UPIV · Second-year to early third-year undergraduate · Modern physics and introductory quantum mechanics
Where classical physics runs out—and what replaced it.
A fifteen-week course on the transition from classical physics to twentieth- and twenty-first-century physics. It opens where classical models measurably fail -- the blackbody spectrum and the photoelectric effect -- builds special relativity and then quantum mechanics as the replacements, solves the Schrodinger equation for wells, barriers, oscillators and the hydrogen atom, and applies the result to atoms, molecules, solids, nuclei and particles before closing on the questions still open.
Independent pathway notice. GioPhysics University & College Physics is an independent learning pathway. GioPhysics is not a university, does not award academic credit or degrees, and is not accredited by or affiliated with any university or professional body. This is not a universal programme specification. Course coverage and degree requirements vary by institution; confirm requirements with your university.
Before week 01
Bring the whole introductory sequence. Add differential equations.
This is the first course in the pathway where the mathematics is the physics. Separation of variables, boundary conditions, and complex exponentials are not background here — they are how every quantisation result in the course is obtained.Readiness, not gatekeeping
Know which mathematics must be fluent.
A student who can separate a partial differential equation, apply boundary conditions, and normalise an integral will find this course demanding but tractable. Without those three, the physics is hidden behind the algebra.
- University Physics I–III, or equivalent calculus-based mechanics, electromagnetism, waves, and thermal physics
- Calculus I–III, including partial derivatives and multiple integrals
- Basic ordinary differential equations; separation of variables is used from unit 6 onward
- Complex numbers in exponential form, and comfort with linear algebra notation is an advantage
How calculus enters: the Schrödinger equation is a partial differential equation, and almost every quantum result in this course comes from separating it, imposing a boundary condition, and normalising what survives. Quantisation is not assumed anywhere in this map — it falls out of the boundary conditions.
Mathematical tools used explicitly
- Second-order differential equations and separation of variables for the Schrödinger equation
- Complex exponentials, probability densities, normalisation, and expectation-value integrals
- Operators, eigenvalue equations, commutators, and Hermitian operators as observables
- Spherical polar coordinates and separation into radial and angular parts
- Series solutions and boundary conditions as the origin of quantisation
- Lorentz transformations and four-vector bookkeeping for relativistic kinematics
- Exponential decay laws, and statistical reasoning for counting and half-life data
- Correspondence checks: every quantum or relativistic result must reduce to the classical one in its limit
Fifteen units · one per teaching week
A single argument, run over fifteen weeks: the classical account fails, and here is what works.
The sequence starts at the measurements classical physics gets wrong, builds relativity and quantum mechanics as the replacements, gives the quantum replacement a workable formalism, applies it to matter from the hydrogen atom to semiconductors, and ends where physics is still open.Units 1–4 establish why classical physics had to be replaced, and build special relativity and the photon and matter-wave evidence.
Units 5–8 are the quantum-mechanics core: wave functions, the Schrödinger equation, bound states, barriers, and the oscillator.
Units 9–12 apply quantum mechanics to angular momentum, the hydrogen atom, many-electron atoms, molecules, and solids.
Units 13–15 cover nuclear and particle physics and close with an orientation toward current research, not assessed mastery of it.
Course scope: This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. It is written for a typical three- to four-credit lecture course with an optional one-credit laboratory over fourteen to fifteen weeks, and it deliberately re-covers the modern-physics ground that University Physics III surveys, at the depth a dedicated course allows. Departments vary widely in how much quantum formalism they expect at this stage; follow your institution's published scope, notation, laboratory programme, and assessment rules.
Complete interactive course map
All fifteen units, searchable by concept and evidence.
Search 135 numbered subsections, inspect outcomes and prerequisites, and expand the laboratory and problem-practice plan for each unit.Mapped lesson links: linked topics open relevant existing GioPhysics material. Many topics at this level are mapped but not yet written; nothing here links to a page that does not teach it at or near this level.
Interactive course map
Choose one phase. Open one unit. Learn by doing.
Search includes unit titles, topics, skills, outcomes, prerequisites, and laboratory questions across all 15 units.
