There exists a considerable amount of work

both theoretical and experimental

on the system of electrons on the surface of liquid helium.Because of the combined effect of the attractive image potential and the surface barrier

electrons are confined to move on the outside surface of the liquid and form an ideal two dimensional(2D) system.In recent years the ripplon problem has been studied by many theoretical methods and the properties of the ripplon are measured using experimental method by many investigators. A polaron is a system of an electron on the surface of liquid He film coupled to the ripplon field

an elementary excitation of the liquid helium surface.The system is of particular interest

because the strength of the electron-ripplon coupling can be changed by adjusting the thickness of the film and by changing the substrate material.However

because the forms of the ripplon dispersion relation and the electron ripplon coupling are more complex

hence the polaron problem of an electron-phonon coupling in crystals is different from an electron ripplon coupling on the liquid helium surface.The theory was established by Jackson and Platzman.They studied the ground state of the system by means of Feynman formulation of polarons

and found the ground-state energy

in both the weak-and strong coupling limits.Stan et al. studied the variation of the electron effective mass and the rate of ripplon scattering from the electron arrays as a function of temperature.Jin et al.investigated the properties of strong coupling system of the electron ripplon on liquid helium film by Landau-Pekar variational theory.In fact

so far research of the effective mass of an electron ripplon coupling system on the surface of liquid helium film is very scarce.The purpose of this paper is to explore the effect of the electron interaction with both the weak-and strong coupling ripplons on the effective mass of 2-D electron coupled to the ripplons on the surface of liquid helium film by using a linear combination operator and Lagrange multiplier method. The results show that for the weak coupling the additional part of the effective mass of the polaron is proportional to the coupling constant α and for the strong coupling the vibration frequency λ of polaron depends not only on the cutoff frequency

K

c

but also on the coupling constant α

thereas the frequency λ will increase with increasing

K

c

and α.The effective mass of the polaron M

*

will increase with increasing electron ripplon coupling constant α

thereas the self energy of the polaron will decrease with increasing coupling constant α.