Dual Nature of Radiation and Matter
'Light behaves like a wave in some experiments and like a particle in others. Matter does the same. Welcome to QUANTUM PHYSICS.'
1. Chapter Overview
This chapter introduces the DUAL NATURE of radiation and matter — the paradigm-shifting idea that WAVES can behave like PARTICLES and PARTICLES can behave like WAVES. Topics include: the PHOTOELECTRIC EFFECT (Einstein's Nobel Prize-winning explanation — light as PHOTONS), the WAVE NATURE OF MATTER (de Broglie's hypothesis — every moving particle has a wavelength), DAVISSON-GERMER EXPERIMENT (experimental confirmation of matter waves), and the PHOTOELECTRIC EFFECT EQUATION (hν = φ + K_max).
2. The Photoelectric Effect
Experimental Observations
- There is a THRESHOLD FREQUENCY (ν₀) below which NO photoelectrons are emitted — even if intensity is very high.
- KINETIC ENERGY of emitted electrons depends on FREQUENCY, NOT intensity.
- NUMBER of emitted electrons depends on INTENSITY (for ν > ν₀).
- Emission is INSTANTANEOUS (no time lag).
'Classical wave theory FAILED to explain all four observations — especially the existence of a threshold frequency (since wave theory predicted that ANY frequency would eventually eject electrons given enough intensity).'
Einstein's Explanation — Photons
- Light consists of PACKETS of energy called PHOTONS. Each photon has energy E = hf.
- Photoelectric equation: hf = φ + K_max — where φ = work function, K_max = ½ mv²_max = hf − φ.
- 'Einstein said: light energy comes in DISCRETE packets. A single photon KNOCKS OUT a single electron.'
3. Experimental Setup for Photoelectric Effect
- Evacuated tube with two electrodes (emitter C and collector A).
- Light of known frequency and intensity is incident on the EMITTER.
- A VARIABLE POTENTIAL is applied between C and A — to measure the stopping potential.
- The stopping potential V₀ is the minimum reverse potential that STOPS even the fastest photoelectrons.
Key Formulas
- K_max = eV₀ (stopping potential × electron charge).
- K_max = hf − φ ⇒ eV₀ = hf − φ.
- Threshold frequency: ν₀ = φ/h.
- Threshold wavelength: λ₀ = hc/φ.
Worked Example 1
Problem: The work function of a metal is 2.3 eV. Light of λ = 400 nm falls on it. Find (a) threshold wavelength, (b) maximum KE of photoelectrons, (c) stopping potential. Solution: (a) λ₀ = hc/φ = (6.63×10⁻³⁴×3×10⁸)/(2.3×1.6×10⁻¹⁹) = 19.89×10⁻²⁶/(3.68×10⁻¹⁹) = 5.4×10⁻⁷ m = 540 nm. (b) hf = hc/λ = 19.89×10⁻²⁶/(400×10⁻⁹) = 4.97×10⁻¹⁹ J = 3.1 eV. K_max = hf − φ = 3.1 − 2.3 = 0.8 eV = 1.28×10⁻¹⁹ J. (c) V₀ = K_max/e = 0.8 V.
4. Wave Nature of Matter — de Broglie's Hypothesis
- de Broglie wavelength: λ = h/p = h/(mv).
- 'If light (a wave) can behave like a particle (photon), then matter (particles) should also behave like waves — with wavelength given by h/p.'
- For an electron accelerated through voltage V: λ = h/√(2meV) = 12.27/√V Angstroms.
Davisson-Germer Experiment
- 'Davisson and Germer confirmed de Broglie's hypothesis by firing electrons at a nickel crystal and observing DIFFRACTION PATTERNS — proving that electrons ARE waves.'
- The observed diffraction pattern matched the de Broglie wavelength formula.
Worked Example 2
Problem: Find the de Broglie wavelength of an electron accelerated through 100 V. Solution: Using λ = 12.27/√V = 12.27/√100 = 12.27/10 = 1.227 Å = 1.227×10⁻¹⁰ m.