Private study checklist
0 of 15 units reviewedWeek 1 · UPIV-01 · Heavy
Limits of Classical Physics
Carry the classical account of radiation to the point where it breaks: cavity thermodynamics, mode counting, and equipartition give the Rayleigh-Jeans divergence at long wavelengths while Wien's exponential law covers the short ones, Planck's quantised oscillator derives both limits from one constant, and photoelectric and Compton evidence establishes that energy and momentum are exchanged in quanta of h nu and h over λ - without yet proving the field itself is quantised.
9 topics5 outcomes2 lab directions190 min practiceOpen Limits of Classical Physics: topics, evidence, and practice
Topics and mapped lessons
- The classical synthesis and where it stops
Newtonian dynamics, Maxwell's equations, and statistical mechanics as one closed account, and the ratios v/c, h nu over kT, and action measured in ℏ that decide where it holds; correspondence binds any successor to reproduce it in its own domain. Naming a boundary supplies no replacement.
- Cavity radiation, Kirchhoff's law, and the pre-Planck laws
Kirchhoff's emissivity-absorptivity equality and the universality it forces, Stefan-Boltzmann from the radiation pressure u/3 with the first two laws, and Wien's scaling u = ν-cubed times f(ν/T) from adiabatic compression. Thermodynamics never fixes f, so Wien's exponential guess u proportional to ν-cubed times exp(minus h nu over kT) was fitted, not derived: excellent at short wavelengths, wrong in the infrared.
- Mode counting, equipartition, and the ultraviolet catastrophe
Standing modes counted to 8 pi ν-squared over c-cubed per unit volume per unit frequency interval, both polarisations included, kT assigned to each by equipartition, the result exact as h nu over kT goes to zero, and the divergent integral over frequency. The count is geometry, so only equipartition can be at fault.
- Planck's quantised oscillator and the radiation law
Energies restricted to n h ν, Boltzmann-weighted geometric sums giving a mean energy of h nu divided by the quantity exp(h nu over kT) minus one, and a mean mode occupancy of one divided by that same quantity. The law returns Rayleigh-Jeans as h nu over kT goes to zero and Wien's exponential form when it is large, and yields Wien's displacement constant and sigma in terms of h, c, and k; Planck quantised the wall oscillators, not the field itself.
- Heat capacities and the freezing out of degrees of freedom
Dulong-Petit at 3R per mole and the diatomic-gas plateaus are what equipartition predicts and where it succeeds; the second failure is their collapse on cooling. Einstein's single-frequency solid freezes out with a Boltzmann factor, Debye's phonon spectrum gives the low-temperature T-cubed law. Both models fit data without deriving the spectrum they assume.
- The photoelectric effect and Einstein's light quantum
Instantaneous emission, a threshold frequency, and intensity setting saturation current alone; maximum kinetic energy h nu minus φ, the stopping potential, and Millikan's slope h/e. The relation assumes one photon per electron and a cold, clean surface, and the contact potential difference between cathode and anode shifts the intercept, so the slope measures h while the intercept does not measure the cathode's phi.
- Compton scattering and photon momentum
Conserving relativistic energy and momentum for a photon of momentum h over lambda on a free electron gives an absolute shift of h over m-sub-e c times one minus cosine θ, with the electron Compton wavelength h over m-sub-e c equal to 2.43 pm. The shift is independent of the incident wavelength, so only X-rays make the fractional change visible; tightly bound electrons violate the free-electron assumption and return the unshifted line.
- What the evidence establishes, and what it does not
A classical field driving quantised atoms reproduces the photoelectric threshold and the Compton shift, so neither result proves the electromagnetic field is quantised; photon antibunching and sub-Poissonian counting statistics do. Historical evidence is not modern proof, and the distinction is the point of the unit.
- Old quantum theory and the correspondence principle
Action quantised by the Wilson-Sommerfeld condition, adiabatic invariants, and the correspondence principle fixing transition intensities at large quantum numbers. The rules work only for separable, multiply periodic systems and fail on helium, forcing a wave equation instead.
Learning outcomes
- Evaluate the dimensionless ratio that decides a regime - h nu over kT, v/c, or an action in units of ℏ - before selecting any relation.
- Derive the Rayleigh-Jeans law from cavity mode counting and equipartition, and locate the failure in equipartition rather than in the mode count.
- Derive the Planck distribution from quantised oscillator energies, recover the Rayleigh-Jeans and Wien exponential laws as its two limits, and obtain Wien's displacement constant and the Stefan-Boltzmann constant in terms of h, c, and k.
