Understanding how the de Broglie wavelength relates to kinetic energy for a charged particle in a potential difference

Outline (skeleton)

• Hook: Waves in the lab aren’t just a metaphor; they’re real in quantum physics, and de Broglie’s idea shows up in a simple formula.

• Core idea: The de Broglie wavelength relates to momentum, λ = h/p. Then connect momentum to kinetic energy: p = √(2 m KE).

• Special case for charged particles: When a particle is accelerated through a voltage V, KE = qV, so λ = h / √(2 m q V). For an electron, q = e, giving λ = h / √(2 e m V).

• Quick contrast with other options: Why A is right, and B, C, D don’t fit this context.

• Example to ground the concept: plug in numbers for electrons at 100 V and 1000 V, showing how wavelength shrinks as voltage rises.

• Why this matters: real-world intuition—electron diffraction, electron microscopes, and what wavelength means for resolution.

• Simple memory aids and caveats: non-relativistic caveat, and a few tips to remember the relation.

• Closing thought: the wave-particle duality is practical, not just philosophical.

Article

If you’ve ever watched a beam of electrons and wondered why it can act like a wave, you’re in good company. The de Broglie wavelength is the bridge between the particle-side and the wave-side of quantum reality. It’s the tiny ruler you use to measure how a particle’s wave nature shows up in experiments.

Let’s start with the clean, fundamental relation: the de Broglie wavelength λ is connected to momentum p by a simple equation:

• λ = h / p

Here, h is Planck’s constant, about 6.626 × 10^-34 J·s. Momentum p is the product of mass and velocity for a non-relativistic particle, but there’s a handy way to tie momentum to the energy the particle carries.

Momentum and kinetic energy sit in a tidy relationship. If a particle has kinetic energy KE and mass m, then:

• p = √(2 m KE)

That means you can rewrite the de Broglie relation in terms of energy:

• λ = h / √(2 m KE)

This is already pretty useful. But many NEET physics questions involve charged particles—like electrons—being sped up through a voltage. When a charged particle passes through a potential difference V, it gains kinetic energy equal to the charge times the voltage:

• KE = qV

For an electron, q = e (the elementary charge, about 1.602 × 10^-19 C). Substituting KE = eV into the wavelength formula gives a neatly packaged expression:

• λ = h / √(2 m e V)

If you’re looking at the options in a multiple-choice style, this is the expression that matches the description “in terms of kinetic energy” for a charged particle accelerated by a voltage. In many textbooks or exams, you’ll see it written with e for the charge of the electron, so:

• λ = h / √(2 e m V)

This is a non-relativistic result. It’s perfectly good for voltages that don’t push the particle toward speeds where Einstein’s relativity would matter. At very high voltages, you’d refine the formula to keep pace with the slower-than-expected speed-up near light speed. But for typical NEET-level questions, the non-relativistic form serves nicely.

So why is A the right pick, and why do the others not fit?

• A. λ = h / √(2 e m V) — This comes directly from λ = h/p, with p = √(2 m KE) and KE = eV, giving the clean dependency on the mass, charge, and voltage. It’s the one that expresses wavelength through kinetic energy in this charged-particle context.

• B. E = -13.6 × (Z² / n²) × (μ / mₑ) — That’s a hydrogenic energy level formula (Rydberg-style). It’s about bound-state energies, not de Broglie wavelengths.

• C. R(1/n1² − 1/n2²) — That’s a difference in energy levels in hydrogen-like systems; again, not a wavelength-energy relation.

• D. K.E. = 2E — That’s not a general correspondence you use to connect energy and wavelength. It doesn’t follow from the de Broglie idea.

To ground this with a concrete feel, let’s do a quick calculation with electrons:

• Take an electron accelerated through V = 100 volts. Its kinetic energy is KE = eV = (1.6 × 10^-19 C)(100 V) = 1.6 × 10^-17 J.

• Mass of electron m = 9.11 × 10^-31 kg.

• Momentum p = √(2 m KE) = √(2 × 9.11 × 10^-31 kg × 1.6 × 10^-17 J) ≈ √(2.92 × 10^-47) ≈ 5.4 × 10^-24 kg·m/s.

• Wavelength λ = h / p ≈ (6.626 × 10^-34 J·s) / (5.4 × 10^-24 kg·m/s) ≈ 1.23 × 10^-10 m, or about 0.12 nanometers.

Now push the voltage up to 1000 V:

• KE = eV = 1.6 × 10^-16 J.

• p = √(2 m KE) ≈ √(2 × 9.11 × 10^-31 × 1.6 × 10^-16) ≈ √(2.91 × 10^-46) ≈ 1.7 × 10^-23 kg·m/s.

