Look at Flow of Probability

We can intergrate the wave function to get the total probability that a wave is within between x=ax=a and x=bx=b

P(a,b)=∫ab∣ψ(x,t)∣2 dxP(a,b) = \int_a^b |\psi(x,t)|^2\,dx

We integrate

dP(a,b)dt=∫ab∂∂t∣ψ∣2 dx=−∫ab∂J∂x dx\frac{dP(a,b)}{dt} = \int_a^b \frac{\partial}{\partial t}|\psi|^2\,dx = -\int_a^b \frac{\partial J}{\partial x}\,dx =−[J(b,t)−J(a,t)]=J(a,t)−J(b,t)= -\big[J(b,t) - J(a,t)\big] = J(a,t) - J(b,t)

hence

dP(a,b)dt=J(a,t)−J(b,t)\boxed{\frac{dP(a,b)}{dt}=J(a,t)-J(b,t)}

so that means in a stionary state which doesn't depend on time, the change in probability dP(a,b)dt=0\frac{dP(a,b)}{dt}=0 so

J(a,t)=J(b,t)J(a,t) = J(b,t)

So jj is constant everywhere in a Stationary States

In 3D,

∂ρ∂t=−∇⋅J⃗\boxed{\frac{\partial \rho}{\partial t} = -\nabla\cdot\vec{J}}