I found this question in a quantum mechanics exam:
What is the physical interpretation of the continuity equation $\frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0$? Here $\rho(x,t)$ is the probability density and $j(x,t)$ is the probability current.
I assume they want a one liner like "probability is conserved". But to be honest I do not understand this. Can any one help me here? What's the one liner they are looking for and why? Many thanks!
The continuity equation in 3-dimensions is $$\frac{\partial \rho}{\partial t} + \vec{\nabla}·\vec{j}=0$$ where the second term is the divergence of $\vec{j}$. By integrating this equation within a fixed volume $V$ whose boundary is $\partial V$, and applying the divergence theorem, we get the integral form of the continuity equation: $$\frac{d}{dt}\iiint\limits_{V}{\rho dV} + \iint\limits_{\partial V}{\vec{j}·\vec{dS}} =0$$ where the surface integral is over the closed surface $\partial V$ with $\vec{dS}$ defined as pointing normally outward. This equation states that the time rate of change of the probability within volume V is equal to the probability flux entering volume V across the boundary $\partial V$. This is a statement of conservation of probability.
You're actually right, it stems from the "conservation" of probability, or the fact that probability sums to 1. It is literally the equation that says if $\rho$ changes, then that must be due to $j$.
Consider the integral version of this equation. In 3D the space derivative is a divergence,
$$\int \left[\frac{\partial \rho}{\partial t} + \nabla\cdot j\right] dV$$ $$\frac{\partial P}{\partial t} = - \oint j \cdot dA$$
The time rate of change of probability in an area is equal to the amount of probability "leaving" the area in any direction (through the surface that defines the area).
In fact, this is the same as the differential form continuity equation in fluids, charge (electromagnetism), heat, etc.
This is a point that's not super subtle but is often only partially expressed in popular conversations in my experience. All the current answers are "correct but not complete", as Einstein liked to say. ;)
OK, so a continuity equation implies a conservation law, sure, but it implies something much stronger. It implies a local conservation law. The difference between the two is beautifully explained by Griffiths in their Electrodynamics book, in Chapter $8$, which I'll borrow. Suppose a quantity, say electric charge, is conserved. Given this much information, one can imagine that $5$ Coulomb of charge suddenly disappears in New York and $5$ Coulomb of charge suddenly appears in Vegas. This is perfectly consistent with the conservation of charge because the total amount of charge remains unchanged. But, Maxwell's equations imply a much stronger conservation law for charges, in particular, for a charge to disappear in New York and reappear in Vegas, it would have to travel in space from New York to Vegas. This is the local conservation of charge. A continuity equation implies local conservation of charge, not just a global one.
Now, let's get to the continuity equation for probability density in quantum mechanics. As with any continuity equation, it implies local conservation of probability. But it is important to ask as to why! The unitarity of quantum mechanics which is implied by the time translational symmetry of the universe (cf. Wigner's theorem) says that the probability in quantum mechanics is conserved. However, this only implies global conservation of probability. And since the Schrödinger equation is simply another way of saying that the evolution of a quantum state is unitary, it also shouldn't imply anything stronger. Then why do we get local conservation for probability using the Schrödinger equation? Well, it is because we smuggle in a specific form of the Hamiltonian. In particular, we usually consider a Hamiltonian of the form $\hat{H} = \hat{p}^2/2m + V(\hat{x})$. This kind of Hamiltonian represents interactions that are local in position basis. This is the key as to why we obtain a local conservation law for probability in the position basis. For example, you wouldn't get a continuity equation for the probability density in the momentum basis because interactions are not local in momentum.
So, to summarize, the continuity equation for probability density in position basis implies that the probability is locally conserved in the position basis, which is because the Hermitian Hamiltonian which governs the unitary time evolution of a state is taken to be local in position basis.
I would like to add a few more elaborations over the previous answer.
The Continuity Equation appears in many areas of Physics; for example, the same equation appears in Electrodynamics, Quantum Mechanics, Fluid Dynamics and Heat conduction but with different physical interpretations of $\rho$ and $\boldsymbol j$. It is indeed a mathematical form of conservation of charge, probability, mass or energy(respectively in the areas mentioned above). You have written it in one dimension but its most general form in three dimensions is $$\frac{\partial\rho}{\partial t}+\nabla\cdot\boldsymbol j=0\tag{1}$$
First let us see this equation in the light of Electrodynamics. Here, $\rho(\textbf r,t)$ is charge density, i.e., $\rho(\textbf r,t) dV$ denotes the amount of charge present in an infinitesimal volume $dV$ around the point $\textbf r$ at time $t$ . $\textbf j$ is the volume current density which is defined as 'charge flowing per unit time per unit area held perpendicular to the flow'; so if the velocity of the charge is $\textbf v$, then in unit time, the volume traversed is $v$ and so $\textbf j=\rho\textbf v$ . The equation $(1)$ can also be written as $$\nabla\cdot\textbf j=-\frac{\partial \rho}{\partial t}\Rightarrow \int_V \nabla\cdot\textbf j\,dV=-\frac{\partial}{\partial t}\int_V \rho \,dV\tag{2}$$
The physical interpretation of the divergence of a vector is that, its integral over a closed volume gives the net outward flux of the vector through the entire closed surface of the volume (in accordance with Gauss' Divergence Theorem). Now here, charge is conserved. So if charge flows out of a closed volume, then it must be at the expense of the charge inside it; in other words, the net outward flux of the volume current density throughout the surface of the volume is equal to the rate of decrease of charge density inside the volume. This is the physical interpretation of the Continuity Equation which is evidently a manifestation of conservation of charge.
There is a beautiful discussion regarding this in the book 'Introduction to Electrodynamics' (4th Edition, Chapter 8) by David J. Griffiths.
In Quantum Mechanics, we speak in terms of probabilities which according to me is not easy to 'feel' at first glance. Now that you have known the physical meaning of the Continuity Equation in the context of charges which is easy to imagine intuitively, you can understand the same equation in the light of Quantum Mechanics at ease. In Electrodynamics, charge flows but here what 'flows' is 'probability' (that's the weird nature of Quantum Mechanics!). $\rho(\textbf r,t)=\Psi^*\Psi$ is probability density which is interpreted as: the probability that a particle exists in an infinitesimal volume $dV$ around the point $\textbf r$ at time $t$ is given by $\rho dV=\Psi^*\Psi dV$ where $\Psi(\textbf r,t)$ is the wave-function of the particle. The probability current is given as $$\textbf j=\frac{\hbar}{2mi}(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*)$$ which describes the flow of probability per unit time per unit area. Now, as the particle will always exist somewhere, so the total probability is conserved. So what does the Continuity Equation $(2)$ says now? That, the net outward flux of the probability current throughout the surface of a closed volume $V$ is equal to the rate of decrease of the probability density inside the volume $V$. It reflects that the more the outward flux, the less probable is the particle to be found inside the volume $V$.
Similarly, you can also find the meaning of $(1)$ in the light of diffusion in Fluid Dynamics or heat conduction. In the former, it means that the net outward flux of a fluid throughout a volume occurs at the expense of the mass of fluid inside the volume, thus reflecting the conservation of mass. In the latter, the quantity analogous to mass is energy.
You can read more about the Continuity Equation from various sources like : https://en.wikipedia.org/wiki/Continuity_equation#Energy_and_heat
Lastly, I would make a remark. When one becomes able to see how the same equation drops in so wonderfully in such a wide variety of areas, one realizes the underlying beauty of it explaining different natural phenomena in the same spirit. That is when, I think, the joy of Physics becomes utmost! Happy learning!