- Fit stopping potential against frequency to extract h from the slope with propagated uncertainty, and explain why the intercept returns the anode work function rather than the emitting cathode's.
- Analyse Compton scattering as relativistic two-body kinematics, and distinguish the quantisation each experiment establishes from the quantisation it merely permits.
Prerequisite thread
- Boltzmann factors, kT energy scales, and equipartition from statistical mechanics
- Maxwell's equations, electromagnetic energy density, and radiation pressure
- Relativistic energy-momentum conservation, including E = pc for a photon
- Geometric series, definite integrals, and linear least-squares fitting with uncertainty
Laboratory directions
hands-onPlanck's constant from a filtered photocell Does stopping potential vary linearly with frequency across five filtered lines, and what value of h does a weighted fit on the slope support?
Evidence: Stopping potentials at five source frequencies, a weighted fit whose slope h/e yields h with propagated uncertainty, residuals, the reverse-current systematic that sets the stopping-potential floor, and an explicit statement that the intercept carries the anode work function and contact potential difference, so no cathode work function may be quoted from itcomputationalFitting a measured thermal spectrum Which temperature and emissivity does a calibrated lamp spectrum support, and over what range do the Rayleigh-Jeans and Wien limiting forms each survive?
Evidence: Calibrated spectral radiance, a two-parameter Planck fit, residuals, the fitted T checked against Wien's displacement law, the bands of h nu over kT where the Rayleigh-Jeans form and the Wien exponential form are each within one percent, and the greybody assumption namedProblem practice
Evaluating the ratio that decides the regime before choosing a classical, semiclassical, or photon relation
Capstone: Extract h independently from one thermal spectrum and one photoemission data set, compare the two values against their uncertainties, then state for each experiment which quantisation it establishes and which it only permits.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimLimits of Classical Physics 02 · RepresentationLog-log spectral radiance curves Cavity mode counting
03 · TestPrediction before measurement Does stopping potential vary linearly with frequency across five filtered lines, and what value of h does a weighted fit on the slope support?
04 · Evidence & boundaryDecide, then qualify Interactive diagram for Limits of Classical Physics: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelCavity mode counting turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Limits of Classical Physics, explain how a physicist can: Evaluate the dimensionless ratio that decides a regime - h nu over kT, v/c, or an action in units of ℏ - before selecting any relation.
- The classical synthesis and where it stops
Week 2 · UPIV-02 · Heavy
Special Relativity I
Rebuild kinematics from two postulates with the algebra carried out: where Galilean covariance fails for Maxwell, the boost derived from homogeneity, isotropy and group closure, rapidity as the parameter that actually adds, and the simultaneity offset from which contraction and the paradoxes follow.
9 topics5 outcomes2 lab directions200 min practiceOpen Special Relativity I: topics, evidence, and practice
Topics and mapped lessons
- Inertial frames and the Galilean group
The operational test for an inertial frame, and the boost x' = x - vt with t' = t as a one-parameter subgroup of the ten-parameter Galilean group, closed under composition and inversion; the covariance of Newton's laws, and the absolute time and frame-free simultaneity the group silently assumes.
- Electrodynamics under a Galilean boost
Substituting the Galilean chain rule into the wave equation leaves (1 - β-squared) ψₓₓ + (2v/c-squared) ψₓₜ - (1/c-squared) ψₜₜ = 0: a first-order cross term that no further Galilean change of coordinates removes, so Maxwell is not Galilean-covariant against a c = 1/√(μ0 eps0) that carries no frame. Aether-drift and emission models as the two classical escapes, and what one null result alone cannot exclude.
- The postulates and clock synchronisation
The relativity principle extended past mechanics, invariant c, and Einstein synchronisation as a stipulation: a distant clock is set to the midpoint of a round-trip light signal, the ε = 1/2 choice. Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell as three separate constraints - isotropy of the two-way speed, independence of the frame's velocity, and time dilation - and the one-way speed that stays conventional whatever they return.
- Deriving the boost from symmetry
Linearity from homogeneity, reciprocity from isotropy, and closure under composition force a one-parameter family carrying a single invariant speed K; causality excludes the Euclidean branch, the second postulate sets K = c, and K to infinity returns Galileo. Flat spacetime, inertial frames only.
