Planck's theory, photoelectric effect, dual nature of electrons

Planck's Theory, Photoelectric Effect, Dual Nature of Electrons

1. Black-body radiation and the failure of classical physics

A hot body emits radiation at all wavelengths. Classical Rayleigh–Jeans law predicted intensity $I(\lambda, T) \to \infty$ as $\lambda \to 0$ — the ultraviolet catastrophe. Experiments instead showed intensity peaks at some $\lambda_{max}$ that shifts with temperature (Wien's displacement law):

2. Planck's quantum theory (1900)

Planck proposed energy is emitted/absorbed in discrete packets (quanta):

where $h = 6.626 \times 10^{-34}$ J·s (Planck's constant), $
u$= frequency,$c = 3 \times 10^8$ m/s.

Key points:

• Energy is quantized, not continuous: $E = nh
  u$for$n = 1, 2, 3, \ldots$
• Perfectly fits the black-body curve at all wavelengths.
• Birth of quantum theory.

Numerical: Energy of a photon of $\lambda = 500$ nm:

3. Photoelectric effect (Einstein, 1905)

When light of frequency above a threshold falls on a metal, electrons are ejected instantly.

Observation	Classical prediction	Quantum explanation
No emission below threshold $
u_0$, whatever the intensity | Any frequency should work if intense enough | Photon must have$h
u \ge$work function$\phi$
KE of electrons rises with frequency, not intensity	KE should rise with intensity	$KE_{max} = h
u - \phi$
Emission is instantaneous	Energy should accumulate slowly at low intensity	One photon → one electron, single collision

Einstein's photoelectric equation:

where $\phi = h
u_0$ is the work function (minimum energy to free an electron).

Numericals:

• Threshold wavelength for Na ($\phi = 2.28$ eV): $\lambda_0 = hc/\phi = 1240/2.28 \approx 544$ nm.
• Light of 400 nm on Na ejects electrons with $KE = 3.10 - 2.28 = 0.82$ eV.

4. Dual nature of electrons (de Broglie, 1924)

If light (a wave) has particle character, then particles like electrons should have wave character:

• Verified by Davisson–Germer electron diffraction experiment.
• An electron accelerated through $V$ volts has $\lambda \approx \frac{1.227}{\sqrt{V}}$ nm (e.g. 100 V → 0.123 nm, comparable to atomic spacings — hence diffraction).
• Bohr orbits are standing waves: $2\pi r = n\lambda$ gives quantized angular momentum $mvr = nh/2\pi$.

Exam Points

• State Planck's theory + formula $E = h
  u$ (2 marks); derive/apply Einstein equation (5 marks).
• Numericals on photon energy, threshold frequency/wavelength, de Broglie wavelength are guaranteed 3–5 markers — practice unit conversion eV $\leftrightarrow$ J ($1$ eV $= 1.602 \times 10^{-19}$ J) and shortcut $E(\text{eV}) = 1240/\lambda(\text{nm})$.