Physics 25 Study Guide

Why Modern Physics?

By 1900, "classical" physics (Newton + Maxwell + thermodynamics) was wildly successful — and broken in two places that turned out to change everything:

• Things that move very fast (approaching the speed of light) → Special Relativity
• Things that are very small (atoms, photons, electrons) → Quantum Mechanics

Both eras produced modern technology: GPS requires relativity; computers, lasers, LEDs, solar cells, and MRI machines all require quantum mechanics. This course surveys both, plus their offspring: atomic physics, solid-state physics, and nuclear physics.

§28.1 — Einstein's Postulates of Special Relativity

An inertial reference frame is one in which Newton's 1^st law holds — a frame moving at constant velocity with no rotation. Accelerating or rotating frames are non-inertial.

Postulate 2 is the strange one. If a spaceship at 0.5c flashes a headlight, Newton would say the beam moves at 1.5c relative to the ground — but every measurement ever performed says it moves at exactly c. The Michelson–Morley experiment (1887) ruled out a luminiferous ether, confirming there is no medium against which to measure absolute motion of light.

§28.2 — Simultaneity & Time Dilation

Simultaneity is Frame-Dependent

Imagine a railcar moving at speed v. Two lamps at the front and back flash. A ground observer (Bob), equidistant from both lamps, sees the flashes arrive at the same moment — so for him they were simultaneous.

A rider on the car (Alice), seated at the middle, is moving toward the front lamp and away from the back lamp. She sees the front flash arrive first. In her frame, the front lamp flashed before the back one.

The Light Clock and Time Dilation

Build a clock by bouncing a photon vertically between two mirrors separated by distance D. One tick = round-trip time.

• In the rest frame of the clock, the photon travels straight up and back: Δt₀ = 2D/c.
• In the ground frame, the same clock moves sideways at speed v. The photon traces a longer, diagonal path — but it still travels at c. So the ground clock reads a larger time Δt > Δt₀.

Pythagoras on the ground-frame triangle (height D, base vΔt/2, hypotenuse cΔt/2) gives:

Lorentz Factor Cheat Table

Experimental Verification: Cosmic-Ray Muons

Muons have a proper half-life of ~1.52 μs. They are created ~10 km up in the atmosphere by cosmic rays and travel toward Earth at v ≈ 0.98c. Classically, they'd decay long before reaching the ground; relativistically, γ ≈ 5 so lab-frame lifetime is ~7.6 μs and they reach us in droves. (OpenStax Example 28.1 runs the same calculation.)

The Twin Paradox

If Alice rockets off at v = 0.9994c (γ = 30) for 2 years of her proper time, she returns to find 60 years have passed on Earth. Her twin Bob is now 60 years older than she is — this is real, not apparent.

§28.3 — Length Contraction

If moving clocks tick slow, then — by consistency — moving rulers must be shorter. Consider the muon again: in the muon's frame, it lives for Δt₀ ~ 1.52 μs and travels only a short distance L = vΔt₀. In Earth's frame, it lives γΔt₀ and covers L₀ = vγΔt₀. The two observers measure different distances.

Length contraction and time dilation are two sides of the same coin — they always conspire so that v = L/Δt is the same in both frames.

§28.4 — Relativistic Velocity Addition & Doppler Effect

Classical velocity addition — if your train moves at v and you throw a ball forward at u′, the ground sees it at u = v + u′ — fails near light speed. If u′ = c, ground would see c + v > c, violating Postulate 2.

Signs: pick an axis; v is the relative velocity between frames, u′ is object velocity in the primed frame, u is object velocity in the unprimed frame. Both u′ and v can be ±.

Sanity Checks

• Low v: denominator → 1, recovering u ≈ v + u′ (Galilean).
• u′ = c: u = (v + c)/(1 + v/c) = c · (v + c)/(c + v) = c ✓
• Combining two sub-c velocities always gives a sub-c result.

Worked Example (OpenStax 28.4)

A ship approaches Earth at v = 0.500c and fires a canister forward at u′ = 0.750c.

Not 1.25c — even when both parts of the sum point the same way, relativity keeps the result below c.

Relativistic Doppler Effect

For light, the classical Doppler formula fails — there's no medium, and time itself is dilated. Let u be the relative speed between source and observer (u > 0 if moving apart).

• Recession (u > 0): λ_obs > λ_s — red shift.
• Approach (u < 0): λ_obs < λ_s — blue shift.
• Galaxies recede from us with u ∝ distance (Hubble's Law) → cosmological red-shift → expanding universe.

§28.5 — Relativistic Momentum

Classical p = mu is not conserved in all inertial frames at relativistic speeds. Momentum conservation is too important to lose, so the definition of momentum must be modified to preserve it under Lorentz transformations.

• m is the invariant rest mass — the same for every observer. Modern convention: no "relativistic mass."
• u is the object's speed in the observer's frame.
• At low u: p → mu (classical). At u → c: p → ∞.

§28.6 — Relativistic Energy & E = mc²

Rest Energy

Every object with rest mass m has an intrinsic energy just by existing:

Mass and energy are interconvertible. A 1-gram mass has E₀ = (10⁻³)(3×10⁸)² = 9 × 10¹³ J ≈ 21 kilotons TNT — roughly twice the Hiroshima bomb.

Total Energy & Kinetic Energy

Low-v limit: γ ≈ 1 + ½u²/c², so KE → ½mu² (classical). High-v limit: KE → ∞ as u → c.

Energy–Momentum Relation (the one to memorize)

Units for the Relativistic World

§13.1 — Temperature & the Kelvin Scale

Temperature is operationally "what a thermometer measures," but microscopically it's a measure of the average translational kinetic energy per molecule. Three scales you must be fluent in:

Always use Kelvin in thermal-radiation formulas — the T⁴ dependence is extremely sensitive to the zero point.