- The boost in standard configuration and matrix form
ct' = γ(ct - β x) and x' = γ(x - β ct) with transverse coordinates unchanged, written as a matrix of determinant γ-squared(1 - β-squared) = 1 whose inverse is β reversed; standard configuration assumes collinear motion, aligned axes, and origins meeting at t = t' = 0.
- Rapidity and the composition of boosts
β = tanh(φ) and γ = cosh(φ) recast the boost as a hyperbolic rotation through rapidity, which simply adds for collinear boosts and puts c out of reach because φ is unbounded while tanh is not; additivity fails for non-collinear boosts, whose composition carries a residual Wigner rotation.
- The relativity of simultaneity
The offset dt' = −γ v dx/c-squared read straight off t' = γ(t - vx/c-squared), simultaneity hyperplanes ct = β x tilting toward the light line as β grows, and the pole-and-barn resolved by naming which event pair each frame calls simultaneous. Order reverses only for pairs with (c dt)-squared - dx-squared negative, that combination being the one the boost leaves unchanged.
- Time dilation and length contraction as corollaries
Imposing dx' = 0 gives dt = γ d-τ; marking both ends at one time in the measuring frame gives L = L0/γ along the motion only, transverse lengths fixed. Contraction is the simultaneity offset converted into a number, and neither result survives without its constraint stated and constant relative velocity.
- Transforming velocities and the c ceiling
Dividing the differentials of the boost gives ux' = (ux - v)/(1 - ux v/c-squared), transverse components carrying an extra 1/γ over that same denominator, and c as a fixed point of the map; composed sub-luminal boosts stay under c, and the law transforms one particle's velocity, not a closing rate a third frame assigns.
Learning outcomes
- Show that the wave equation is not covariant under a Galilean boost, and state what a null aether-drift result does and does not exclude.
- Derive the boost from homogeneity, isotropy, and group closure, obtaining an invariant speed whose value the second postulate fixes.
- Write a boost in matrix form, compose and invert boosts using rapidity, and recover the Galilean forms as an expansion in beta.
- Compute the simultaneity offset for a frame pair and use it to resolve the pole-and-barn and train paradoxes without contradiction.
- Read time dilation and length contraction off the transformation by imposing the correct constraint - dx' = 0 for the clock, simultaneous end-marking for the rod - and naming the proper quantity in each case.
Prerequisite thread
- Maxwell's equations and c = 1/√(μ0 eps0) from calculus-based electromagnetism
- The one-dimensional wave equation and partial differentiation by the chain rule
- Matrix algebra: linear maps, determinants, inverses, and composition
- Hyperbolic functions, their identities, and their series expansions
- The longitudinal Doppler factor √((1-β)/(1+β)) as met in the Physics III survey
Laboratory directions
computationalTime dilation in published ion-beam Doppler data Do the forward- and backward-shifted line frequencies from a fast ion beam multiply to the squared rest frequency, as the relativistic factor √((1-β)/(1+β)) requires and a classical moving-source model - which returns γ-squared times f0-squared - does not?
Evidence: Digitised line centres with uncertainty, the stated Doppler relation, the product test against f0 squared, an extracted Lorentz factor, residuals, and the dominant systematic nameddesign investigationWhat a one-way light-speed measurement can decide Which parts of a proposed one-way light-speed measurement are fixed by physics, and which by the synchronisation convention the design has already chosen?
Evidence: A declared synchronisation procedure with its epsilon stated, predicted results under two conventions, an error budget, and an explicit statement of what the design cannot decideProblem practice
Fixing the frame pair and the constraint that defines each measurement before applying the transformation to anything
Capstone: Assign coordinates to all four end-events of a rod passing through a shorter barn in both frames using the boost, show that each frame's account is internally consistent, and name the single assumption whose removal dissolves the apparent paradox.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimSpecial Relativity I 02 · RepresentationEvent tables with frame and coordinate labels Lorentz transformation algebra
03 · TestPrediction before measurement Do the forward- and backward-shifted line frequencies from a fast ion beam multiply to the squared rest frequency, as the relativistic factor √((1-β)/(1+β)) requires and a classical moving-source model - which returns γ-squared times f0-squared - does not?
