One-Dimensional Scattering and Tunneling

Scattering asks how an incoming wave divides into reflected and transmitted parts. A finite barrier can transmit even below its height, and a potential step can reflect even above its height. This chapter derives those effects by matching wavefunctions and comparing probability currents.

Transmission measures current

For a real, time-independent scalar potential, constant mass, and open channels on both sides, write a left-incident solution as

$$\psi_L=e^{ik_Lx}+r e^{-ik_Lx},\qquad \psi_R=t e^{ik_Rx},\qquad k_L,k_R>0.$$

A wave $Ae^{ikx}$ carries current $\hbar k|A|^2/m$. Consequently,

$$R=|r|^2,\qquad T=\frac{k_R}{k_L}|t|^2,\qquad R+T=1.$$

The velocity factor matters: squared amplitudes alone are not generally the probabilities. If the right asymptotic region is forbidden, its admissible decaying tail carries no outgoing current. A finite forbidden barrier differs because there is a propagating region beyond its second interface.

For a rectangular barrier of width $a$, the opaque-barrier transmission has the leading exponential $T\propto e^{-2\kappa a}$, with $\kappa=\sqrt{2m(V_0-E)}/\hbar$. The amplitude decay and the probability exponent differ by a factor of two. No energy borrowing occurs in this stationary process.

Reading sequence

Page	Central question
Potential Step	How can one finite interface reflect an above-threshold wave?
Finite Potential Barrier	What do two interfaces add to exact scattering?
Rectangular Barrier Tunneling	How does forbidden-region opacity control transmission?
Transfer Matrix Method	How are many interfaces composed systematically and stably?
Reflection and Transmission Coefficients	Why are scattering probabilities current ratios?
Resonant Transmission	How do quasi-bound states create narrow transparency windows?
Quantum Tunneling	Which phenomena count as tunneling, and what common stories are misleading?
Scattering from a Delta Potential	How does a point interaction encode scattering and a bound-state pole?
Wave Packets and Scattering	How do stationary amplitudes become late-time particle probabilities?
Tunneling Applications: First Encounters	Which part of elementary tunneling survives in real applications?
One-Dimensional Scattering Revisited	How do these models fit into $S$-matrix, pole, resonance, and semiclassical language?

From one interface to resonances

Read current ratios and the potential step before the finite barrier. Transfer matrices organize multiple interfaces, while resonant transmission explains how a leaky intermediate state can produce a sharp peak. For a prepared packet, average over its incident spectrum; one monochromatic transmission value is accurate only over a sufficiently narrow bandwidth.

The scattering volume develops channel matrices, poles, and general collision theory. Semiclassical barrier penetration extends the elementary exponential to smoothly varying barriers.

Exercises

At a step with $k_R=k_L/2$, matching gives $t=4/3$ and $r=1/3$. Are these amplitudes physically allowed?

Solution

Yes. $R=1/9$ and $T=(1/2)(16/9)=8/9$, so current is conserved. A transmitted amplitude larger than one is compatible with a smaller transmitted velocity. Calling $|t|^2$ the transmission probability would give the wrong answer.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.