§14.7 — Radiation & the Stefan–Boltzmann Law

Heat can transfer by conduction (contact), convection (fluid flow), or radiation (electromagnetic waves through vacuum). Radiation is how the Sun heats Earth and how a hot stove coil heats a pan from across an air gap.

• σ = 5.67 × 10⁻⁸ W/(m²·K⁴) — the Stefan–Boltzmann constant.
• e = emissivity, dimensionless (0 ≤ e ≤ 1). Perfect blackbody: e = 1.
• A = radiating surface area.
• T = absolute temperature in Kelvin.

Emissivity Table

Worked Example (OpenStax 14.9) — Heat Loss from a Person

Skin at T = 306 K, room at T = 295 K, A = 1.40 m², e = 0.97.

About the size of basal metabolism (~125 W) — which is why you feel cold in a cool room even in still air.

The Ultraviolet Catastrophe

A blackbody is an idealized perfect absorber (e = 1). Real objects (hot stoves, lightbulb filaments, the Sun's photosphere, stars) radiate very close to blackbody curves. Experimentally observed features:

• Intensity depends only on T — not on composition.
• Spectrum is continuous, peaked at a wavelength λ_max that shifts with T.
• Hotter → shorter λ_max (red → orange → yellow → white → blue).
• Total area under the curve ∝ T⁴ (Stefan–Boltzmann).

Examples: room-temperature objects (T ≈ 300 K) peak in mid-IR (λ ≈ 10 μm); incandescent bulb filament (~3000 K) peaks in near-IR (~1 μm), so most energy is wasted as heat; Sun (~5800 K) peaks in green visible (~500 nm).

Classical physics (Rayleigh–Jeans) predicts each mode of the EM field gets average energy kT, and counting modes per wavelength interval gives

This diverges as λ → 0 — total radiated energy would be infinite. Clearly wrong. Needed: a new idea. Enter Planck (Lecture 5).

§29.1 — Planck's Quantum Hypothesis

In 1900, Max Planck fit the observed blackbody curve with a desperate, ad-hoc assumption:

At high frequencies (short λ), hf ≫ kT, so exciting even one quantum is unlikely — Boltzmann factor kills the divergence. At low f, hf ≪ kT and many quanta are excited, recovering Rayleigh–Jeans. Planck's full formula:

Atomic line spectra (the narrow bright wavelengths emitted by hot gases — neon, hydrogen, sodium lamps) were also unexplained by classical physics and pointed to quantization of atomic energy levels, not just oscillators in walls.

§29.2 — The Photoelectric Effect

Shine light on a metal surface and electrons are ejected ("photoelectrons"). Measured facts that classical wave theory cannot explain:

1. Threshold frequency f₀: below a material-specific f₀, no electrons are ejected no matter how bright the light.
2. No time delay: above f₀, ejection is instantaneous even at extremely dim intensity.
3. KE_max depends on f, not intensity: brighter light ejects more electrons but not more energetic ones.
4. KE_max rises linearly with f above threshold.

Einstein's 1905 Resolution

Einstein proposed that light itself is quantized — delivered in particle-like photons of energy E = hf. One photon ejects one electron; no photon, no electron, regardless of total light power.

Work Function Table (typical values)

Worked Example (OpenStax 29.1) — Violet Light on Calcium

λ = 420 nm, BE(Ca) = 2.71 eV. Using hc = 1240 eV·nm:

§29.3 — Photon Energies & the EM Spectrum

The single most useful relation in this chapter for quick calculations:

The Energy Ladder

Visible light corresponds to 1.63 eV (700 nm red) – 3.26 eV (380 nm violet) — exactly the right scale to drive outer-shell electron transitions, which is why human eyes evolved for that band.

X-ray Tubes

Accelerate an electron through V volts. Its KE = eV in electron-volts. Hitting an anode, it can radiate a photon with E ≤ eV. So an X-ray tube at 50 kV produces photons with E_max ≈ 50 keV (λ_min ≈ 0.025 nm) — plus characteristic lines specific to the anode material.

Photons per Second from a Lightbulb

100 W bulb, ~10% visible at average λ = 580 nm: each photon is E = 1240/580 ≈ 2.14 eV = 3.43 × 10⁻¹⁹ J. Rate: (10 W)/(3.43 × 10⁻¹⁹ J) ≈ 2.9 × 10¹⁹ photons/s. That's why light seems continuous in everyday life — individual photons are too many and too small to notice.

§27.3 — Young's Double-Slit Experiment

Thomas Young (1801) passed light through two narrow, closely-spaced slits and saw alternating bright and dark fringes on a screen — a pattern only possible for waves. This was the decisive 19th-century evidence that light is a wave (which Einstein's photons would complicate a century later).

For small angles (slit spacing d ≪ screen distance L) the m-th bright fringe sits at y_m ≈ mλL/d on the screen.

Worked Example (OpenStax 27.1)

He–Ne laser, d = 0.0100 mm, m = 3 bright fringe at θ = 10.95°.

§29.4 — Photon Momentum

A photon carries energy E = hf. From the relativistic mass-shell E² = (pc)² + (mc²)² with m = 0:

Comet Tails and Solar Sails

Solar radiation pressure always points away from the Sun. Comet dust tails are pushed into the classic "anti-Sun" direction. Proposed solar sails (LightSail-2 demonstrated this) use large mirrored sheets to accumulate momentum from billions of photons per second — slow but free acceleration in deep space.

