Dual Nature of Radiation & Matter— Notes, Formulas & Revision

When light of sufficient frequency strikes a metal surface, electrons are ejected — the photoelectric effect, discovered by Hertz (1887) and explained by Einstein (1905).

Einstein's photon hypothesis: light comes in quanta of energy E = hf. Each photon interacts with ONE electron.

Einstein's equation: K_max = hf − φ, where φ is the work function (minimum energy to release an electron from the metal surface).

Below the threshold frequency f₀ = φ/h, NO electrons are emitted, regardless of intensity. Above, electrons emerge instantaneously.

Maximum KE depends on the frequency, NOT intensity. Intensity affects only the NUMBER of photoelectrons per second.

Stopping potential V_s is the retarding voltage that just stops the most energetic photoelectrons: eV_s = K_max = hf − φ.

Quantum nature: classical wave theory cannot explain (i) threshold frequency, (ii) zero time-delay, (iii) K_max independent of intensity.

Result: light has particle-like (photon) behaviour even though it's a wave — laid the foundation for quantum mechanics.

Threshold frequency f₀ is the minimum frequency of light below which no photoemission occurs, regardless of intensity.

Defined by the work function: hf₀ = φ ⇒ f₀ = φ/h.

Equivalently, threshold wavelength λ₀ = hc/φ. Below f₀ (or above λ₀): no photoelectrons.

Threshold is metal-specific. Cesium (φ ≈ 2.14 eV) responds to visible red. Copper (φ ≈ 4.7 eV) needs UV.

Above f₀: any non-zero intensity produces photoelectrons instantaneously (no measurable delay even at extremely low intensities — classical theory predicted ~minutes of delay).

Threshold is sharp — there is no 'gradual onset'. It's a step function in classical experiments.

Threshold frequency was a primary mystery for classical physics — solved by Einstein's photon model.

Maximum KE of photoelectrons varies LINEARLY with frequency: K_max = hf − φ.

Slope = h (Planck's constant). x-intercept = f₀ = φ/h. y-intercept (at f=0) = −φ.

Below threshold (f < f₀): K_max would be negative — physically means no emission.

Crucially: K_max does NOT depend on intensity. Same f and metal ⇒ same K_max regardless of how bright the light is.

This linearity was a key prediction of Einstein's photon theory — confirmed by Millikan's experiments.

Different metals: parallel lines (same slope h) but different intercepts (different φ).

Photon picture: each photon transfers its full energy hf to ONE electron. The electron uses φ to escape, keeps the rest as KE.

Photoelectric current (number of electrons per unit time × charge) is DIRECTLY proportional to intensity, for given frequency above threshold.

Doubling intensity doubles the rate of photons hitting the surface ⇒ doubles the rate of photoelectron emission ⇒ doubles the saturation current.

Intensity does NOT affect: K_max, threshold frequency, stopping potential.

This proportionality is exact in the photon picture: I = (photon rate)·hf. Doubling I doubles photon rate, hence emission rate.

Saturation current: at high anode voltage, every photoelectron is collected. Saturation level scales with I.

Below saturation (low voltage), some electrons return to the cathode without crossing — current is sub-saturation.

Photodetectors, solar cells, light meters — all rely on the intensity-current proportionality.

Stopping potential V_s is the retarding voltage (anode held NEGATIVE w.r.t. cathode) at which photoelectric current drops to ZERO.

At V = V_s, even the MOST energetic photoelectron (K = K_max) is just barely stopped: eV_s = K_max.

Stopping-potential equation: eV_s = hf − φ.

V_s is INDEPENDENT of intensity — depends only on frequency and metal.

Plotting V_s vs f: straight line with slope h/e and x-intercept f₀.

Slope h/e ≈ 4.136 × 10⁻¹⁵ V·s — same for all metals. Y-intercept (at f = 0): −φ/e — depends on metal.

Millikan's 1914 experiment measured V_s vs f for several metals — confirmed Einstein's equation and gave a precise value of h.

