Guided reading
Scattering states are not square-normalizable over all space. Their physical normalization is instead expressed through probability current: the incident flux splits into reflected and transmitted flux.
Probability current
\[j=\frac{\hbar}{2mi}\left(\psi^*\frac{d\psi}{dx}-\psi\frac{d\psi^*}{dx}\right)\]
For a plane wave \(Ae^{ikx}\), \(j=+(\hbar k/m)|A|^2\); for \(Be^{-ikx}\), \(j=-(\hbar k/m)|B|^2\).
Continuity equation
\[\frac{\partial |\Psi|^2}{\partial t}+\frac{\partial j}{\partial x}=0\]
In a stationary, source-free one-dimensional problem, total current is constant across the interaction region.
Scattering ansatz
\[\psi_I=Ae^{ik_1x}+Be^{-ik_1x},\qquad \psi_{III}=Fe^{ik_3x}\]
No left-moving term appears on the far right when particles are incident only from the left.
Reflection and transmission
\[R=\frac{|j_{\rm ref}|}{j_{\rm inc}}=\frac{|B|^2}{|A|^2}\]
\[T=\frac{j_{\rm trans}}{j_{\rm inc}}=\frac{k_3}{k_1}\frac{|F|^2}{|A|^2}\]
For real, time-independent potentials, current conservation gives \(R+T=1\).
Conservation at the interface
For a real potential, Schrödinger evolution preserves probability. In a stationary scattering state, \(\partial|\Psi|^2/\partial t=0\), so the continuity equation requires \(dj/dx=0\). Evaluating far from the interaction region gives \(j_{in}+j_{ref}=j_{trans}\). Dividing by positive incident current produces \(R+T=1\).
Exercise-ready boundary
This page is designed to support short guided exercises on: Incident, reflected and transmitted waves and the current-based definitions of R and T.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Practice anchors
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: Incident, reflected and transmitted waves and the current-based definitions of R and T.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[j=\frac{\hbar}{2mi}\left(\psi^*\frac{d\psi}{dx}-\psi\frac{d\psi^*}{dx}\right)\]
- Equation: \[T=\frac{j_{\rm trans}}{j_{\rm inc}}=\frac{k_3}{k_1}\frac{|F|^2}{|A|^2}\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
Original book and previews:
Source note: Original auxiliary summary for this book-app, based on Chapter 4 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, derivations and exercises.