanics the mechanical quantities was described by the operator. However, the operator was
arXiv:1201.0136v5 [math-ph] 25 Mar 2013
Yongqin Wang1 , Lifeng Kang2
1
Department of Physics, Nanjing University, Nanjing 210008, China e-mail:yhwnju@hotmail.com
2
Faculty of Science, National University of Singapore, Singapore 117543
Abstract A new mathematical method is established to represent the operator, wave functions and square matrix in the same representation. We can obtain the specific square matrices corresponding to the angular momentum and RungeLenz vector operators with invoking assistance from the operator relations and the orthonormal wave functions. Furthermore, the first-order differential equations will be given to deduce the specific wave functions without using the solution of the second order Schr¨dinger equation. As a result, we will o unify the descriptions of the matrix mechanics and the wave mechanics on hydrogen atom. By using matrix transformations, we will also deduce the specific matrix representations of the operators in the SO(4,2) group. Keywords: Matrix Operator Wave function Hydrogen atom Quantum mechanics 1. Introduction In 1925, based on Niels Bohr’s correspondence principle, Werner Heisenberg represented the spatial coordinate q and the momentum p by the following form [1] q = [q(nm)e2πiν(nm)t ], p = [p(nm)e2πiν(nm)t ] (1)
Preprint submitted to Elsevier
March 26, 2013
They abandoned the representation (1) in favor of the shorter notation q = q(nm) p = p(nm)
Max Born and Pascual Jordan then wrote q substituted for q(nm) as a matrix [2] 0 q(01) 0 0 0 ··· q(10) 0 q(12) 0 0 ··· 0 q(21) 0 q(23) 0 · · · ··· ··· ··· ··· ··· ···
The founders of matrix mechanics tried to describe the mechanics quantum by the square matrix. They had not been successful because the source of the matrix could not be explained. In modern quantum mechanics the mechanical quantities was described by the operator. However, the operator was