At one instant a particle confined between and is described by (x) = cases Ae^ ikx &…

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 The Schrödinger equation and the Born rule

Problem

At one instant a particle confined between

x=0

and

x=L

is described by

\Psi(x) = \begin{cases}Ae^{ikx} & 0<x<L\\ 0 & \text{otherwise}\end{cases}

with

A

real and positive and

k

a real constant. A worked attempt at normalising it runs like this:“Normalisation means

\int|\Psi|^2dx=1

, and

|\Psi|^2

means

\Psi^2

, so I need

\int_0^L A^2e^{2ikx}dx=1

. Carrying this out,

\int_0^L e^{2ikx}dx = \dfrac{e^{2ikL}-1}{2ik}

, so

A^2 = \dfrac{2ik}{e^{2ikL}-1}

.”Identify the error, explain why this result should have raised suspicion on sight, and find the correct value of

A