An electron and a photon have same de Broglie wavelength. It implies that they have the same

Correct — A, linear momentum. The de Broglie relation is λ = h/p, and it holds for the electron as a matter wave and for the photon alike, since a photon's momentum is also h/λ. Equal wavelengths therefore force equal momenta, and that is the only quantity the condition fixes. Energy, speed and angular momentum are all free to differ, because the two particles relate energy to momentum by completely different rules.

• (b)energy — A photon's energy is pc, while a non-relativistic electron's is p²/2m. At the same momentum these give quite different numbers — the photon's energy is far the larger for the momenta involved in ordinary experiments.
• (c)speed — A photon always travels at the speed of light. An electron with the same momentum moves much more slowly, since v = p/m and the electron has mass.
• (d)angular momentum — Angular momentum describes rotation or orbital motion and does not follow from the de Broglie wavelength at all. A photon's intrinsic spin angular momentum is fixed regardless of its wavelength.

Louis de Broglie proposed that every particle has a wavelength λ = h/p, with h the Planck constant. The relation gives measurable wavelengths only when the momentum is very small, which is why electron diffraction is observable and cricket-ball diffraction is not. Since the same expression governs the photon, wavelength and momentum are locked together for both, whatever else differs between them.

Working the item is a matter of writing λ = h/p and reading it backwards: fixing λ fixes p, and nothing more. The tempting error is to slide from equal wavelength to equal energy, because for photons alone energy and wavelength are tied by E = hc/λ. That relation does not carry over to a massive particle. Davisson and Germer confirmed the electron's wave nature by diffracting electrons from a nickel crystal, and the electron microscope is the practical consequence — short wavelengths give resolution far beyond anything visible light allows.

• The de Broglie relation is λ = h/p, so equal wavelength means equal linear momentum for any two particles.
• A photon's energy is E = pc; a non-relativistic electron's is E = p²/2m, so equal momentum does not give equal energy.
• A photon travels at c while an electron of the same momentum moves far slower, since v = p/m.
• Davisson and Germer demonstrated electron diffraction from a nickel crystal, confirming matter waves.
• For an electron accelerated through a potential V, the wavelength is about 1.227 nanometres divided by the square root of V in volts.

• Carrying E = hc/λ across to a massive particle, where it does not hold.
• Assuming equal wavelength implies equal speed; it implies equal momentum instead.
• Confusing spin angular momentum with the orbital motion the wavelength describes.

A one-relation inference. Write the formula, see which quantity it constrains, and refuse to grant the others.

• practice — not a real PYQ

  The de Broglie wavelength associated with a moving particle is

  • (a)directly proportional to its momentum
  • (b)inversely proportional to its momentum
  • (c)independent of its momentum
  • (d)proportional to the square of its momentum

  Answer(b) inversely proportional to its momentum — λ = h/p, which is why heavy everyday objects have unmeasurably short wavelengths.

• practice — not a real PYQ

  An electron and a proton are moving with the same speed. Which one has the longer de Broglie wavelength?

  • (a)The proton
  • (b)The electron
  • (c)Both have equal wavelengths
  • (d)It depends on the direction of motion

  Answer(b) The electron — at equal speed the lighter particle has the smaller momentum, and λ = h/p is largest when p is smallest.