Scattering Examples

Scattering examples are convention-heavy because states are not usually normalized like bound states. The central observable is flux, or in higher dimensions a cross section, not a raw wave amplitude.

Example Routes

Need	Learn	Main Check
One-dimensional current	Probability Current	Current direction and sign
Potential step	Potential Step	Velocity factor in $T$
Reflection and transmission	Reflection and Transmission Coefficients	Flux ratios
Rectangular barrier	Rectangular Barrier Tunneling	Resonance and opaque-barrier limits
Transfer matrices	Transfer Matrix Method	Matrix ordering
WKB tunneling	WKB Barrier Tunneling	Turning-point assumptions
Born approximation	Gaussian Potential Born	Weak potential and normalization convention

Core One-Dimensional Check

For a right-moving plane wave $\psi=Ae^{ikx}$,

$$j=\frac{\hbar k}{m}\lvert A\rvert^2.$$

Thus a transmitted amplitude must be converted into a current before forming a probability. If the incident wave number is $k$ and the transmitted wave number is $q$, then the common one-dimensional structure is

$$T=\frac{q}{k}\lvert t\rvert^2,$$

with convention-dependent definitions of $t$.

Higher-Dimensional Routes

For three-dimensional scattering, use the First Born Approximation and Scattering Cross Section pages. The worked Gaussian Potential Born example is the cleanest route for seeing the Fourier-transform structure without Coulomb long-range complications.

Common Mistakes

• Calling $\lvert t\rvert^2$ a transmission probability when velocities differ.
• Mixing left-incident and right-incident transfer-matrix conventions.
• Forgetting that bound-state normalization and scattering normalization are different.
• Using the Born approximation for a potential that is too strong or too long-ranged without qualification.
• Interpreting evanescent waves as carrying ordinary propagating current inside a barrier.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. A potential step has incident wave number $k$ and transmitted wave number $q$. Why can $T$ differ from $\lvert t\rvert^2$?

Solution

Transmission is a ratio of transmitted current to incident current. Since current for a plane wave is proportional to wave number times amplitude squared, $T=(q/k)\lvert t\rvert^2$ in the common convention.