Compton Scattering (1923)

X-ray photons scattering off nearly-free electrons emerge at longer wavelength, with shift Δλ = (h/m_ec)(1 − cos θ). Wave theory predicts no wavelength shift — the result makes sense only if the photon is a particle carrying p = h/λ that collides elastically with the electron. Compton won the 1927 Nobel Prize.

Worked Example (OpenStax 29.5)

500-nm photon: p = h/λ = 6.63×10⁻³⁴ / 5×10⁻⁷ = 1.33 × 10⁻²⁷ kg·m/s. An electron with that same momentum moves at only v = p/m_e ≈ 1460 m/s and has KE ≈ 6 × 10⁻⁶ eV — five orders of magnitude less than the photon's 2.48 eV. So "momentum equal" doesn't mean "energy equal."

§29.5 — Particle–Wave Duality

Light is both a wave (double-slit, diffraction grating, interference) and a particle (photoelectric effect, Compton scattering). Which face you see depends on what you measure:

• Propagation (going from A to B, interfering, diffracting) → wave behavior.
• Emission/absorption (atom giving off or absorbing energy) → particle behavior.

The next surprise (Lecture 7): if light is both wave and particle, maybe electrons are too. And indeed — they are.

§29.6 — de Broglie & the Wave Nature of Matter

In his 1924 doctoral thesis, Louis de Broglie (pronounced "de broy") made a radical proposal: if photons have p = h/λ, then every particle has an associated wavelength:

Why We Don't See It Macroscopically

• A 3-kg bowling ball at 10 m/s: λ ≈ 2 × 10⁻³⁵ m — 20 orders of magnitude smaller than an atom. Utterly undetectable.
• An electron at 1% of c: λ ≈ 2.4 × 10⁻¹⁰ m = 2.4 Å — comparable to atomic spacings in crystals. Observable!

Experimental Confirmation: Davisson–Germer (1927)

Electrons scattered off a nickel crystal produced clear diffraction peaks obeying the Bragg condition:

The measured λ matched h/p exactly. Every kind of particle has since been shown to diffract — neutrons, protons, whole atoms, and even buckminsterfullerene (C₆₀ — "buckyballs"). Matter waves are universal.

Electron Microscopes

Optical microscopes are limited by diffraction to ~200 nm resolution (Rayleigh's 1.22λ/D criterion from P23). Electrons at 100 keV have λ ≈ 4 × 10⁻³ nm — ~50,000× shorter than visible light. Modern transmission electron microscopes (TEM) resolve individual atoms; scanning electron microscopes (SEM) provide 3D-like surface images.

§29.7 — Heisenberg Uncertainty Principle

In the double-slit experiment with electrons fired one at a time, individual electrons land at definite points — but accumulate into an interference pattern. Each electron "explores both slits" as a probability wave, then "lands" as a particle. If you add a detector to determine which slit the electron went through, the pattern vanishes: measurement disturbs the system.

Position–Momentum Uncertainty

The more precisely you localize a particle (small Δx), the more uncertain its momentum must be (large Δp). Not a measurement flaw — it's a fundamental property of the wavefunction.

Energy–Time Uncertainty

Short-lived excited atomic states have intrinsically broadened spectral lines; unstable particles have a mass-energy spread Γ related to their lifetime τ by Γ ≈ ℏ/τ.

Worked Example (OpenStax 29.8)

Localize an electron to Δx = 0.01 nm = 10⁻¹¹ m (a typical atomic size):

That Δv is comparable to actual electron speeds in atoms — so you can't "track" the electron in a classical orbit, only describe a probability cloud (the orbital).

§29.8 — Duality, Reviewed

By 1927, the picture had snapped into focus:

• Every "thing" — photon, electron, atom — has both wave and particle aspects.
• The wave aspect gives probability amplitudes; squaring gives observable probabilities.
• Wavelike behavior (interference, diffraction) dominates when λ is comparable to the apparatus; particle behavior dominates when individual quanta are counted.
• The correspondence principle explains why classical physics works in the everyday: Planck's constant is tiny relative to macroscopic actions, so quantum effects vanish in the classical limit.

With this toolbox in hand, we can now build atoms (Lecture 8 onward).

Topics You Must Own

• Special Relativity (Ch 28): two postulates, γ factor, proper vs. observer time/length, time dilation Δt = γΔt₀, length contraction L = L₀/γ, relativistic velocity addition, Doppler shift, p = γmu, E = γmc², KE = (γ−1)mc², E² = (pc)² + (mc²)².
• Thermal Radiation (Ch 13.1, 14.7): Kelvin conversions, emissivity, P = σeAT⁴, net transfer σeA(T₂⁴ − T₁⁴), Wien's λ_max T = 2.898 × 10⁻³ m·K.
• Quantization (Ch 29.1): Planck E = nhf, blackbody curves, why classical R–J diverged at short λ.
• Photoelectric (Ch 29.2): KE_max = hf − BE, threshold f₀ = BE/h, slope-h KE-vs-f graph, why intensity doesn't change KE_max.
• Photon Properties (Ch 29.3–4): E = hf = hc/λ, p = h/λ, hc = 1240 eV·nm, radiation pressure.
• Interference (Ch 27.3): d sin θ = mλ (bright), (m+½)λ (dark), fringe position y = mλL/d.
• Matter Waves & Uncertainty (Ch 29.6–7): λ = h/p, Bragg 2d sin θ = mλ, Δx·Δp ≥ h/(4π), ΔE·Δt ≥ h/(4π).

