  We use the variational approximation with double Gaussian type trial
wave-functional approximation, in which we use the square root of the
dispersion of the zero-mode wave-function as an order parameter, to study the
out of equilibrium quantum dynamics of time-dependent second order phase
transitions in (3+1) dimensions. We study the time evolution of symmetric
states of scalar $\lambda \phi^4$ theory in several situations by properly
treating the effect of the $\lambda\phi^4$ interaction. We also calculate the
effective action and the effective potential of the theory with the precarious
renormalization. We show that the presence of a quenching of the mass-squared
leads to second order phase transition nontrivially since the vacuum structure
changes by absorbing the energy required for quenching, even though there is no
symmetry breaking in the effective potential of the theory without quenching
process. We also calculate the equal time correlation function, and then
evaluate the correlation length as a function of the mass-squared. The time
dependence of the correlation length varies depending on how the mass-squared
changes in time. For constant mass-squared it gives the classical Cahn-Allen
relation, and it leads to different relations for other time-dependence of the
mass-squared. We also show that there exists a propagating spatial correlation
after termination of the phase transition process in addition to the
correlation corresponding to the formation and growth of domains.