Wave-particle duality: light and matter both exhibit wave-like AND particle-like properties depending on the experiment.

Light: behaves as wave in interference, diffraction, polarisation. Behaves as particle (photon) in photoelectric effect, Compton scattering, pair production.

Matter: behaves as particle in collision/Newtonian experiments. Behaves as wave in electron diffraction (Davisson-Germer, 1927), neutron interferometry, etc.

The two pictures are COMPLEMENTARY — you can't see both at once. Bohr's complementarity principle.

Photon energy E = hf, momentum p = h/λ. Matter wavelength λ_dB = h/p (de Broglie).

For everyday objects (e.g., a thrown ball), de Broglie wavelength is fantastically tiny (~10⁻³⁴ m) — wave nature is undetectable.

Wave-particle duality is a foundational pillar of quantum mechanics — it eventually led to the Schrödinger equation, where particles are described by wavefunctions.

Louis de Broglie (1924): every moving particle of momentum p has an associated wave of wavelength λ = h/p.

For a particle of mass m and speed v (non-relativistic): λ = h/(mv).

For an electron accelerated through V volts: λ = h/√(2meV) = 12.27/√(V[volts]) Å.

Larger momentum (heavier or faster) ⇒ smaller wavelength.

Matter waves were experimentally verified by Davisson-Germer (electron diffraction off Ni crystal, 1927) and G.P. Thomson (electron diffraction through thin foils).

Atom-interference experiments now confirm de Broglie waves for atoms and even C₆₀ buckyballs.

de Broglie's hypothesis led directly to Schrödinger's wave equation and modern quantum mechanics.

Electrons can diffract off crystal lattices — Davisson and Germer's 1927 experiment was the first direct evidence of matter waves.

An electron beam incident on a single-crystal nickel target produces a diffraction pattern, analogous to X-ray diffraction.

Bragg's law applies: 2d·sin θ = nλ — the same as for X-rays — but here λ = h/p (de Broglie).

G.P. Thomson independently showed electron diffraction through THIN foils, getting Debye-Scherrer rings.

Confirmed de Broglie's hypothesis quantitatively: the measured λ matched h/p for the electron's KE.

Modern applications: electron microscopy (resolution depends on small λ), low-energy electron diffraction (LEED) for surface analysis.

Neutrons also diffract — used in neutron crystallography (better for light atoms like H than X-rays).

All matter has wave-like properties: every moving particle has a de Broglie wave λ = h/p.

Quantum mechanics replaces the classical concept of trajectory with the wave function ψ(x,t) — |ψ|² is the probability density.

Bound states (atoms, nuclei) are described by standing matter waves — explains discrete energy levels.

Matter waves can interfere and diffract like light. Experimental demos: electron diffraction (Davisson-Germer), neutron interferometry, atom interferometers.

Matter waves do NOT travel at v_particle in general — phase velocity v_p = E/p, group velocity v_g = dE/dp = v_particle.

For a relativistic particle: v_p · v_g = c² (so v_p > c for massive particles — but this carries no information).

Macroscopic objects have λ_dB so small that wave behaviour is hidden — but it is THERE, in principle.

Heisenberg's uncertainty principle (1927): you cannot simultaneously know a particle's position and momentum to arbitrary precision.

Quantitatively: Δx · Δp_x ≥ ℏ/2, where ℏ = h/(2π).

Similarly for energy and time: ΔE · Δt ≥ ℏ/2 — has implications for line widths and short-lived excited states.

NOT a statement about measurement disturbance — it's a fundamental property of wave packets.

Smaller Δx requires a sharper wave packet, which has a broader range of k (= 2π/λ) — and therefore a broader range of momentum.

Macroscopic objects: ℏ is so small that Δx·Δp can be far below any measurable scale ⇒ classical mechanics is recovered.

Implications: there is NO classical 'orbit' for electrons in atoms — only probability clouds.