6B (Quantum Mechanics)

H = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} .

Since

χ^{''} = - k^{2} χ

Hχ = \frac{ℏ ^{2} k ^{2}}{2 m} χ .

Thus

E = \frac{ℏ ^{2} k ^{2}}{2 m} \geq 0, p = ℏ k,

where the momentum follows because

- i ℏ d χ / d x = ℏ k χ

b

ρ = ∣ ψ ∣^{2}, J = \frac{ℏ}{2 mi} (ψ^{*} \frac{\partial ψ}{\partial x} - ψ \frac{\partial ψ ^{*}}{\partial x}) .

ψ (x, t) = χ (x) e^{- i Et /ℏ}

, so

ρ = ∣ χ ∣^{2}

is independent of time. The continuity equation then gives

\partial J / \partial x = 0

. Hence

J is independent of x .

A e^{ik x}

carries current

(ℏ k / m) ∣ A ∣^{2}

. The incident, reflected, and transmitted currents are therefore

J_{in} = \frac{ℏ k}{m}, J_{ref} = - \frac{ℏ k}{m} ∣ R ∣^{2}, J_{tr} = \frac{ℏ k}{m} ∣ T ∣^{2} .

Spatial constancy of the current gives

J_{in} + J_{ref} = J_{tr},

and hence

∣ R ∣^{2} + ∣ T ∣^{2} = 1 .

This is conservation of probability flux: every incident particle is either reflected or transmitted.

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