Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $$\Psi^*\Psi$$ is probability density function.

Consider the two statements below

1. Probability will change with time. If probability of finding a particle at one place os increased, then the probability of finding this particle in some other place is decreased. So,we can define a probability flow, which gave the continue equation in QM.
2. If wave function is normalised once to one. It can be derived that the wave function is normalised for ever. Means, it's independent of time.

Question:

As per 2nd statement probability is independent of time and as first, probability is dependent of time. What is actually correct? Or is that implies that we can't apply continue equation, once it's normalised?

• Normalization is a global property. There is no contradiction in a normalized function which locally varies with time. This si something which has nothing to do with QM. – GiorgioP Feb 2 at 9:30

The probability density is dependent on time. This is different from what does means for the wavefunction to be normalized forever. We cannot conclude from $$\frac{d}{dt}\int_{-\infty}^{\infty}|\Psi(x, t)|^2dx = 0$$ that $$\frac{\partial}{\partial t} |\Psi(x, t)|^2 = 0$$. As an example, momentum and energy being both conserved doesn't tell us much about how the distribution of momentum and energy change with time: It just says that the total, wherever it is, stays the same. In the same way, the probability density may change with time, so long as the total is conserved.