Modern Physics: Photoelectric Effect & Matter Waves

Dual Nature of Radiation

Light exhibits both wave nature (interference, diffraction, polarization) and particle nature (photoelectric effect, Compton effect). Einstein proposed that light consists of quanta called photons, each carrying energy E = hf = hc/λ, where h = 6.63 × 10⁻³⁴ J·s is Planck's constant.

Photon properties:

• Energy: E = hf = hc/λ
• Momentum: p = h/λ = E/c = hf/c
• Rest mass: zero (photons always travel at c)
• Equivalent mass (relativistic): m = E/c² = hf/c² = h/(λc)

Useful conversions: E (eV) = 12400/λ(Å) = 1240/λ(nm). 1 eV = 1.6 × 10⁻¹⁹ J.

Photoelectric Effect

When light of sufficient frequency strikes a metal surface, electrons are ejected. Key observations:

1. Threshold frequency (f₀): Minimum frequency below which no electrons are emitted, regardless of intensity. f₀ = φ/h, where φ is the work function.

2. Instantaneous emission: Electrons are emitted within ~10⁻⁹ s of illumination (contradicts wave theory prediction of gradual energy accumulation).

3. Intensity dependence: Photocurrent is proportional to light intensity (more photons = more electrons), but maximum KE of electrons is independent of intensity.

4. Frequency dependence: Maximum KE increases linearly with frequency: $\text{KE}_{max}$ = hf - φ.

Einstein's Photoelectric Equation:

$KE_{max} = hf - \phi = h(f - f_0) = eV_0$

where φ = hf₀ is the work function, V₀ is the stopping potential.

Stopping potential: V₀ = (hf - φ)/e = (h/e)f - φ/e. The V₀ vs f graph is a straight line with slope h/e, x-intercept f₀, and y-intercept -φ/e.

Graphs in Photoelectric Effect

1. Photocurrent vs Intensity: Linear (at constant frequency above f₀)
2. Photocurrent vs Collector potential: Saturation current at large V; zero current at V = -V₀
3. V₀ vs Frequency: Straight line, slope = h/e, threshold at f₀
4. $\text{KE}_{max}$ vs Frequency: Straight line, slope = h, threshold at f₀

de Broglie Hypothesis

Every moving particle has an associated wave with wavelength:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

For a particle accelerated through potential V:

$\lambda = \frac{h}{\sqrt{2mKE}} = \frac{h}{\sqrt{2meV}}$

For electrons: λ = 12.27/√V Å (V in volts)

For thermal neutrons: λ = h/√(3mkT), where T is temperature and k is Boltzmann's constant.

Davisson-Germer Experiment

Confirmed de Broglie hypothesis by observing electron diffraction from nickel crystal. The diffracted beam showed a peak at specific angles, with the wavelength matching the de Broglie prediction λ = h/p.

Heisenberg's Uncertainty Principle

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi} = \frac{\hbar}{2}$

$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$

One cannot simultaneously know both position and momentum (or energy and time) with arbitrary precision. This is a fundamental limit, not a measurement limitation.

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