recall and use λ = h / p

Wave‑Particle Duality 🌊✨

In physics, many objects behave like both waves and particles. Think of a water ripple (wave) and a ball (particle). This dual nature is key to understanding quantum mechanics.

Why It Matters

It explains why electrons can interfere like waves, yet also hit a detector as individual spots.

Key Formula

De Broglie wavelength: \$\lambda = \frac{h}{p}\$

Where h is Planck’s constant (\$6.626\times10^{-34}\,\text{J·s}\$) and p is momentum (\$p = mv\$).

Analogy: The “Bouncing Ball”

Imagine a ball thrown at a wall. If the ball is very light and fast, it behaves like a tiny wave that can bend around obstacles. If it’s heavy, it behaves like a classic particle that just bounces straight back.

Quick Example

Calculate the wavelength of an electron moving at \$v = 1.0\times10^6\,\text{m/s}\$.

  1. Find momentum: \$p = mv\$ (\$m = 9.11\times10^{-31}\,\text{kg}\$).

  2. Compute \$p = 9.11\times10^{-31}\,\text{kg} \times 1.0\times10^6\,\text{m/s} = 9.11\times10^{-25}\,\text{kg·m/s}\$.

  3. Use \$\lambda = h/p\$:

    \$\lambda = \frac{6.626\times10^{-34}}{9.11\times10^{-25}} \approx 7.3\times10^{-10}\,\text{m}.\$

That’s about 0.73 nanometres – smaller than a cell!

Exam Tips 📚

  • Remember the formula: \$\lambda = h/p\$.

  • Always check units – \$h\$ in J·s, \$p\$ in kg·m/s, giving \$\lambda\$ in metres.

  • When given energy \$E\$, convert to momentum using \$p = \sqrt{2mE}\$ for non‑relativistic particles.

  • Use the De Broglie wavelength to explain diffraction patterns in exams.

Quick Quiz

QuestionAnswer
What is the De Broglie wavelength of a photon with energy \$E = 2.0\,\text{eV}\$?Use \$E = h\nu\$ and \$\lambda = c/\nu\$. Result: \$\lambda \approx 620\,\text{nm}\$.
If an electron has \$\lambda = 0.5\,\text{nm}\$, what is its momentum?\$p = h/\lambda \approx 1.32\times10^{-24}\,\text{kg·m/s}\$.