• λ ≈ 6.626 × 10^-34 / 1.7 × 10^-23 ≈ 3.9 × 10^-11 m, or about 0.039 nm.

See the trend? When you crank up the voltage, the electron speeds up, its momentum goes up, and its wavelength gets shorter. The wave aspect becomes finer—think of a tighter ripple in a pond as you throw a bigger rock: the wavelength shrinks as energy climbs.

Why does this matter beyond math trivia?

• Electron diffraction: The wave nature of electrons is not just a novelty. In crystal lattices, electrons interfere in ways that reveal the spacing of atoms. The wavelength sets the scale of those interference patterns. As you increase voltage, the wavelength shortens, and the diffraction pattern changes accordingly. This is a direct, tangible demonstration of wave-particle duality.

• Electron microscopes: The resolving power of an electron microscope hinges on the electron wavelength. Shorter wavelengths (higher voltages) can resolve finer details, which is why transmission electron microscopes often work with electrons accelerated to quite high voltages.

• Conceptual intuition: The idea that a particle’s energy controls its wavelength helps connect multiple NEET topics—kinetic energy, momentum, and wave phenomena—into one coherent picture. It’s a nice example of how energy scales translate into observable phenomena in the lab.

A few quick notes to keep in mind, so you can recall this on the fly:

• The core idea is λ = h/p. If you can get p or KE, you’re almost there.

• For a charged particle accelerated through V, KE = qV. For an electron, q = e.

• Substituting p = √(2 m KE) gives λ = h / √(2 m e V). That’s the neat, compact form.

• This is non-relativistic. At very high voltages, you’d need to adjust for relativity; the basic logic—energy increases momentum, which reduces wavelength—still holds, but the exact numbers shift a bit.

• The physics behind the other options isn’t wrong in their own realms; they just describe different phenomena (hydrogen energy levels, transitions, or unrelated energy relations). The de Broglie wavelength formula in terms of kinetic energy is the right fit when you’re dealing with particles speeding up in an electric field.

If you’re trying to memorize it, here’s a simple way to anchor it: think of a wave ruler (the de Broglie ruler) that the particle carries. The ruler length, λ, shrinks as the particle’s momentum grows. Since momentum grows with the square root of kinetic energy, a rise in energy only modestly tightens the ruler at first, but as energy climbs, the wavelength keeps shrinking. For electrons, voltage is a very handy handle on that energy—so λ = h / √(2 e m V) becomes the go-to expression you’ll reach for when you want a quick estimate.

A tiny digression that fits naturally here: the deeper you go into quantum mechanics, the more you realize how often simple constants and small relations carry heavy meaning. Planck’s constant isn’t just a weird number; it’s a bridge between the everyday world of energy we can measure and the wavelike behavior that reveals itself under the right conditions. Debates about wave-particle duality aren’t just philosophy; they’re about whether a beam of electrons will blur into an interference pattern or stay as a crisp particle. In math terms, it’s all about λ and p, and how one maps to the other with your preferred energy or voltage.

For a quick mental recap:

• Start with λ = h/p.

• If KE = ½ m v², then p = √(2 m KE).

• If you’re dealing with a charged particle through voltage V, KE = qV, so λ = h / √(2 m q V). For an electron, q = e: λ = h / √(2 e m V).

• Higher voltage = higher KE = higher p = shorter λ.

And a gentle reminder: this relationship is most reliable in the non-relativistic regime. If you ever push electrons toward speeds where relativistic effects matter, you’ll see the math bend a little to keep the physics honest.

If you’re curious to explore further, you can try a few quick experiments or simulations: sketch how the interference pattern in an electron diffraction setup shifts as you vary the accelerating voltage, or look up historical electron diffraction photographs from crystal lattices to visualize what a changing wavelength looks like in real data. It’s one of those topics where the math is clean, but the pictures you get can be surprisingly striking.

So, in one crisp line: when you want the de Broglie wavelength of a charged particle in terms of kinetic energy, especially for electrons accelerated by voltage, the right expression is λ = h / √(2 e m V). It’s a compact formula, but it opens a window into how the tiny world refuses to be pinned down as just a “particle” or just a “wave”—it’s both, and with a wavelength you can actually measure.

If you’d like, we can walk through more numerical examples or connect this to other NEET physics ideas, like the link between wavelength and resolution in diffraction experiments, or how this ties into modern instruments. The more you play with the numbers, the more the wave-particle dance starts to feel like a familiar, almost everyday phenomenon.