Strategy

• Always convert temperatures to Kelvin in P = σeAT⁴.
• Use hc = 1240 eV·nm for photon energies — one line, no scientific notation.
• Identify who measures proper time vs. proper length before picking a formula. Clock at rest → Δt₀; meterstick at rest → L₀.
• Pick a clear + axis for velocity addition so signs work out; then u = (v + u′)/(1 + vu′/c²).
• Check reasonable limits: at v ≪ c, formulas should reduce to classical; at v → c, γ diverges; KE → 0 as v → 0.
• Work out units in eV, MeV for atomic and sub-atomic problems; convert to J only at the end if needed (1 eV = 1.6 × 10⁻¹⁹ J).

§30.1 — Discovery of the Atom

The word "atom" (Greek atomos, "uncuttable") goes back to Democritus, but the experimental case for atoms is recent. The 19th century gave us:

• Definite proportions in chemistry (Dalton, ca. 1808) — elements combine in fixed integer ratios, suggesting discrete units.
• Brownian motion (Einstein, 1905) — pollen grains jiggle because they are kicked by invisible molecules. Quantitative match between observed jiggle and Avogadro's number.
• Cathode rays & the electron (J. J. Thomson, 1897) — found a particle thousands of times lighter than hydrogen, with a fixed charge-to-mass ratio.

Knowing e and e/m together determined the electron mass m_e = 9.11 × 10⁻³¹ kg = 0.511 MeV/c².

§30.2 — The Nucleus & Rutherford's Experiment

Thomson proposed the plum-pudding model: electrons embedded in a positively-charged sphere the size of the whole atom (~10⁻¹⁰ m). It was wrong.

In 1909–1911, Geiger & Marsden (working for Rutherford) shot α particles (He²⁺ nuclei, KE ≈ 5 MeV) at a thin gold foil and watched where they went. Most passed straight through, but ~1 in 8000 deflected by > 90°. Rutherford famously said it was "as if you fired a 15-inch shell at tissue paper and it bounced back at you."

§30.3 — Bohr's Theory of the Hydrogen Atom

By 1885 spectroscopists had cataloged hydrogen's discrete emission lines. The wavelengths fit Balmer's empirical formula and Rydberg's generalization:

R = 1.097 × 10⁷ m⁻¹ (Rydberg constant). The integer pairs label series:

Bohr's Three Postulates (1913)

1. The electron travels in certain allowed circular orbits without radiating.
2. The allowed orbits have quantized angular momentum: L = n ℏ, n = 1, 2, 3, …
3. A photon is emitted (or absorbed) only when the electron jumps between allowed levels: hf = E_i − E_f.

Deriving the Bohr Results

Set the Coulomb force equal to centripetal force, then plug in L = mvr = nℏ:

The same calculation reproduces the empirical Rydberg formula and predicts R from fundamentals — a stunning success for a desperately ad-hoc model.

Worked Example: Balmer Hα

Find the wavelength of the n = 3 → 2 transition.

§30.4 — X Rays

X rays were discovered by Röntgen in 1895 — high-energy photons (λ ~ 0.01 to 10 nm; E ~ 100 eV to 100 keV) able to expose photographic plates through opaque matter. Two production mechanisms in an x-ray tube:

• Bremsstrahlung ("braking radiation"): energetic electrons decelerate when they slam into a target, emitting a continuous spectrum. The shortest wavelength corresponds to losing all the KE in one photon: λ_min = hc/(eV_tube).
• Characteristic x-rays: incoming electrons knock out a deep core electron (e.g., 1s "K-shell"); an outer electron drops down to fill the hole, emitting a sharp-line photon (K_α, K_β, …). The energies are unique to each element — basis of XRF (X-ray fluorescence) elemental analysis.

The (Z − 1)² scaling let Moseley correctly order the periodic table by Z (instead of by atomic mass) — and pointed out the "missing" elements that hadn't been discovered yet.

§30.5 — Atomic Excitations & De-Excitations

An electron can absorb a photon of just the right energy and jump up to a higher level (absorption). It can later drop down again, emitting one or more photons whose total energy equals the original gap.

• Fluorescence: absorb a UV photon, emit visible photon(s) almost immediately (~ns). Highlighters, blacklight posters, jellyfish GFP.
• Phosphorescence: the excited state is "metastable" (lifetime seconds to hours); the glow lingers. Glow-in-the-dark stars.
• Resonance fluorescence: absorb and re-emit at the same wavelength — used in atomic clocks and laser cooling.
• Stimulated emission (preview): an incoming photon can cause an excited atom to dump another identical photon, doubling the beam. Foundation of the laser (Lecture 14).

§30.6 — Why Quantization? de Broglie Standing Waves

Bohr postulated L = nℏ. de Broglie (1924) derived it from a single new idea: the electron is a wave with λ = h/p. For a wave to fit smoothly around a circular orbit (avoiding destructive self-interference), an integer number of wavelengths must equal the circumference:

Substituting λ = h/(mv):

Bohr's quantization rule falls out for free. Quantization is a consequence of wave physics applied to a confined system — exactly like the discrete modes on a guitar string.

Confirmation: Davisson–Germer (1927)

An electron beam scattered off a nickel crystal showed Bragg diffraction maxima at angles consistent with λ = h/p. Electrons are waves. So is everything else: protons, neutrons, atoms, even C₆₀ buckyballs (interferometry experiments in the late 1990s). The wave nature is hidden by the tiny λ for everyday objects.

§30.8 — The Four Quantum Numbers

Solving Schrödinger's equation for the hydrogen atom (a 3-D problem) yields four labels per electron. Only certain combinations are allowed.

Spectroscopists name the ℓ values with letters (historical, from "sharp/principal/diffuse/fundamental"):

Counting States

How many distinct (n, ℓ, m_ℓ, m_s) states are there for n = 3?

