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Physics · Advanced university

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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Prerequisites

Wave mechanics, complex amplitudes and boundary matching.

Study 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.

Worked examples

What determines transmission probability in one-dimensional scattering?

Current accounts for both amplitude and propagation factors.

The transmitted-to-incident probability-current ratio

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

Conservation gives R+T=1.

0.7

Check your understanding

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

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

    1. Only the incident phase
    2. A factor of zero
    3. Only squared amplitude with no correction
    4. Their appropriate current ratio

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

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

Answers and marking guidance

Answers and marking notes
  1. 0.25

    With equal flux factors, T is squared amplitude magnitude.

  2. Their appropriate current ratio

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

  3. 0.4

    Multiply the flux factor by squared amplitude ratio.

  4. 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.

Further reading and review references

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