Modern Physics — Study Notes

1. The Quantum Nature of Light — Photons

Classical wave theory predicted that light should continuously transfer energy to electrons in a metal surface. However, experiments with the photoelectric effect revealed that energy is transferred in discrete packets — quanta. This led to the modern understanding that electromagnetic radiation behaves as both a wave and a particle.

A photon is a quantum (a discrete packet) of electromagnetic radiation. The energy of a single photon: $$E = hf = \frac{hc}{\lambda}$$

Where: - $h = 6.63 \times 10^{-34}$ J s (Planck's constant) - $f$ = frequency (Hz) - $c = 3.00 \times 10^8$ m s⁻¹ (speed of light in vacuum) - $\lambda$ = wavelength (m)

Higher frequency radiation (e.g., gamma rays) carries more energy per photon than lower frequency radiation (e.g., radio waves). The electron-volt (eV) is a convenient unit for photon energies at the atomic scale: 1 eV = 1.6 × 10⁻¹⁹ J.

2. The Photoelectric Effect

When electromagnetic radiation of sufficient frequency illuminates a metal surface, electrons are emitted. These are called photoelectrons. Key experimental observations:

1. Threshold frequency ($f_0$): No electrons are emitted if the radiation frequency is below $f_0$, regardless of intensity. This cannot be explained by wave theory.
2. Instantaneous emission: Above $f_0$, electrons are emitted immediately, even at very low intensities.
3. Maximum KE independent of intensity: The maximum kinetic energy of emitted electrons depends only on frequency, not intensity. Greater intensity means more electrons per second, not more energetic electrons.

Einstein's explanation: Each electron absorbs exactly one photon. The photon's energy is used first to overcome the work function ($\Phi$) — the minimum energy needed to liberate an electron from the metal surface — and the rest appears as kinetic energy:

$$hf = \Phi + E_{k,max}$$

The work function relates to the threshold frequency: $\Phi = hf_0$

Stopping potential ($V_s$): The minimum reverse potential needed to stop all emitted electrons: $eV_s = E_{k,max}$

3. Wave–Particle Duality

Light demonstrates duality — sometimes behaving as a wave (interference, diffraction) and sometimes as a particle (photoelectric effect). In 1924, Louis de Broglie extended this idea to matter: every moving particle has an associated wave (matter wave).

The de Broglie wavelength: $$\lambda = \frac{h}{p} = \frac{h}{mv}$$

Where $p = mv$ is the momentum of the particle. A fast-moving electron with $v = 1 \times 10^7$ m s⁻¹ has: $\lambda = \frac{6.63\times10^{-34}}{9.11\times10^{-31} \times 10^7} \approx 7.3 \times 10^{-11}$ m — comparable to atomic spacings!

This explains why electron diffraction is observable when electrons are directed at a crystal lattice — the lattice spacing acts as a diffraction grating for the electron waves. Confirmed by Davisson and Germer (1927).

Heavier, faster particles have smaller wavelengths and exhibit less obvious wave properties. A cricket ball at 30 m s⁻¹ has $\lambda \approx 10^{-34}$ m — utterly undetectable.

graph TD
    A[Electromagnetic Radiation] --> B[Wave behaviour: interference, diffraction]
    A --> C[Particle behaviour: photoelectric effect, photon]
    D[Matter, e.g. electrons] --> E[Particle behaviour: momentum, KE]
    D --> F[Wave behaviour: electron diffraction, de Broglie λ = h/p]

4. Atomic Energy Levels

In the Bohr model, electrons occupy discrete energy levels around the nucleus. Each level has a specific energy value (negative, because energy must be supplied to remove the electron). The ground state (n=1) is the lowest.

• Excitation: An electron absorbs exactly the right amount of energy (a photon or via collision) to jump to a higher level.
• De-excitation: An electron drops to a lower level, emitting a photon with energy equal to the energy difference: $$\Delta E = E_{higher} - E_{lower} = hf$$

This explains emission spectra (bright lines on a dark background): excited atoms emit photons of specific frequencies when electrons de-excite, producing a unique spectral "fingerprint" for each element.

Absorption spectra (dark lines on a continuous spectrum): When white light passes through a cool gas, atoms absorb photons of specific frequencies, causing dark lines at exactly the same wavelengths as the emission lines.

5. Radioactivity and Nuclear Structure

The nucleus contains protons (charge +e) and neutrons (neutral), collectively called nucleons. Notation: $^A_Z X$ where $A$ = mass number (nucleons), $Z$ = atomic number (protons), and $A - Z$ = number of neutrons.

Isotopes have the same $Z$ but different $A$ — same element, different neutron count. Some isotopes are unstable and undergo radioactive decay.

Types of Radiation

6. Radioactive Decay Laws

Radioactive decay is spontaneous (unaffected by physical/chemical conditions) and random (impossible to predict which nucleus decays next). However, statistically, large samples follow an exponential decay law.

The activity $A$ (in Becquerels, Bq) = decays per second: $$A = -\frac{dN}{dt} = \lambda N$$

The decay constant $\lambda$ (s⁻¹) is the probability of decay per unit time for a single nucleus.

Exponential decay of nuclei: $$N = N_0 e^{-\lambda t}$$

Similarly: $A = A_0 e^{-\lambda t}$

The half-life ($t_{1/2}$) is the time for activity (or number of undecayed nuclei) to halve: $$t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$

7. Mass–Energy Equivalence and Nuclear Energy

Einstein's famous equation: $E = mc^2$ ($c = 3 \times 10^8$ m s⁻¹)

The mass defect ($\Delta m$) of a nucleus is the difference between the mass of its separated nucleons and its actual nuclear mass. This "missing" mass was converted to energy when the nucleus formed: $$E_{binding} = \Delta m \cdot c^2$$

The binding energy per nucleon peaks for iron-56 (~8.8 MeV/nucleon). Elements lighter than iron release energy by nuclear fusion (combining nuclei); elements heavier than iron release energy by nuclear fission (splitting nuclei).

Both processes release enormous energy — nuclear power stations and stars operate on these principles.