5. Comparison Table: Wave Theory vs Particle Theory of Light
| Phenomenon | Wave Theory Explains | Particle Theory Explains |
|---|---|---|
| Interference | YES — wave superposition | NO |
| Diffraction | YES — wave bending | NO |
| Polarisation | YES — transverse wave | NO |
| Photoelectric effect | NO — cannot explain threshold frequency | YES — photon model |
| Black body radiation | NO — ultraviolet catastrophe | YES — Planck's quanta |
6. Photon Characteristics
| Property | Formula/Value |
|---|---|
| Energy | E = hf = hc/λ |
| Momentum | p = h/λ = E/c |
| Mass | ZERO rest mass |
| Speed | c (3×10⁸ m/s in vacuum) |
| Charge | ZERO |
| Planck's constant | h = 6.63×10⁻³⁴ J·s = 4.14×10⁻¹⁵ eV·s |
7. Common Mistakes
- Stopping potential is NOT the maximum KE: K_max = eV₀ (in eV). The stopping potential is the VOLTAGE, not the energy.
- Threshold frequency vs threshold wavelength: ν₀ = φ/h. λ₀ = hc/φ. Know both formulas.
- de Broglie wavelength formula: λ = h/p = h/(mv). For electrons, you can use the shortcut λ = 12.27/√V.
- Photoelectric effect is instantaneous: There is ZERO time lag — if ν > ν₀, electrons are emitted immediately. Classical wave theory predicted a measurable time lag.
8. CBSE Exam Focus
- Photoelectric effect — experimental observations (all four)
- Einstein's photoelectric equation — hf = φ + K_max
- Stopping potential — eV₀ = hf − φ
- de Broglie wavelength — λ = h/p, electron wavelength formula
- Davisson-Germer experiment — confirmation of matter waves
- Graphs — K_max vs f (straight line with slope h), I vs V for different intensities/frequencies
9. Self-Test
Q1: The threshold frequency of a metal is 5×10¹⁴ Hz. Find the work function in eV. (h = 4.14×10⁻¹⁵ eV·s) A1: φ = hν₀ = 4.14×10⁻¹⁵×5×10¹⁴ = 2.07 eV.
Q2: The maximum KE of photoelectrons from a metal is 1.5 eV for λ = 300 nm. Find the work function. A2: hf = hc/λ = 1240/300 = 4.13 eV (using hc ≈ 1240 eV·nm). φ = hf − K_max = 4.13 − 1.5 = 2.63 eV.
Q3: Find the de Broglie wavelength of a proton (m = 1.67×10⁻²⁷ kg) moving at 2×10⁶ m/s. A3: λ = h/(mv) = 6.63×10⁻³⁴/(1.67×10⁻²⁷×2×10⁶) = 6.63×10⁻³⁴/(3.34×10⁻²¹) = 1.98×10⁻¹³ m.
Q4: In a photoelectric experiment, the stopping potential is 1.2 V for λ = 400 nm. Find the work function. A4: hf = 1240/400 = 3.1 eV. K_max = eV₀ = 1.2 eV. φ = hf − K_max = 3.1 − 1.2 = 1.9 eV.
Q5: The wavelength of a photon and an electron are both 1 nm. Which has greater energy? (Mass of electron = 9.1×10⁻³¹ kg) A5: Photon energy: E = hc/λ = 1240/1 = 1240 eV. Electron energy: E = p²/(2m) = (h/λ)²/(2m) = (6.63×10⁻³⁴/10⁻⁹)²/(2×9.1×10⁻³¹) = (6.63×10⁻²⁵)²/(1.82×10⁻³⁰) = 4.4×10⁻⁴⁹/(1.82×10⁻³⁰) = 2.42×10⁻¹⁹ J = 1.51 eV. The PHOTON has MUCH greater energy.
10. Conclusion
The dual nature of radiation and matter CHANGED physics forever:
- PHOTONS: 'Light is quantised — energy comes in packets. Einstein used this to explain the photoelectric effect.'
- MATTER WAVES: 'Electrons, protons, atoms — everything has a wavelength. de Broglie extended wave-particle duality to ALL matter.'
- DAVISSON-GERMER: 'Experimental proof that electrons DIFFRACT — they ARE waves.'
- 'The principle of COMPLEMENTARITY: wave and particle are COMPLEMENTARY descriptions of the same reality. Which aspect you see depends on the experiment you perform.'
'Wave-particle duality is the MOST profound discovery of modern physics — it forces us to accept that nature is fundamentally DUAL.'
