Chapter 4 defined the probability current for a general wave function

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Free particles, wave packets and tunnelling

Problem

Chapter 4 defined the probability current

j(x,t) = \dfrac{\hbar}{2mi}\left(\Psi^*\dfrac{\partial\Psi}{\partial x} - \Psi\dfrac{\partial\Psi^*}{\partial x}\right)

for a general wave function. For

x < 0

at a potential step,

\psi = Ae^{ik_1x} + Be^{-ik_1x}

. Apply Chapter 4's formula directly to this

\psi

and show that the net current reduces to

j = (\hbar k_1/m)(|A|^2 - |B|^2)

, with no leftover term mixing

A

and

B