• ℓ = 0: m_ℓ = 0 → 1 spatial state × 2 spins = 2.
• ℓ = 1: m_ℓ = −1, 0, +1 → 3 × 2 = 6.
• ℓ = 2: m_ℓ = −2, −1, 0, +1, +2 → 5 × 2 = 10.
• Total for n = 3: 2 n² = 18 electrons.

Spin and Stern–Gerlach

Spin is an intrinsic angular momentum that has no classical analog (it is not the electron literally rotating). Each elementary particle carries a fixed spin: electrons, protons, neutrons all have s = ½, photons have s = 1.

The 1922 Stern–Gerlach experiment shot a beam of silver atoms through an inhomogeneous magnetic field. The beam split into exactly two spots — confirming that the angular momentum projection is quantized into m_s = +½ ℏ and −½ ℏ. Spin was a shock: nothing classical could explain a half-integer "rotation."

§30.7 — Patterns in Spectra: Zeeman & Fine Structure

If you place a hydrogen lamp in an external magnetic field, single emission lines split into multiple closely-spaced lines — the Zeeman effect. Each m_ℓ sub-level has a slightly different energy in B:

This directly proves m_ℓ takes (2ℓ + 1) discrete values. Even with no external field, the electron's spin couples weakly to its own orbital motion (spin–orbit coupling) producing tiny "fine structure" splittings — visible as doublets in the sodium D-line.

§30.9 — The Pauli Exclusion Principle & the Periodic Table

This single statement organizes the periodic table. Electrons fill from low energy upward (the Aufbau principle), two per spatial state.

Order of Filling (Approximate Energy Order)

Note the ordering swaps (4s before 3d, 5s before 4d, 6s before 4f) — these are the famous "diagonal rule" details that shape the shape of the periodic table (transition metals, lanthanides, actinides).

Filling the First Row

The Layout of the Table

• Columns (groups) share the same outer-shell configuration — and therefore the same chemistry.
• Group 1 (alkali metals): ns¹ — soft, reactive metals.
• Group 2 (alkaline earths): ns² — harder, less reactive metals.
• Group 17 (halogens): ns² np⁵ — one electron short of closed.
• Group 18 (noble gases): ns² np⁶ — closed-shell, chemically inert.
• Transition metals (d-block): partially filled d subshell — colors, magnetism, catalysis.
• Lanthanides & actinides (f-block): 4f / 5f filling — chemically very similar within each row (hard to separate).

Worked: Writing Electron Configurations

Example 1 — Iron (Z = 26)

Fill in order: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶. Check: 2 + 2 + 6 + 2 + 6 + 2 + 6 = 26 ✓.

Shorthand: [Ar] 4s² 3d⁶ — start from the previous noble gas core.

Example 2 — Copper (Z = 29)

Naive Aufbau predicts [Ar] 4s² 3d⁹. Reality: [Ar] 4s¹ 3d¹⁰ (the full d shell wins energetically).

Example 3 — Iodine (Z = 53)

[Kr] 5s² 4d¹⁰ 5p⁵. Halogen — wants one more electron to close the 5p shell, hence I⁻ is common.

Periodic Trends Worth Knowing

• Atomic radius grows down a column (more shells), shrinks across a row (more nuclear pull on the same outer shell).
• Ionization energy opposite: peaks at noble gases (closed shell), dips at alkali metals.
• Electron affinity largest for halogens (one slot from full).
• Electronegativity: F is most electronegative, Cs/Fr the least.

Angular Momentum Coupling

The total angular momentum of a multi-electron atom comes from coupling all the individual orbital and spin angular momenta. Two limiting schemes:

• L-S (Russell–Saunders) coupling: useful for light atoms. Add all ℓ's into a total L, all s's into a total S, then couple L + S = J.
• j-j coupling: useful for heavy atoms where spin–orbit is large. Couple each electron's ℓ and s into j first, then combine all j's.

The total magnitude is always |J| = √(J(J+1)) ℏ, with J_z = m_J ℏ taking values m_J = −J, −J+1, …, +J. This is the same structure you've already seen for ℓ — just at the whole-atom level.

Spectroscopic "Term Symbols"

An atomic state is written ^2S+1L_J. For example, hydrogen's ground state is ²S_½ (S = ½, L = 0, J = ½). Sodium's ground state is ²S_½; its first excited p state splits into ²P_½ and ²P_3/2 — these are the famous yellow D-lines at 589.0 and 589.6 nm.

The Three Photon–Atom Processes (Einstein, 1917)

Consider an atom with two levels separated by ΔE = hf. There are exactly three ways the atom and a photon can interact:

Why Stimulated Emission Matters

The output photon in stimulated emission is identical to the trigger: same frequency, same direction, same phase, same polarization. Cascading this process amplifies a coherent beam — this is light amplification by stimulated emission of radiation, or L A S E R (Lecture 14).

Einstein A and B Coefficients (Conceptual)

Let n₁ and n₂ be the number of atoms in lower and upper states, and let ρ(f) be the photon energy density at the transition frequency. The three rates are:

Einstein showed B₁₂ = B₂₁ and A₂₁/B₂₁ = 8πhf³/c³. In thermal equilibrium n₂/n₁ follows the Boltzmann factor:

For visible light at room temperature, ΔE/kT ~ 80, so n₂/n₁ ~ e⁻⁸⁰ — astronomically tiny. Almost everything is in the ground state, and absorption dominates over stimulated emission. That's why ordinary matter absorbs more than it emits.

Selection Rules & Lifetimes

• "Allowed" electric-dipole transitions follow Δℓ = ±1, etc. (Lecture 12). Lifetimes are typically a few nanoseconds.
• "Forbidden" magnetic-dipole or electric-quadrupole transitions can have lifetimes from milliseconds to hours — these are metastable states. Phosphorescent paint stores energy in metastable triplet states.