04 · Evidence & boundaryDecide, then qualify Digitised line centres with uncertainty, the stated Doppler relation, the product test against f0 squared, an extracted Lorentz factor, residuals, and the dominant systematic named
Interactive diagram for Special Relativity I: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelLorentz transformation algebra turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Special Relativity I, explain how a physicist can: Show that the wave equation is not covariant under a Galilean boost, and state what a null aether-drift result does and does not exclude.
- Inertial frames and the Galilean group
Week 3 · UPIV-03 · Heavy
Special Relativity II
Carry the two postulates into measurement and dynamics: proper time as an integral along a worldline, the twin asymmetry, length contraction as a simultaneity convention, then p = γmv, E = γmc², and E² = (pc)² + (mc²)² applied to decays, collisions, and fixed-target thresholds, all in flat spacetime between inertial frames.
9 topics5 outcomes2 lab directions200 min practiceOpen Special Relativity II: topics, evidence, and practice
Topics and mapped lessons
- Proper time and time dilation
Imposing Δx′ = 0 — the condition that one clock in the primed frame is present at both events — and reading the inverse transformation t = γ(t′ + vx′/c²) to get Δt = γΔτ; proper time as the shortest reading, mutual reciprocity without contradiction, and muon evidence; it fixes an interval between two named events at constant relative velocity.
- Proper time along an arbitrary worldline
Chopping a journey into instants so each gives dτ = dt/γ(t), evaluating τ = ∫√(1 − v(t)²/c²) dt leg by leg, and the inertial path maximising proper time between two timelike-separated events; the clock hypothesis, that rate depends on speed alone and not on acceleration, is measured, not derived.
- The twin paradox and the asymmetry that resolves it
Why mutual dilation is not a contradiction, resolved three ways: the bent worldline's shorter proper time, the simultaneity slice sweeping across the stay-at-home twin at turnaround, and a signal count using the Doppler factor k = √((1+β)/(1−β)) outbound and 1/k on the return; the asymmetry is which twin changed frames, not the size of her g-force.
- Length contraction and the measurement it rests on
Marking both ends at one time in the measuring frame — Δt = 0 there — turns x′ = γ(x − vt) into L₀ = γL, so L = L₀/γ, with transverse dimensions untouched and proper length the rest-frame value; contraction is therefore the relativity of simultaneity converted into a number, not a stress in the material.
- Pole and barn, rigidity, and what a fast object looks like
Tabulating the two door-closing events in both frames to resolve the pole-and-barn paradox, why no body can be rigid once a push cannot outrun c, and the Terrell–Penrose rotation that makes a photograph of a fast object differ from its measured length; the analysis stays between inertial frames throughout.
- Relativistic momentum and dynamics
Why p = mv fails to be conserved in every frame once velocities compose relativistically, p = m dx/dτ = γmv as the repair, F = dp/dt giving γ³ma along the motion and γma across it, and c as an asymptote under constant force; mass is invariant, so both F = ma and relativistic mass go.
- Work, kinetic energy, and rest energy
Integrating W = ∫v dp by parts to K = (γ − 1)mc², reading E = γmc² with rest energy mc² at v = 0, and expanding γ to recover ½mv² with a leading fractional correction of ¾β²; within mechanics alone mc² is only an additive constant, and it shows itself when the invariant mass changes.
- The energy-momentum four-vector and its invariant
Assembling (E/c, p) as a four-vector that boosts exactly like (ct, x), so E′ = γ(E − vpₓ) and p′ₓ = γ(pₓ − vE/c²) while transverse components pass through; the invariant E² − (pc)² = (mc²)² holds the same value in every frame, β = pc/E, and m = 0 gives E = pc; invariant mass belongs to a system and is not additive over its parts.
- Collisions, decays, and threshold energies
Conserving total four-momentum through a decay or collision and squaring it to read the invariant mass, working in the centre-of-momentum frame, and converting proper lifetime to a lab decay length βγcτ; for a fixed target the available energy is only √s ≈ √(2mtarget c²Elab), so the beam energy a threshold demands climbs as the square of the mass produced, which is the whole case for colliders. Conservation fixes what is allowed, never a rate or a branching ratio.
Learning outcomes
- Derive Δt = γΔτ and L = L₀/γ from the Lorentz transformation, naming the event pair and the frame that owns the proper quantity.
- Evaluate proper time as τ = ∫√(1 − v(t)²/c²) dt along a segmented worldline and show the inertial path maximises it.
