  The method of the effective action for the composite operators $\Phi^2(x)$
and $\Phi^4(x)$ is applied to the termodynamics of the scalar quantum field
with $\lambda\Phi^4$ interaction. An expansion of the finite temperature
effective potential in powers of $\hbar$ provides successive approximations to
the free energy with an effective mass and an effective coupling determined by
the gap equations. The numerical results are studied in the space-time of one
dimension, when the theory is equivalent to the quantum mechanics of an
anharmonic oscillator. The approximations to the free energy show quick
convergence to the exact result.
