# Scattering, currents and probability conservation

> Scattering models compare incoming and outgoing waves far from a localized interaction. Boundary conditions identify which waves are physically incident.

Course: [Advanced physics methods and spacetime](https://studybrix.com/en/courses/advanced-physics-methods/index.md)
Level: advanced
Prerequisites: Wave mechanics, complex amplitudes and boundary matching.

## Notes

### Asymptotic waves

Scattering models compare incoming and outgoing waves far from a localized interaction. Boundary conditions identify which waves are physically incident.

### Current

For a nonrelativistic one-dimensional wave, j=(ℏ/m) Im(ψ* ∂ψ/∂x). For a plane wave, current includes wave number as well as amplitude.

### Transmission

Transmission probability is a transmitted-to-incident flux ratio. It is not always the squared transmission-amplitude ratio when wave numbers differ.

### Conservation

For a real, time-independent nonabsorbing potential with the usual scattering channels, R+T=1. Absorption or additional channels require fuller accounting.

## Practice questions

### 1. What determines transmission probability in one-dimensional scattering?

A. The sum of amplitudes
B. Only potential colour
C. The transmitted-to-incident probability-current ratio
D. Always the phase alone

Answer: The transmitted-to-incident probability-current ratio

Explanation: Current accounts for both amplitude and propagation factors.

### 2. In a nonabsorbing two-channel problem, R=0.3. Find T.

Answer: 0.7

Explanation: Conservation gives R+T=1.

### 3. If transmitted and incident wave numbers are equal and amplitude ratio is 0.5, find T.

Answer: 0.25

Explanation: With equal flux factors, T is squared amplitude magnitude.

### 4. When unequal wave numbers occur, what must a transmission calculation include?

A. Only the incident phase
B. A factor of zero
C. Only squared amplitude with no correction
D. Their appropriate current ratio

Answer: Their appropriate current ratio

Explanation: For equal masses, the plane-wave current ratio includes kt/ki.

### 5. Use kt/ki=2 and squared transmission-amplitude ratio 0.2. Find T.

Answer: 0.4

Explanation: Multiply the flux factor by squared amplitude ratio.

### 6. Why can a wavefunction be nonzero in a classically forbidden finite barrier?

Sample answer / marking guidance: The Schrodinger equation admits an evanescent solution in the barrier. Matching through a finite region can produce transmitted flux beyond it.

Explanation: Credit finite-barrier matching, not a classical trajectory with negative kinetic energy.

### 7. Explain why R+T=1 must not be imposed blindly on a model with absorption.

Sample answer / marking guidance: Absorption removes flux from the explicitly represented outgoing channels. A complete conservation equation must include the absorbed or other-channel contribution.

Explanation: Credit the missing channel accounting.

### 8. For ψ=A exp(ikx), what determines the sign of current when mass is positive?

A. The absolute position
B. The sign of k
C. The normalization label
D. The sign of abs(A)²

Answer: The sign of k

Explanation: j=ℏk abs(A)²/m.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.