Introduction to Quantum Mechanics · Problem 1.14

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · probability_current

Problem

Let Pab(t)P_{ab}(t) be the probability of finding the particle in the range (a<x<ba < x < b), at time tt. (a) Show that dPabdt=J(a,t)−J(b,t),\frac{dP_{ab}}{dt} = J(a, t) - J(b, t), where J(x,t)≡iℏ2m(Ψ∂Ψ∗∂x−Ψ∗∂Ψ∂x).J(x, t) \equiv \frac{i\hbar}{2m}\left(\Psi\frac{\partial \Psi^*}{\partial x} - \Psi^*\frac{\partial \Psi}{\partial x}\right). What are the units of J(x,t)J(x, t)? Comment: JJ is called the probability current, because it tells you the rate at which probability is "flowing" past the point xx. If Pab(t)P_{ab}(t) is increasing, then more probability is flowing into the region at one end than flows out at the other. (b) Find the probability current for the wave function in Problem 1.9. (This is not a very pithy example, I'm afraid; we'll encounter more substantial ones in due course.)

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