Topics You Must Own

• Discovery of the atom (30.1–2): Thomson's e/m, Millikan's e, Rutherford gold-foil → tiny dense nucleus.
• Bohr model (30.3): postulates, L = nℏ, r_n = n²a₀, E_n = −13.6 eV/n², photon energies, Rydberg formula, Lyman/Balmer/Paschen series.
• X rays (30.4): Bremsstrahlung continuous spectrum (λ_min = hc/eV), characteristic K_α lines, Moseley.
• Atomic excitations (30.5): absorption ↔ emission, fluorescence vs. phosphorescence, metastable states.
• de Broglie standing waves (30.6): nλ = 2πr derives Bohr's L = nℏ; Davisson–Germer confirmation.
• Quantum numbers (30.8): n, ℓ, m_ℓ, m_s; ranges; subshell capacities (s2, p6, d10, f14); 2n² per shell; Stern–Gerlach.
• Patterns (30.7): Zeeman ΔE = m_ℓ μ_B B; spin–orbit fine structure.
• Pauli & periodic table (30.9): exclusion principle; Aufbau filling order; Hund's rule; alkali / halogen / noble gas behavior.
• Light–matter interaction: three Einstein processes (absorption, spontaneous, stimulated), Boltzmann population, why population inversion is needed for lasing.

Strategy

• Use the Bohr formula liberally: E_n = −13.6 eV/n². Photon energy = |E_i − E_f|. Then λ (nm) = 1240/E (eV).
• For hydrogen-like ions (He⁺, Li²⁺), multiply by Z².
• Know the Lyman/Balmer/Paschen end-states (1, 2, 3) cold — they give you the spectral region instantly.
• Counting electrons: a shell n holds 2n²; a subshell ℓ holds 2(2ℓ+1).
• Periodic-table chemistry: identify the outermost partially-filled subshell to predict valence behavior.
• Selection rules + Boltzmann: the only "free" lasers happen at very low temperatures or very narrow lines — you must work to produce inversion.

How a Laser Works

You need three ingredients:

1. A gain medium with at least three energy levels and a long-lived (metastable) upper state.
2. A pump (flashlamp, electrical discharge, another laser) that drives the population to the upper state — creating population inversion.
3. An optical cavity: two mirrors (one fully reflective, one ~99% reflective) bouncing the light through the medium to amplify it on each pass.

3-Level vs 4-Level Schemes

The 4-level scheme is easier to keep inverted because level 1 is empty (it dumps quickly to GS), so even a small upper population (n₂) outnumbers it.

Properties of Laser Light

• Monochromatic — narrow line set by the cavity resonance.
• Coherent — every photon in step (temporally and spatially).
• Directional — collimated beam, low divergence.
• Bright — vastly higher intensity (W/m²/sr/Hz) than thermal sources.

Common Laser Types

From Atoms to Solids: Bonding

When atoms get close, their outer-shell wavefunctions overlap, and electrons are shared in ways that lower total energy. The four primary bond types:

Hydrogen bonding (a special vdW-style attraction involving H–O, H–N) is what holds water together and folds DNA — strong by vdW standards, weak by covalent standards.

From Discrete Levels to Continuous Bands

Pauli forbids two electrons from occupying the same state. When N atoms come together to form a solid, each atomic level splits into N closely-spaced levels — and for N ~ 10²³, those levels merge into a continuous band.

Two bands matter most:

• Valence band — derived from the outermost filled atomic orbital. Holds the bonding electrons.
• Conduction band — the next available band above. Electrons here can move freely under an applied field.
• Between them is a band gap E_g — a forbidden range of energies.

• Conductor (Cu, Ag, Au): partially filled band, electrons free to drift in any tiny field. Conductivity ~ 10⁸ S/m.
• Insulator (diamond, glass): full valence band, large gap (~5 eV) — at room temperature kT ≈ 0.025 eV is too small to bridge the gap. Conductivity ~ 10⁻¹⁵ S/m.
• Semiconductor (Si, Ge, GaAs): small gap (~0.7–1.5 eV). At 300 K some electrons thermally jump into the conduction band; conductivity is intermediate and tunable.

Doping & the p–n Junction

Pure silicon (Z = 14, 4 valence electrons) is a poor semiconductor. Adding tiny impurities transforms its conductivity by orders of magnitude:

A hole behaves like a positive mobile charge — when an electron jumps to fill it, the hole effectively moves the other way.

The p–n Junction (Diode)

Place p-type and n-type silicon in contact. Electrons from the n-side and holes from the p-side meet at the boundary and recombine, leaving a depletion region with a built-in electric field that opposes further diffusion.

• Forward bias (+ on p, − on n): external field cancels the built-in field, current flows.
• Reverse bias (opposite): field strengthens, depletion widens, almost no current — diode rectifies.
• LED: an electron in the conduction band (n-side) recombines with a hole (p-side) in the depletion region, releasing E_g as a photon. λ ≈ 1240/E_g(eV) nm.
• Photodiode / solar cell: reverse — a photon excites an electron across the gap, the built-in field separates the e⁻–hole pair, current flows.
• Transistor: a third terminal modulates the junction, providing voltage-controlled gain — the foundation of all digital electronics.

Anatomy of a Nucleus

The nucleus contains:

• Z protons (charge +e, mass ≈ 938 MeV/c² ≈ 1.6726 × 10⁻²⁷ kg) — defines the element.
• N neutrons (charge 0, mass ≈ 939.6 MeV/c²) — sets the isotope.
• Total A = Z + N "nucleons."

