cyclotron radius. Mass The mass of composite fermions has been determined from the measurements of: the effective cyclotron energy of composite fermions; the temperature dependence of Shubnikov–de Haas oscillations; energy of the cyclotron resonance; spin polarization of the Fermi sea; and quantum phase transitions between states with different spin polarizations. Its typical value in GaAs systems is on the order of the electron mass in vacuum. (It is unrelated to the electron band mass in GaAs, which is 0.07 of the electron mass in vacuum.) Theoretical formulations Much of the experimental phenomenology can be understood from the qualitative picture of composite fermions in an effective magnetic field. In addition, composite fermions also lead to a detailed and accurate microscopic theory of this quantum liquid. Two approaches have proved useful. Trial wave functions The following trial wave functions embody the composite fermion physics: Here is the wave function of interacting electrons at filling factor ; is the wave function for weakly interacting electrons at ; is the number of electrons or composite fermions; is the coordinate of the th particle; and is an operator that projects the wave function into the lowest Landau level. This provides an explicit mapping between the integer and the fractional quantum Hall effects. Multiplication by attaches vortices to each electron to convert it into a composite fermion. The right hand side is thus interpreted as describing composite fermions at filling factor . The above mapping gives wave functions for both the ground and excited states of the fractional quantum Hall states in terms of the corresponding known wave functions for the integral quantum Hall states. The latter do not contain any adjustable parameters for , so the FQHE wave functions do not contain any adjustable parameters at . Comparisons with exact results show that these wave functions are quantitatively accurate. They can be used to compute a number of measurable quantities, such as the excitation gaps and exciton dispersions, the phase diagram of composite fermions with spin, the composite fermion mass, etc. For they reduce to the Laughlin wavefunction at fillings . Chern–Simons field theory Another formulation of the composite fermion physics is through a Chern–Simons field theory, wherein flux quanta are attached to electrons by a singular gauge transformation. At the mean field approximation the physics of free fermions in an effective field is recovered. Perturbation theory at the level of the random phase approximation