Quantum Mechanics in Matrix Form

Zhi Ren

In this paper the analysis and discussion about the contribution made by matrix mechanics, i.e. the establishment of complex form of quantum mechanics and the proposal of mathematic formalism of uncertainty principle, to the development of quantum mechanics are conducted, thereby to disclose how the complex functions enter the quantum mechanics by accident and turn to be a must of it. The evolution of mathematic formula, which implies the uncertainty principle, is also outlined to reflect the contribution of three founders of matrix mechanics. Also disclosed is the physical meaning of coordinate matrix in matrix mechanics and its connection to propagator in terms of modern quantum mechanics terminology. This analysis underlines the essentials of quantum mechanics in physics: discontinuous and statistical elementary processes which are the current research hotspot.

Lampros A. A. Nikolopoulos

Mario S. Reis

F.J. Duarte

Using the vector direct product , the probability amplitudes for quantum entanglement, ψ + , ψ − , ψ + , and ψ − are expressed in a matrix form and shown to be represented by the Pauli matrices plus the identity matrix. The Hadamard matrix is derived from a beam splitter Hong-Ou-Mandel (HOM) configuration. Numerous mathematical operations with the polarization rotator matrix ( R = σ x ) and the Hadamard matrix operating on quantum entanglement states are demonstrated. This chapter concludes with a discussion of reversibility in quantum mechanics between quantum interference and quantum entanglement.