An isotope is a nucleus with the same Z but different N. Carbon-12 (6p, 6n) and carbon-14 (6p, 8n) are both carbon — chemically identical, different masses, different nuclear stability.

Nuclear Size

This A^1/3 dependence means nuclear matter has roughly constant density (~2.3 × 10¹⁷ kg/m³) — a teaspoon of nuclear matter weighs about a billion tons.

Atomic Mass Unit

Mass Defect & Binding Energy

Pull a nucleus apart into its constituent protons and neutrons. The free-nucleon mass total is more than the bound nucleus — the difference is the energy that was holding it together:

BE is positive — you have to put in energy to separate the nucleus.

Worked Example: Helium-4

2 m_p + 2 m_n = 2(1.00728) + 2(1.00867) = 4.03190 u. M(⁴He) = 4.00260 u. Δm = 0.02930 u. BE = 0.02930 × 931.494 = 27.3 MeV, or ~7 MeV per nucleon — a remarkably tightly bound nucleus, which is why α emission is favored.

Binding Energy per Nucleon — the Master Curve

Nuclear Stability — the Valley of Stability

• Light stable nuclei have N ≈ Z.
• Heavier stable nuclei have N > Z (more neutrons dilute proton–proton repulsion).
• Beyond Z ≈ 83 (bismuth), no stable isotopes — all radioactive.
• "Magic numbers" (2, 8, 20, 28, 50, 82, 126) for protons or neutrons → extra stability (analog of noble-gas closed shells).

The Strong Nuclear Force

The strong force binds nucleons. Properties:

• Attractive, much stronger than EM (~100×) at very short range.
• Range ~ 1 fm — drops off essentially to zero beyond a few fm.
• Independent of charge: pp, pn, nn all the same strength.
• Repels at extremely short range (< 0.5 fm) — keeps nucleons from collapsing.

The Three Main Modes of Radioactive Decay

Decays must conserve charge, baryon number, mass-energy, and momentum — that's why β decay needed an "invisible" neutrino (Pauli's 1930 prediction, confirmed in 1956): the observed e⁻ energy spectrum is continuous, so a third particle must carry off the missing energy.

Examples

The Decay Law

Each unstable nucleus has a fixed probability per unit time of decaying — independent of its history (no "aging"). Let λ be that decay constant. The number remaining obeys:

Worked: Carbon-14 Dating

Atmospheric carbon has a steady ¹⁴C/¹²C ratio (~10⁻¹²) because cosmic-ray neutrons constantly create ¹⁴C from ¹⁴N. Once a tree dies it stops exchanging carbon and the ¹⁴C decays away (T_½ = 5730 yr).

If a sample reads 25% of the modern ¹⁴C activity: 0.25 = (½)ⁿ ⇒ n = 2 half-lives ⇒ age ≈ 11,460 yr.

Decay Chains

A heavy radionuclide may decay through a sequence of α and β steps until it lands on a stable isotope. The famous ²³⁸U series ends at ²⁰⁶Pb after 8 α and 6 β decays — the Pb/U ratio is the basis of geological dating (Earth is 4.55 Gyr old).

Fission

A heavy nucleus (e.g., ²³⁵U) absorbs a slow neutron, becomes momentarily ²³⁶U*, and splits into two medium-sized fragments plus 2–3 free neutrons:

The released ~200 MeV per event is roughly 50 million times larger than a typical chemical-bond energy — that's why nuclear fuel is so energy-dense (1 g of U-235 ≈ 3 tons of coal).

Fusion

Two light nuclei merge, releasing energy as the product climbs the BE/A curve toward iron. The Coulomb barrier (~1 MeV) makes fusion much harder to ignite than fission — you need temperatures of ~10⁷ K (kT ~ 1 keV with quantum tunneling helping) before things go.

The Sun's primary process is the p–p chain:

Net mass loss: 0.7% of the proton mass. The Sun converts ~6 × 10¹¹ kg of hydrogen per second into helium, radiating L_☉ ≈ 3.8 × 10²⁶ W. It has ~5 billion years of fuel left.

The Sun & Stellar Nucleosynthesis

The Sun is a roughly steady-state self-gravitating ball of plasma — gravity wants to collapse it, fusion-generated thermal pressure pushes back. The energy released by fusion in the core (T ~ 1.5 × 10⁷ K) takes ~100,000 years to random-walk out to the photosphere as photons, then 8 minutes to reach Earth as sunlight.

• p–p chain (Sun & cooler stars): builds ⁴He from H. Limited by the slow weak-interaction step pp → d + e⁺ + ν.
• CNO cycle (hotter stars): C, N, O act as catalysts; faster than p–p above ~1.7 × 10⁷ K. Dominates in stars heavier than ~1.3 M_☉.
• Triple-α: 3 ⁴He → ¹²C in red giants — the source of all carbon in the universe (and us).
• Heavier elements: built up to iron in successive shell burnings. Beyond iron requires energy input — produced in supernova explosions (s- and r-processes). You are made of stardust.

End States

• Star < 8 M_☉: red giant → planetary nebula → white dwarf (electron-degenerate, M < 1.4 M_☉ Chandrasekhar limit).
• Star 8–25 M_☉: supernova → neutron star (neutron-degenerate, ~1.4–3 M_☉, R ~ 10 km).
• Star > 25 M_☉: supernova → black hole.

Nuclear Reactors & Weapons

Reactors (Controlled Fission)

• Fuel: enriched U (3–5% ²³⁵U; natural U is 0.72% ²³⁵U).
• Moderator (water, graphite, heavy water): slows down neutrons so they're more likely to be captured by ²³⁵U.
• Control rods (B, Cd): absorb neutrons to keep the chain at exactly k = 1.
• Coolant: removes the heat to drive a turbine.
• Containment: protects the public from radioactive fission products.

