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Reflection And Transmission

Key idea: Understand reflection and transmission of a matter wave at a potential barrier, including reflection and transmission probabilities (optional enrichment for A Level Physics).

  • A-level H2 Physics topic extensions
On this page

Learning objectives

  • Explore potential-barrier transmission, tunnelling and X-ray production and spectra beyond the H2 syllabus.
Optional / Enrichment (9478 scope)

Reflection/transmission coefficients for matter waves at barriers are not explicitly required by 9478. Treat this page as enrichment connected to the wavefunction picture used in tunnelling.

1. Definitions (Must Know)

Matter wave at a finite potential barrierA particle with energy E below rectangular barrier height U zero approaches a barrier of width L. The incident and reflected wave components are on the left; the wavefunction decays exponentially within the classically forbidden barrier, and a smaller transmitted wave continues on the right.Potential-energy profileUxparticle energy E < U₀barrier height U₀width LWavefunction amplitude ψ(x)incident + reflectedexponential decaysmaller transmitted waveclassically forbidden, but ψ ≠ 0
Near a finite barrier, an incident matter wave can be partly reflected and partly transmitted; probability current is conserved.

A. Reflected and transmitted waves

At a boundary (e.g. a potential step or barrier), a matter wave can split into:

  • a reflected component (travelling back),
  • a transmitted component (travelling forward).

The incident and reflected components can interfere to form a standing-wave-like pattern on the incident side.

B. Reflection coefficient, R

The reflection coefficient, R, is the probability the particle is reflected by the barrier.

C. Transmission coefficient, T

The transmission coefficient, T, is the probability the particle is transmitted through the barrier.

D. Probability conservation (simple case)

In a simple 1D model with no absorption, the total probability is conserved: R + T = 1

2. Key Ideas (What Earns Marks)

  • At a boundary, matter waves can be partly reflected and partly transmitted (even if E > U₀).
  • For tunnelling (E < U₀), the transmitted wave amplitude is small but non-zero.
  • Transmission probability depends strongly on barrier properties (height, width) and on particle mass.
  • In many simple models: R + T = 1

3. Detailed Explanations

A. Why reflection happens even for particles

In the wave description, a change in potential acts like a change in “medium” for the wavefunction. Matching conditions at the boundary generally requires a reflected component as well as a transmitted component.

B. How to explain a standing-wave-like pattern

On the incident side, the superposition of the incident and reflected waves produces an interference pattern (nodes/antinodes), similar to a standing wave.

4. Common Mistakes

  • Thinking “reflection only happens when E < U₀” (reflection can happen for E > U₀ too, in the wave picture).
  • Interpreting R and T as amplitudes (they are probabilities).
  • Forgetting to state the condition for tunnelling (E < U₀).

5. Exam Tips

  • If asked to define R and T, use “probability” language, not “percentage of the wave”.
  • If the question mentions “no absorption”, state R + T = 1.
  • For qualitative comparisons: “thicker/higher barrier” and “larger mass” both reduce T.

6. Worked Examples

Modelled example 1

Quick definitions

Core

Problem

Define R and T for a matter wave incident on a potential barrier, and state their relation when there is no absorption.
Study the worked solution

  1. Define reflection

    Method

    R is the probability that the particle is reflected.

    Reason

    It describes an outcome probability, not a wave amplitude.

    Working

    0 ≤ R ≤ 1

  2. Define transmission

    Method

    T is the probability that the particle is transmitted.

    Reason

    It likewise refers to the particle outcome.

    Working

    0 ≤ T ≤ 1

  3. Conserve probability

    Method

    Without absorption, R + T = 1.

    Reason

    Reflection and transmission exhaust the possible outcomes in the simple model.

    Working

    R + T = 1

Common misconception 2

Trend question

Find and correct the mistake

Learner claim

For E < U₀, a learner says a wider barrier gives the particle more distance to “borrow energy”, so T increases. Diagnose the claim and state the correct trend.

Try this before viewing the solution

Effect of increasing barrier width

View solution step by step

  1. Reject the mechanism

    Method

    The particle does not borrow energy to cross the barrier.

    Reason

    Its total energy remains E before and after transmission.

    Working

    E conserved

  2. Use wavefunction decay

    Method

    The wavefunction decays through the classically forbidden region.

    Reason

    A wider region produces more decay before the far boundary.

    Working

    T ∝ e^(-2κ L)

  3. State the trend

    Method

    T decreases strongly as barrier width increases.

    Reason

    Probability density beyond the barrier becomes smaller.

    Working

    L↑ ⇒ T↓

7. Mind Stretchers

Mind stretcher 1: Can you have transmission when E < U₀ without violating energy conservation?Extension

Show Answer

Yes. The particle’s total energy E is the same on both sides; tunnelling is about non-zero probability from the wavefunction, not about the particle “gaining” energy.

8. Optional (Enrichment)

A. One common exponential form

Many simplified models give a transmission probability that decays exponentially with barrier width: T ∝ e^(-2κ L) where κ depends on particle mass and the gap (U₀-E).

Continue with the next resource in this course.

Course and syllabus information
Course
A-level H2 Physics topic extensions
Syllabus scope
Beyond the syllabus
Edition
A-level H2 Physics topic extensions