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Physics · lesson 2 of 4

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.

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Learn the ideas

Prerequisites: Wave mechanics, complex amplitudes and boundary matching.

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.

Worked examples

Example 1

What determines transmission probability in one-dimensional scattering?

Current accounts for both amplitude and propagation factors.

The transmitted-to-incident probability-current ratio

Example 2

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

Conservation gives R+T=1.

0.7

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 What determines transmission probability in one-dimensional scattering?

Show answer and explanation

The transmitted-to-incident probability-current ratio

Current accounts for both amplitude and propagation factors.

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

Show answer and explanation

0.7

Conservation gives R+T=1.

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

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0.25

With equal flux factors, T is squared amplitude magnitude.

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

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Their appropriate current ratio

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.

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0.4

Multiply the flux factor by squared amplitude ratio.

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

Sample answer and marking guidance

The Schrodinger equation admits an evanescent solution in the barrier. Matching through a finite region can produce transmitted flux beyond it.

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 and marking guidance

Absorption removes flux from the explicitly represented outgoing channels. A complete conservation equation must include the absorbed or other-channel contribution.

Credit the missing channel accounting.

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

Show answer and explanation

The sign of k

j=ℏk abs(A)²/m.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.