Study On The Microscopic Theory Of Nematic And Cholesteric Liquid Crystals With The Quantum Theory Methods

The liquid crystals is a many-body system which consists of a large number of rod-like(or disc-like) molecule. In this paper, we introduce quantum theory into the liquid crystals in order to research the collective effect (i.e. the physics effect produced by many-body effect and collective excitation). In this way, we research the microscopic theory of nematic phase and cholesteric phase.The characteristic of nematic phase is orientation order of the molecule long axis. The orientation order can be described by scalar order parameter S. In the first part of this paper, we put forward the quantum mechanics expression for molecule state above all. The molecule have two kind of the degree of freedom, which include the orientation vector Ω of the molecule long axis and the position vector r of the molecule center-of-mass. The orientation state of the molecule can be described by angular momentum eigenstate of the molecule rotating around its long axis; The state of the molecule center-of-mass can be described by energy eigenstate of the particle in the harmonic well potential. We use the cell model in liquid statistical physics to deal with the space distribution of the molecule center-of-mass. The Hamiltonian of system can be expressed by the quasi-creation operator and quasi-destruction operator that connect with the two kinds of eigenstate above-mentioned. In the proper approximation, the Hamiltonian can be expressed as the diagonalization. Then we can derive the expression of the scalar order parameter S, and the result is in accordance with the experiment.The microscopic mechanism of the helix structure is one of the most important questions for cholesteric phase, from the general view, which is because there are chiral potential Vx .There are some disputes about the potential Vx. It had not been proved by far that the form of the chiral potential Vx that is generally accepted can induce the cholesteric phase. In thesecond part of this paper, we proved this question with the quantum theory way and derive the expression of the pitch. We use the lattice model to deal with the space distribution of the molecule center-of-mass in there.