Introduction to Quantum Mechanics · Problem 2.32

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · scattering_coefficients

Problem

Derive Equations 2.170 and 2.171. Hint: Use Equations 2.168 and 2.169 to solve for

C

and

D

in terms of

F

C = \left[\sin(la) + i\frac{k}{l}\cos(la)\right]e^{ika}F; \quad D = \left[\cos(la) - i\frac{k}{l}\sin(la)\right]e^{ika}F.

Plug these back into Equations 2.166 and 2.167. Obtain the transmission coefficient, and confirm Equation 2.172.