why the probability density doesn't oscillate with time?
It can do - and usually does, for instance, in a stationary state.
What is the situation.
For a stationary state the probability density does not oscillate in time.
The wavefunction of a stationary state does evolve in time according to ψ(x,t)=f(x)e-iEt, where f(x) is an eigenfunction of the Hamiltonian, and E is the corresponding eigenvalue. However, the probability density is the "square" of the wavefunction, ψψ*, where the multiplication of e-iEt with the complex conjugatate of eiEt gives e0=1, which doesn't change with time.
The probability density of a general state does evolve in time, because it is the superposition of several eigenfunctions.
Oh I get you - I misread.
Yeah - the probability density of a stationary state does not vary with time, which is sort-of why it is a stationary state.
Still need more info to answer OPs question properly.
Here are some nice animations of probability densities varying with time:
Separate names with a comma.