If an electron and a photon have the same wavelength, which physical quantity will definitely have the same value for both?

Introduction / Context:
In modern physics, wave particle duality tells us that both matter particles like electrons and radiation like photons can show wave behaviour. De Broglie and Einstein relations connect momentum and energy with wavelength and frequency. This question checks whether you can pick the correct quantity that must match when an electron and a photon share the same wavelength.

Given Data / Assumptions:

• We are considering one electron and one photon.
• Both are assumed to have the same de Broglie or electromagnetic wavelength.
• The Planck constant h is the same for both.
• The electron has non zero rest mass, while the photon has zero rest mass.
• We ignore external fields and focus on idealised free particles.

Concept / Approach:
For any particle that shows wave behaviour, the de Broglie relation connects linear momentum p with wavelength lambda through p = h / lambda. This relation holds for photons and for matter waves such as electrons. Therefore, if lambda is the same for the electron and the photon, and h is a universal constant, they must have the same linear momentum. Their velocities and energies, however, are determined by additional properties like mass and dispersion relations, so they will generally not match.

Step-by-Step Solution:
Step 1: Write the de Broglie relation for any particle that exhibits wave behaviour as p = h / lambda.Step 2: Note that h, the Planck constant, is the same for the electron and the photon.Step 3: Since the wavelength lambda is given to be the same for both particles, the momentum p computed from h / lambda must also be the same.Step 4: Conclude that linear momentum is the quantity that must match, while velocity and energy can differ because of the difference in rest mass and dispersion relations.

Verification / Alternative check:
For a photon, momentum can also be written as p = E / c, where E is energy and c is the speed of light. For an electron, classical or relativistic formulas relate momentum, mass, and velocity in different ways. Because c is fixed and the electron speed is usually less than c, energy and speed do not generally match between the two particles. However, using the common wavelength and the relation p = h / lambda directly shows that the linear momenta must coincide. This cross check confirms the choice of linear momentum as the correct answer.

Why Other Options Are Wrong:
Option a is wrong because wavelength does not uniquely determine speed for massive particles; the electron speed depends on its kinetic energy and mass. Option c is incorrect since angular momentum is not determined solely by wavelength; it depends on orbital or spin properties. Option d is not valid because energy expressions differ for photons and electrons; for photons E = hc / lambda, but for electrons energy includes rest mass and kinetic terms. Option e is clearly wrong because the photon has zero rest mass, while the electron has non zero rest mass, irrespective of wavelength.

Common Pitfalls:
Students sometimes equate wavelength directly with energy without considering the type of particle, assuming that same wavelength implies same energy. This is true only within a single category such as photons. Another mistake is to forget that massless and massive particles obey different energy momentum relations. Keeping in mind the universal de Broglie relation p = h / lambda helps focus on linear momentum as the common link between wave behaviour of different particles.

Final Answer:
Their linear momentum, because momentum is proportional to h divided by wavelength.