Dual Nature of Radiation and Matter

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Wave-Particle Duality — Key Facts

Light exhibits both wave-like (interference, diffraction, polarisation) and particle-like (photoelectric effect) behaviour. Matter (electrons, protons) also shows wave-like properties (de Broglie wavelength).

Photoelectric Effect:

Einstein’s equation: $E = h\nu - \phi = KE_{max}$

where h = Planck’s constant = 6.63 × 10⁻³⁴ J·s, ν = frequency, φ = work function

Threshold frequency: $\nu_0 = \frac{\phi}{h}$

Work function: $\phi = h\nu_0 = \frac{hc}{\lambda_0}$ (λ₀ = threshold wavelength)

$$KE_{max} = \frac{1}{2}mv_{max}^2 = eV_0$$

where V₀ = stopping potential (voltage needed to stop fastest electrons)

⚡ JEE Exam Tip: For photoelectric effect, intensity doesn’t affect electron energy — it only affects number of electrons. Doubling intensity doubles photoelectron count but doesn’t change maximum kinetic energy.

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

de Broglie Hypothesis:

Every particle has wave-like properties: $$\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}}$$

where p = momentum, m = mass, K = kinetic energy

For electron accelerated by potential V: $$KE = eV \Rightarrow p = \sqrt{2meV}$$ $$\lambda = \frac{h}{\sqrt{2meV}} = \frac{1.227}{\sqrt{V}} \text{ nm}$$

For thermal particles (at temperature T): $$KE \approx \frac{3}{2}k_B T \Rightarrow \lambda = \frac{h}{\sqrt{3mk_BT}}$$

Matter Waves:

• Not electromagnetic waves
• No charge (electrically neutral particles have de Broglie wavelength too)
• Probability interpretation: |ψ|² gives probability density (Born’s interpretation)

Davisson-Germer Experiment (1927):

Confirmed electron wave nature by showing electron diffraction from nickel crystal. Observed diffraction pattern matching Bragg’s law for λ = h/p.

⚡ JEE Exam Tip: de Broglie wavelength of electron is typically sub-nanometer (for 150 V, λ ≈ 0.1 nm), comparable to X-ray wavelengths. This is why electron microscopes can resolve much smaller features than optical microscopes.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Photon Picture of Light:

Energy: $E = h\nu = \frac{hc}{\lambda}$ Momentum: $p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda}$

For photon: $E = pc$ always (massless particle) For massive particle: $E^2 = p^2c^2 + m^2c^4$

de Broglie Relation (Advanced):

From $E = h\nu$ and $p = h/\lambda$: $$\vec{p} = \hbar \vec{k}$$ $$E = \hbar \omega$$

where $\hbar = \frac{h}{2\pi}$ and $\vec{k}$ is wave vector (magnitude $k = 2\pi/\lambda$)

Phase and Group Velocity:

• Phase velocity: $v_p = \frac{\omega}{k} = \frac{E}{p} = \frac{c^2}{v}$ (can exceed c!)
• Group velocity: $v_g = \frac{d\omega}{dk} = \frac{dE}{dp} = v$ (velocity of particle)

For photon: $v_p = c$, $v_g = c$ For matter wave: $v_p > c$, $v_g < c$

Wave Function and Uncertainty:

Heisenberg Uncertainty Principle: $$\Delta x \cdot \Delta p_x \geq \frac{\hbar}{2}$$

Other forms: $$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$ $$\Delta L_z \cdot \Delta \phi \geq \frac{\hbar}{2}$$

where Δx = uncertainty in position, Δpₓ = uncertainty in momentum

Photon Diffraction:

Single photon going through double slit: builds up interference pattern over many photons. This confirms wave-particle duality.

Photoelectric Effect — Complete Analysis:

When light hits metal:

1. Photon is absorbed by electron
2. Electron uses some energy to escape metal (work function φ)
3. Remaining energy becomes kinetic energy of electron

Einstein’s equation derivation: $$h\nu = \phi + \frac{1}{2}mv_{max}^2 = \phi + eV_0$$

Work Function for Different Metals:

Metal	φ (eV)	λ₀ (nm)
Na	2.3	540
K	2.3	540
Ca	3.2	388
Cu	4.5	276
Pt	6.4	194

Einstein’s Contribution (Nobel Prize 1921):

Explained photoelectric effect using photon concept. Key insight: light energy comes in discrete quanta (photons), not continuous waves. This dual nature was revolutionary.

⚡ JEE Advanced 2022 Analysis: Questions on photoelectric effect equations and de Broglie wavelength appeared in 2022. The relationship between stopping potential, frequency, and work function is frequently tested. Also prepare for questions involving the Davisson-Germer experiment confirming wave nature of electrons.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.