- Resolve the twin and pole-and-barn paradoxes by event bookkeeping in both frames rather than by declaring one frame correct.
- Derive p = γmv from dx/dτ, obtain F = γ³ma along the motion and γma across it, and integrate the work to K = (γ − 1)mc².
- Transform (E/c, p) between frames, apply E² = (pc)² + (mc²)² and invariant mass to decays, collisions, and fixed-target thresholds, and recover the Newtonian forms as β → 0.
Prerequisite thread
- Lorentz transformation, velocity composition, and the invariant interval from Unit 2
- Momentum, energy, and the work-energy theorem from calculus-based mechanics
- Binomial and Taylor expansion, definite integrals, and integration by parts
- Vector components and the dot product; index notation used descriptively only
Laboratory directions
computationalStorage-ring muons and the clock hypothesis Do published decay curves for muons circulating in a storage ring at γ ≈ 29 scale with speed alone, or does the enormous transverse proper acceleration shift the decay rate?
Evidence: Decay constant fitted to the counting histogram with its uncertainty, γ inferred from the ring momentum, τlab/τ₀ compared with γ, residuals, the upper bound those residuals place on any acceleration-dependent term, and the dominant systematic named.hands-onBeta momentum against kinetic energy Do magnetically selected β-particle momenta and their measured kinetic energies follow E² = (pc)² + (mc²)² rather than K = p²/2m?
Evidence: Magnetic rigidity Bρ = p/q for each selected momentum, detector energies with calibration uncertainty, both model curves plotted over the data, residuals, and the β range below which the two predictions become indistinguishable within those uncertainties.Problem practice
Naming the two events and the frame that owns the proper quantity before writing any γ, building every answer from an invariant where one exists, and testing each result against its β → 0 limit
Capstone: Take one two-body decay of a known parent at relativistic speed, such as Λ → pπ⁻: fix the daughter momenta in the centre-of-momentum frame from the three invariant masses alone, boost to the lab with E′ = γ(E − vpₓ) and p′ₓ = γ(pₓ − vE/c²), and show the two frames disagree on every energy, momentum, angle, and time interval while the squared total four-momentum is unchanged. Then compute the parent's lab decay length βγcτ, state the fixed-target beam energy that would be needed to produce it, and name the step in your argument that fails the moment one of the frames accelerates.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimSpecial Relativity II 02 · RepresentationEvent tables labelled by frame Worldline proper-time integration
03 · TestPrediction before measurement Do published decay curves for muons circulating in a storage ring at γ ≈ 29 scale with speed alone, or does the enormous transverse proper acceleration shift the decay rate?
04 · Evidence & boundaryDecide, then qualify Decay constant fitted to the counting histogram with its uncertainty, γ inferred from the ring momentum, τlab/τ₀ compared with γ, residuals, the upper bound those residuals place on any acceleration-dependent term, and the dominant systematic named.
Interactive diagram for Special Relativity II: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelWorldline proper-time integration turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Special Relativity II, explain how a physicist can: Derive Δt = γΔτ and L = L₀/γ from the Lorentz transformation, naming the event pair and the frame that owns the proper quantity.
- Proper time and time dilation
Week 4 · UPIV-04 · Heavy
Photons and Matter Waves
Give the photon a momentum and the electron a wavelength, then carry the consequences out: the Compton shift from a relativistic two-body ledger, diffraction geometry from λ = h/p, group velocity against a phase velocity that turns out not to be convention-independent at all, and Fourier reciprocity as the quantitative form of duality.
9 topics5 outcomes2 lab directions200 min practiceOpen Photons and Matter Waves: topics, evidence, and practice
Topics and mapped lessons
- Photon energy, momentum, and the massless limit
E = hf with p = E/c = h/λ read off the m = 0 branch of E-squared = (pc)-squared + (mc-squared)-squared, and (E/c, p-vector) carried as a four-vector so the Doppler shift is nothing but an energy transformation between frames. The same ledger gives emission recoil: an atom of mass M shedding a photon of energy Eγ keeps Eγ-squared over 2Mc-squared and emits that much red of the line centre. A photon has no rest frame and no position operator.
