where:

• h = Planck's constant = $6.626 \times 10^{-34}$ J·s
• f = frequency (Hz)
• c = speed of light
• λ = wavelength

Higher frequency → higher energy

• Gamma rays: Very high E
• Radio waves: Very low E

Photoelectric Effect

Light shining on metal can eject electrons!

Key Observations:

1. Threshold frequency $f_0$: Below this, NO electrons (even intense light!)
2. Instantaneous: Electrons ejected immediately
3. KE depends on f, not intensity
4. Intensity affects number of electrons, not their energy

Cannot be explained by classical wave theory!

Einstein's Photoelectric Equation

$KE_{max} = hf - \phi$

where:

• $KE_{max}$ = maximum kinetic energy of ejected electron
• $hf$ = photon energy
• $\phi$ = work function (minimum energy to eject electron)

At threshold: $hf_0 = \phi$ (just enough to eject, KE = 0)

Stopping potential $V_s$: $eV_s = KE_{max}$

Voltage needed to stop most energetic electrons.

💡 Won Nobel Prize 1921! Not for relativity, but photoelectric effect.

De Broglie Wavelength

Particles can behave like waves!

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

Matter waves confirmed by electron diffraction experiments!

Larger mass → smaller wavelength

• Electron: λ ~ nm (measurable!)
• Baseball: λ ~ 10⁻³⁴ m (unmeasurable)

Atomic Models

Rutherford Model (1911):

• Nucleus: tiny, massive, positive
• Electrons orbit (like planets)
• Problem: Accelerating charges radiate → atom should collapse!

Bohr Model (1913):

• Electrons in discrete energy levels (orbits)
• Only certain radii allowed: $r_n = n^2 a_0$ where $a_0 = 0.529$ Å
• Quantized angular momentum: $L = n\hbar$ where $\hbar = h/2\pi$

Energy levels (hydrogen): $E_n = -\frac{13.6 \text{ eV}}{n^2}$

where n = 1, 2, 3, ... (principal quantum number)

• n = 1: Ground state, E = -13.6 eV
• n = ∞: Ionization, E = 0

Negative energy means bound (need energy to remove electron).

Atomic Transitions

Electron jumps between levels → photon absorbed or emitted!

Energy of photon: $E_{photon} = |E_f - E_i| = hf$

Emission (high → low): Photon out Absorption (low → high): Photon in

Emission spectrum: Discrete lines (fingerprint of element!)

Hydrogen Series:

• Lyman (UV): n → 1
• Balmer (visible): n → 2
• Paschen (IR): n → 3

Heisenberg Uncertainty Principle

Cannot know position and momentum simultaneously with perfect precision!

$\Delta x \Delta p \geq \frac{\hbar}{2}$

Also for energy and time: $\Delta E \Delta t \geq \frac{\hbar}{2}$

Not limitation of measurement, but fundamental nature of reality!

💡 Key: More certain about position → less certain about momentum

Nuclear Physics

Nucleus Composition:

• Protons: Z (atomic number), charge +e
• Neutrons: N, charge 0
• Mass number: A = Z + N

Isotopes: Same Z, different N (different A)

Example: $^{12}\text{C}$ vs $^{14}\text{C}$

Mass-Energy Equivalence

Einstein's most famous equation:

$E = mc^2$

Mass and energy are interchangeable!

Atomic mass unit: 1 u = $1.66 \times 10^{-27}$ kg = 931.5 MeV/c²

Binding energy: Mass of separated nucleons > mass of nucleus $\Delta m = \text{(mass of parts)} - \text{(mass of whole)}$ $BE = \Delta m c^2$

More binding energy → more stable

Nuclear Reactions

Fission:

• Heavy nucleus splits → lighter nuclei
• Releases energy (BE/nucleon increases)
• Used in nuclear power plants
• Example: $^{235}\text{U}$ + neutron → fission fragments + neutrons + energy

Fusion:

• Light nuclei combine → heavier nucleus
• Releases enormous energy
• Powers the sun!
• Example: $^2\text{H}$ + $^3\text{H}$ → $^4\text{He}$ + neutron + 17.6 MeV

Fusion releases MORE energy per nucleon than fission!

Radioactive Decay

Unstable nuclei decay spontaneously:

Types:

• Alpha (α): $^4\text{He}$ nucleus, A↓4, Z↓2
• Beta (β⁻): Electron, neutron→proton, A same, Z↑1
• Gamma (γ): High-energy photon, A and Z same

Half-life $t_{1/2}$: $N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

After one half-life: 50% remain After two: 25% remain After three: 12.5% remain

Conservation Laws

All nuclear reactions must conserve:

1. Mass-energy: Total E (including mc²) conserved
2. Charge: Total Z conserved
3. Mass number: Total A conserved
4. Momentum: Total p conserved

Problem-Solving Strategy

Photoelectric:

1. Find photon energy: $E = hf$ or $E = hc/\lambda$
2. Apply: $KE_{max} = hf - \phi$
3. Check threshold: if $f < f_0$, no electrons!

Atomic transitions:

1. Find energy levels: $E_n = -13.6/n^2$ eV
2. Energy difference: $\Delta E = |E_f - E_i|$
3. Photon: $\lambda = hc/\Delta E$

Nuclear:

1. Check conservation (A and Z)
2. Calculate mass defect: Δm
3. Energy: $E = \Delta m c^2$

Common Mistakes

❌ Using wavelength in meters with h in J·s (watch units!) ❌ Thinking intensity affects electron KE (only frequency does!) ❌ Forgetting negative sign in atomic energy levels ❌ Not converting eV to Joules (or vice versa): 1 eV = 1.60×10⁻¹⁹ J ❌ Confusing emission (high→low) with absorption (low→high) ❌ Using half-life formula incorrectly (power is t/t_1/2, not just t!)

Solution:

Photon energy: $E = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{5.00 \times 10^{-7}}$ $E = \frac{1.988 \times 10^{-25}}{5.00 \times 10^{-7}} = 3.98 \times 10^{-19} \text{ J}$

Convert to eV: $E = \frac{3.98 \times 10^{-19}}{1.60 \times 10^{-19}} = 2.49 \text{ eV}$

Answer:

• E = 3.98 × 10⁻¹⁹ J
• E = 2.49 eV (green light)