A typical 1 GW reactor consumes ~1 kg of ²³⁵U per day. The spent fuel contains long-lived fission products and transuranic elements — the long-term storage problem.

Bombs (Uncontrolled Chain Reaction)

• Critical mass = the smallest amount of fissile material that sustains a chain reaction. For pure ²³⁵U: ~52 kg bare, ~15 kg with a beryllium reflector.
• Gun-type (Hiroshima, "Little Boy"): two subcritical pieces of ²³⁵U slammed together — simple, only works for U.
• Implosion-type (Nagasaki, "Fat Man"): a hollow shell of ²³⁹Pu compressed by symmetric explosive lenses to supercritical density — works for both U and Pu.
• Thermonuclear (H-bomb): a small fission "primary" creates the temperatures needed to ignite a fusion "secondary" of D + T + Li-6. Yields can be 100s of times the fission limit.

Radiation Dose & Biological Effect

• Background radiation: ~3 mSv/yr (mostly radon + cosmic + medical).
• Trans-Atlantic flight: ~0.05 mSv (cosmic).
• CT scan: ~10 mSv.
• Acute effects: nausea ~1 Sv; LD_50/30 ~4 Sv (without medical care).

The Two Cracks in 1900 Physics — and What Each Became

The Through-Lines of PHYS 25

1. Quantization is the new normal. Energy (E = nhf), angular momentum (L = nℏ), spin (½ℏ), charge (e), and even spectra (line vs. continuous) are all discrete.
2. Wave–particle duality is universal. Light has p = h/λ; matter has λ = h/p. Both diffract, both can be photons or particles depending on the question you ask.
3. Pauli + quantization → structure. The periodic table, chemistry, the rigidity of solids, white dwarfs and neutron stars all follow from "no two fermions in the same state."
4. E = mc². Mass is just one form of energy. Nuclear reactions release the largest available energy density in the everyday universe (~10⁶× chemistry).
5. Conservation laws survive everything. Energy, momentum, angular momentum, charge, baryon & lepton number — these still hold even when classical intuition fails.

What's Beyond PHYS 25

• General relativity: gravity as curvature of spacetime; black holes; cosmology.
• Full quantum mechanics: Schrödinger and Dirac equations; entanglement, decoherence, quantum information.
• Particle physics & the Standard Model: quarks, leptons, gauge bosons, the Higgs.
• Condensed matter: superconductors, topological insulators, the quantum Hall effect.
• Open puzzles: dark matter, dark energy, the matter–antimatter asymmetry, quantum gravity.

Coverage Map

Strategy for a 2-Hour Cumulative Final

• Skim the entire exam first to set time per problem. Knock out the easy ones to bank points.
• Identify which "big idea" a problem is testing — relativity, quantization, Pauli, decay law — before reaching for a formula.
• Use shortcuts: hc = 1240 eV·nm for photon energies; E_n(H) = −13.6 eV/n²; T_½ = ln 2 / λ; 1 u = 931.5 MeV/c².
• Sanity-check limiting cases: v ≪ c → classical; large quantum numbers → classical; long times → equilibrium.
• Keep units consistent. Convert eV ↔ J only if necessary; otherwise stay in eV.
• Write what you know on every problem, even if you can't finish — partial credit is real.

Topics to Drill the Night Before

1. γ factor + time dilation + length contraction + p = γmu + E = γmc².
2. P = σeAT⁴ and λ_maxT = 2.898 × 10⁻³ m·K.
3. Photoelectric: KE_max = hf − BE.
4. Bohr: E_n = −13.6 Z²/n² eV, Rydberg formula, Lyman/Balmer/Paschen.
5. Quantum numbers and capacities (s2, p6, d10, f14; 2n² per shell).
6. Aufbau filling order; Hund's rule; alkali / halogen / noble gas patterns.
7. Three Einstein processes; population inversion; how a 4-level laser works.
8. Conductor / semiconductor / insulator + n/p doping + p–n diode + LED.
9. Mass defect & binding energy; BE/A peaks at Fe; fusion vs fission.
10. α, β⁻, β⁺, γ — what changes; N(t) = N₀e^−λt; T_½ = ln2/λ.

Chapter 28 — Special Relativity

Chapters 13.1 & 14.7 — Temperature & Thermal Radiation

Chapter 29 — Quantum Mechanics (Photons & Matter Waves)

Chapter 27.3 — Double-Slit Interference

Chapter 30 — Atomic Physics (Bohr, Quantum Numbers)

Light–Matter, Lasers, & Solids

Nuclear Physics (Lectures 16–18)

Essential Constants

Built Into This Site

• Simultaneity — The Spaceship Duel (Lecture 1): two frames, same events, different orderings.
• Relativistic Velocity Addition (Lecture 2): four animated cases of u = (u′ + v)/(1 + u′v/c²) and its inverse.
• Relativistic Momentum & Center of Mass (Lecture 3): watch γmv diverge at v → c.
• Thermal Radiation — Why Hot Things Glow (Lecture 4): Stefan–Boltzmann, Wien, Planck vs. Rayleigh–Jeans.
• The Photoelectric Effect (Lecture 5): tune f and intensity, pick a metal, read V_stop.

PhET (University of Colorado Boulder)

• Simple Relativity — time dilation & length contraction intuition.
• Blackbody Spectrum — move T from a fridge to the Sun to a blue star.
• Photoelectric Effect — the canonical demo, with work function slider.
• Wave Interference — single + double slit with water, sound, or light.
• Davisson–Germer: Electron Diffraction — matter waves off a crystal lattice.
• Quantum Wave Interference — single-photon/electron double-slit built up one particle at a time.