- Radiation pressure, photon flux, and counting statistics
Momentum flux S/c giving I/c on an absorber and 2I/c on a mirror at normal incidence, falling to 2I cos-squared theta over c off-axis; the same beam divided by hf to get a photon arrival rate, and root-N shot noise on the count for a coherent beam, whose arrivals are Poissonian. The two accounts agree in the mean, and only in the mean.
- The Compton shift, derived
Conserving relativistic energy and both momentum components for a photon striking a free electron at rest, then eliminating the recoil angle to leave δ-λ = (h/mₑ c)(1 - cos θ), with h/mₑ c = 2.426 pm and no dependence on the incident wavelength. Because the shift is absolute rather than fractional it is invisible against 500 nm light and a three-percent effect against a 71 pm molybdenum K-α line, which is why the experiment needed X-rays.
- Recoil kinematics, the unshifted line, and scattering regimes
Recoil energy and angle from the same ledger, with the Compton edge Tₘₐₓ = 2E-squared over (mₑ c-squared + 2E) at θ = π - 477 keV for a 662 keV line. The unshifted line is what tightly bound electrons return, because the whole atom recoils and mₑ is replaced by M. Below hf much less than mₑ c-squared it fades into Thomson scattering; angular intensities need Klein-Nishina.
- The de Broglie hypothesis
Running λ = h/p backwards from photons to matter by demanding phase harmony, giving λ = h/√(2 mₑ e V) for an electron accelerated through potential V: 3.88 pm at 100 kV non-relativistically against a true 3.70 pm, an overstatement of about five percent. De Broglie's 2 π r = n λ does recover Bohr's quantisation, but it is a heuristic the three-dimensional theory discards - the hydrogen ground state has l = 0, no circulating standing wave and no orbit to wrap a wavelength around.
- Electron, neutron, and molecular diffraction
Davisson-Germer read as a surface grating, D sin φ = n λ giving 1.65 angstrom from the 54 eV peak at 50 degrees against h/p = 1.67 angstrom; the crystal-plane reading n λ = 2d sin θ needs an inner-potential refraction correction, because the electron speeds up on entering the metal. Thermal-neutron and large-molecule interferometry extend the same geometry, and h/p sets the resolution floor of an electron microscope. Fringes need lambda comparable to the spacing and a coherence length beyond the path difference.
- Wave packets, phase velocity, and group velocity
Superposing de Broglie plane waves into a complex Ψ(x, t) = ∫ φ(k) e-to-the-i(kx - ω t) dk, whose modulus squared is read as a probability density. The group velocity d-ω/dk = ℏ k/m returns the particle speed; the phase velocity ω/k does not and is not even convention-independent, giving v/2 for E = p-squared/2m but c-squared/v for E = γ m c-squared, because shifting the zero of energy only multiplies Ψ by a global phase. A free packet spreads, because ω = ℏ k-squared over 2m is dispersive.
- Fourier reciprocity and the uncertainty relation
Δ-x δ-k at least one half is a theorem about any square-integrable packet; times ℏ it becomes δ-x δ-p at least ℏ/2, saturated only by Gaussians. It bounds the spread of an ensemble of identically prepared systems - not the precision of an instrument, and not a disturbance the instrument inflicts. The energy-time form is a different statement, since t is a parameter and not an operator: δ-E times a lifetime of order ℏ reads as natural linewidth.
- Wave-particle duality and complementarity
Single-quantum interference accumulating count by count with one particle in the apparatus at a time, which-path marking in a Mach-Zehnder interferometer, and the duality bound distinguishability-squared plus visibility-squared at most one. The fringes go because the marker becomes entangled with the path, not because it kicks the particle - erase the marker and they come back in coincidence.
Learning outcomes
- Derive p = h/λ from the massless limit of the energy-momentum invariant and carry photon four-momentum through a Doppler shift and an emission recoil.
- Derive the Compton shift by conserving relativistic energy and both momentum components, then solve for the recoil electron's energy and angle and locate the Compton edge.
- Convert a measured intensity into a radiation pressure and a photon arrival rate, and say where the classical and photon accounts stop agreeing.
- Predict electron and neutron diffraction geometry from λ = h/p, applying the relativistic correction and the coherence condition.
- Build a wave packet by superposition, argue from the energy zero why only the group velocity is physical, and obtain δ-x δ-p at least ℏ/2 as a Fourier property of that packet rather than a disturbance inflicted by an instrument.
