Solution

For a one-dimensional wavefunction,

ρ (x, t) = ∣ ψ (x, t) ∣^{2}, j (x, t) = \frac{ℏ}{2 mi} (ψ^{*} \frac{\partial ψ}{\partial x} - ψ \frac{\partial ψ ^{*}}{\partial x}) = \frac{ℏ}{m} Im (ψ^{*} \partial_{x} ψ) .

\partial_{t} ρ + \partial_{x} j = 0.

ψ (x, t) = ϕ (x) e^{- i Et /ℏ}

, the probability density is independent of time, so

\partial_{x} j = 0

and the probability current is constant in space.

ψ = A e^{ik x - i Et /ℏ},

one obtains

ρ = ∣ A ∣^{2}, j = \frac{ℏ k}{m} ∣ A ∣^{2} .

This is a momentum eigenstate with momentum

p = ℏ k

and, when it satisfies the free Schrodinger equation, energy

E = ℏ^{2} k^{2} / (2 m)

. Its constant density and current describe a spatially uniform beam carrying probability in the sign of

k

. It is not a normalizable wavefunction, so it represents an idealized state rather than a localized particle.

Solved by gpt-5.6-sol high.

 Ancestors (11)