## Section 9.3: The Probability Current Density

The animation depicts a Gaussian wave packet in an infinite square well. The packet has no initial momentum, but after its left edge hits the infinite wall, the packet begins to move to the right.   The animation also shows the probability current density and the total probability current.  In the animation, ħ = 2m = 1.

What is the probability current density?  We can construct a probability relationship from the Schrödinger equation which, for real V(x), yields2

Jx/∂x + ∂ρ/∂t = 0. (9.3)

This relationship takes the same form as the differential statement of charge-current conservation from electromagnetism. Here, however ρ(x,t) and J(x,t) are not the electric charge and electric current densities in the sense of electromagnetism, they are instead

ρ(x,t) = Ψ*(x,t)Ψ(x,t) , (9.4)

and

J(x,t) = (ħ/2mi) {Ψ*(x,t) [∂ Ψ(x,t)/∂x] − [∂ Ψ*(x,t)/∂x] Ψ(x,t)} , (9.5)

which are the probability density and the probability current density, in one dimension, respectively.3

As in Section 7.6, consider an arbitrary complex wave function that we choose to write as

Ψ(x,t) = Aexp[iθ(x,t)]. (9.6)

Consider the probability density and the probability current density in this representation:

ρ(x,t) = Ψ*(x,t)Ψ(x,t) = A2(x,t) (9.7)

and

J(x,t) = (ħ/m) A2(x,t) [∂ θ(x,t)/∂x] . (9.8)

This tells us that the probability density is insensitive to the phase of the wave function, but the probability current density is sensitive to spatial changes in phase. Given that the phase is represented by color in this animation, spatial variations in the color of Ψ(x,t) signify a non-zero probability current density. Restart the animation and watch how the probability current density evolves with time.

The insensitivity of the probability density to phase may suggest that the phase of the wave function is unimportant (even though the probability current does depend on the phase). When we consider wave functions that are superpositions of two or more states, we will find that relative phase does matter.

2Consider the construction: Ψ*(x,t)SE − SE*Ψ(x,t) where the complex conjugate of the Schrödinger equation, SE*, with V(x) real, is

−(ħ2/2m)(∂2/∂x2)Ψ*(x,t) + V(x) Ψ*(x,t) = ∂ Ψ *(x,t)/∂t . (9.1)

This ultimately, after a fair amount of algebra, yields:

∂/∂x [ (ħ/2mi) {Ψ*(x,t) [∂ Ψ (x,t)/∂x] − [∂ Ψ *(x,t)/∂x]Ψ (x,t)}] + ∂/∂t[Ψ *(x,t) Ψ (x,t)] = 0 , (9.2)

again as long as V(x) is real. 3In three-dimensions, Eqs. (9.3), (9.4), and (9.5) can be generalized to ∇ · J + ∂ρ/∂t = 0,   ρ(r,t) = Ψ*(r,t)Ψ (r,t), and  J(r,t) = (ħ/2mi) {Ψ *(r,t) Ψ (r,t)] − [Ψ *(r,t)]Ψ (r,t)}.