Free energy of a particle with saddle point potential in manetic field

Abstract

The free energy and the partition function of the system of a particle with saddle point potential in the presence of magnetic field are calculated exactly using the Feynman path integral method. The technique used for solving the problem is based on the method developed by Sa-yakanit, Choosiri and Robkob ( Phys. Rev. B. 37, 1998) for a particle in the presence of magnetic field with non-local potential. The main idea is to calculate the propagator with the following action, S = ∫Ldt, where L is the Lagrangian L = (m/2)(x² + y² + ω (Ωy - yΩ ) – (Ω²[subscript x]x² - Ω²y y²) ) . Here ω is the cyclotron frequency and Ωx and Ωy are frequencies of the saddle point potential. The variation of the action leads to two coupled equations of motion which can be solved and lead to the classical action Sd . The prefactor F(τ) is also obtained by using Van Vleck-Pauli formula, which gives the propagator of system as K(rb , ra ; τ) = F (τ) exp(iS[subscript cl]/ ħ ). This result can be reduced to several limit cases, i.e. ω = 0 , zero magnetic field. In this case the propagator is reduced to product of two oscillators with one as an inverse potential. For the case of zero saddle point potential, the propagator of a particle in magnetic field is obtained. Using relationship between the propagator and the density matrix, we can express the partition function of the system in terms of trace of density matrix. The free energy is obtained as F = - β -1ln Z. Calculated results for free energy with different limiting cases are presented in the graphic forms.