Index
Chapter 4 · Item 4.11
Delta well: transmission and reflection
The zero-width limit of the positive-energy finite well
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Guided reading

Take the zero-width limit, \(a\to0\), of the finite potential well with \(E>0\). Region II is suppressed and the potential becomes \(V(x)=-\alpha\delta(x)\). The goal is to determine the transmission and reflection coefficients for a particle incident from the left.

Delta potential well
Book diagram of the delta potential well, with regions I and III separated at x equals zero.
Figure 4.9: delta potential well. In the zero-width limit only regions I and III remain.
Regional wave functions
\[\psi_{\mathrm I}(x)=Ae^{ikx}+Be^{-ikx}=\psi_i(x)+\psi_r(x),\qquad\psi_{\mathrm{III}}(x)=Fe^{ikx}=\psi_t(x)\]
\[k^2=\frac{2m}{\hbar^2}|E|>0\]

The coefficients \(A\), \(B\), and \(F\) follow from the boundary conditions at \(x=0\).

Wave-function continuity
\[\psi_{\mathrm I}(0)=\psi_{\mathrm{III}}(0)\qquad\Longrightarrow\qquad A+B=F\]

The derivative relation is obtained by integrating the Schrödinger equation across the interface.

\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}-|\alpha|\delta(x)\psi(x)=|E|\psi(x)\]
Integrated interface relation at \(x=0\)

Integrating from \(-\epsilon\) to \(+\epsilon\) and taking \(\epsilon\to0\) gives the derivative on the left of the interface minus the derivative on the right:

\[\psi'(0^-)-\psi'(0^+)=\frac{2m|\alpha|}{\hbar^2}\psi(0)\]

Before taking the limit, the regional derivatives are

\[\left.\frac{d\psi_{\mathrm I}}{dx}\right|_{-\Delta x}=ik\!\left(Ae^{-ik\Delta x}-Be^{ik\Delta x}\right),\qquad\left.\frac{d\psi_{\mathrm{III}}}{dx}\right|_{\Delta x}=ikFe^{ik\Delta x}\]

Thus, when \(\Delta x\to0\),

\[\psi'_{\mathrm I}(0^-)=ik(A-B),\qquad\psi'_{\mathrm{III}}(0^+)=ikF\]
Coefficients

Using \(\psi(0)=A+B\) in the derivative relation, together with \(A+B=F\), yields

\[B=\frac{i\sqrt{\mathcal E'/|E|}}{1-i\sqrt{\mathcal E'/|E|}}\,A,\qquad F=\frac{A}{1-i\sqrt{\mathcal E'/|E|}}\]
\[\mathcal E'=\frac{m|\alpha|^2}{2\hbar^2}\]
Transmission and reflection
\[T=\frac{1}{1+\mathcal E'/|E|},\qquad R=\frac{1}{1+|E|/\mathcal E'}\]

The coefficients obey \(R+T=1\). The larger the ratio \(|E|/\mathcal E'\), the greater the transmission probability.

Coefficients as functions of energy
Book graph of transmission and reflection coefficients as functions of absolute E divided by script E prime.
Figure 4.10: the transmission \(T\) and reflection \(R\) coefficients as functions of \(|E|/\mathcal E'\). The two curves correspond to the displayed expressions for \(T\) and \(R\).
Delta barrier

For \(V(x)=\alpha\delta(x)\), a delta-type barrier, the transmission and reflection coefficients are the same as for the delta well when the incident particle has \(E>0\).

Exercise-ready boundary

This page supports guided exercises on the positive-energy delta-well scattering calculation.

  • Use from this page: regional wave functions, the interface conditions, \(\mathcal E'\), and the coefficients \(T\) and \(R\).
  • Keep in the book: the complete integral calculation across the delta interface.
  • Good exercise balance: apply continuity and the derivative relation before forming \(T\) and \(R\).
Practice anchors

Use these anchors to build compact exercises from the delta-well scattering calculation.

  • Focus: positive-energy scattering from \(V(x)=-\alpha\delta(x)\).
  • Conceptual check: identify the incident, reflected, and transmitted amplitudes.
  • Key equation: \[T=\frac{1}{1+\mathcal E'/|E|},\qquad\mathcal E'=\frac{m|\alpha|^2}{2\hbar^2}\]
  • Boundary: use the displayed conditions to set up the calculation; use the book for the complete interface integration.
  • Typical task: obtain \(T\), \(R\), and verify \(R+T=1\).
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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.