Prerequisite thread
- Relativistic energy and momentum, including the invariant and its m = 0 branch
- Photon energy hf and h extracted from a stopping-potential fit
- Interference, diffraction, the Bragg condition, and coherence length
- Fourier series and transforms, complex exponentials, and Gaussian integrals
Laboratory directions
hands-onCompton edge from a γ-ray spectrum Does the Compton edge in a measured scintillator spectrum sit at the predicted Tₘₐₓ = 2E-squared over (mₑ c-squared + 2E), and what value of h/(mₑ c) does the fit support?
Evidence: Energy-calibrated pulse-height spectrum, edge position with fitting uncertainty, the edge predicted from the two-body ledger, residuals, detector-resolution limit, and identified escape and backscatter peakscomputationalGroup velocity and packet spreading Does a numerically propagated Gaussian packet travel at d-ω/dk rather than at ω/k, and spread at the rate the free-particle dispersion relation predicts?
Evidence: Packet evolved under ω = ℏ k-squared over 2m, centroid tracked against both candidate velocities, width against the analytic spreading law, conservation of the norm and grid-convergence checks, and the uncertainty product against its ℏ/2 floorProblem practice
Naming the frame, the zero of energy, the conserved quantities, and the free-electron or ideal-packet assumption before reaching for a relation
Capstone: Take one beam - X-rays on graphite or electrons on a crystal - and predict its photon flux, scattering angles, and wavelength shift from conservation laws alone, then state which of those predictions fails first as the target electrons stop being free.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimPhotons and Matter Waves 02 · RepresentationMomentum-conservation vector triangles Relativistic collision bookkeeping
03 · TestPrediction before measurement Does the Compton edge in a measured scintillator spectrum sit at the predicted Tₘₐₓ = 2E-squared over (mₑ c-squared + 2E), and what value of h/(mₑ c) does the fit support?
04 · Evidence & boundaryDecide, then qualify Energy-calibrated pulse-height spectrum, edge position with fitting uncertainty, the edge predicted from the two-body ledger, residuals, detector-resolution limit, and identified escape and backscatter peaks
Interactive diagram for Photons and Matter Waves: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelRelativistic collision bookkeeping turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Photons and Matter Waves, explain how a physicist can: Derive p = h/λ from the massless limit of the energy-momentum invariant and carry photon four-momentum through a Doppler shift and an emission recoil.
- Photon energy, momentum, and the massless limit
Labs and problem solving
Keep the quantum results answerable to measurement.
The 30 laboratory directions and unit-specific problem plans use one repeatable loop from prediction to defensible evidence. Simulation or archival-data analysis supplements—not replaces—the hands-on work required by a learner's institution.Predict
State what the classical model says will happen, and what the experiment actually shows, before writing anything down.
Derive
Carry the mathematics out: separate the equation, apply boundary conditions, normalise, and keep every constant physical.
Measure
Design or interpret a spectral, counting, or interference measurement with calibration and uncertainty visible.
Test
Check the correspondence limit: the result must reduce to the classical one where the classical one is known to work.
Communicate
State the model, the regime it holds in, the evidence behind it, and what remains genuinely unsettled.
Course-level outcomes
What successful study should make possible.
These are learning capabilities, not promises of a grade, academic credit, transfer approval, professional status, or course completion.- Identify where classical models fail quantitatively, and state which experiment forces the replacement.
- Apply Lorentz transformations and relativistic energy-momentum, recovering Newtonian results at low speed.
- Interpret a wave function through the Born rule, and compute probabilities, normalisation, and expectation values.
- Solve the time-independent Schrödinger equation for wells, barriers, and the harmonic oscillator, and read quantisation from the boundary conditions.
- Use angular-momentum and spin quantum numbers, including the Pauli principle, to build atomic structure.
- Explain molecular bonding, band structure, and semiconductor behaviour from quantum states.
- Apply binding energy and decay laws to nuclear processes, including fission and fusion.
- Place particles and interactions within the Standard Model, and state plainly what it does not explain.
Choose the right starting point
Met the photoelectric effect before? Start at week 01 anyway.
Most students arrive having been told the photoelectric effect proves light is quantised. Week 01 asks the harder question this course is built on: what exactly does the classical prediction say, and by how much does the measurement miss it? Everything after depends on taking that failure seriously.