Core Course: What You Should Know 4. Gravity. Circular orbits. Geostationary orbits. Escape velocity.

This set of notes gives an indication of the basic physics you are expected to know from 5. Simple harmonic motion. mẍ + kx = 0
the core course. It is not a comprehensive list of the syllabus. Hopefully, it gives you Recognise and solve the differential equation for SHM in different contexts. Natu-
some perspective of the core course as a whole. ral frequency ω. Derive general form for x(t) and ω 2 = k/m. You should be able to
These notes are supposed to help you with your revision for the Comprehensive recognise the equations of motion for SHO in other situations outside mechanics,
papers. (Some hints about exam technique are included.) However, it is important e.g. in circuits.
to remember that the Comprehensive questions are designed to test how you can apply 6. Damped harmonic oscillator. mẍ − γ ẋ + kx = 0
basic concepts to solve physics problems. They are not designed to test how much detail Underdamping, critical damping, and overdamping. Derive the behaviour of x(t)
you recall from individual courses. in the three regimes (oscillatory vs. exponential decay.) Q factor of a resonance.

7. Special Relativity. Lorentz transformation for 4-vectors: (x, t) and (p, E).
1 General Lorentz contraction and time dilation (e.g., muon decay lifetime). Transforma-
tion for velocity. Rest mass. Derivation of kinetic energy in the non-relativistic
1. Dimensional analysis. Check that your answers have the correct dimensions.
limit (v  c). Notion of simultaneity and causality in special relativity.
This will pick up most errors in algebra.

2. Algebra. To avoid unnecessary mistakes, keep it neat and do not miss out too
many steps. Try to reduce your formulae before evaluating them numerically.
3 Waves
3. Diagram. Draw a diagram to help visualise the problem. Define a suitable 1. Travelling waves and standing waves. sin(kx − ωt), cos(kx − ωt), ei(kx−ωt) .
coordinate system if needed – this helps you get the correct signs. Quantisation of k for standing waves.

4. Vectors. Don’t mix vectors with scalars. Otherwise, you will lose information 2. Principle of superposition. Beats. Relation to Fourier analysis.
about the direction of a vector quantity (e.g., momentum, electric field.) Make
3. Wave velocities. Phase velocity ω/k and group velocity dω/dk. Use latter for
sure you know the direction of a cross product.
flow of physical quantities, e.g. energy/information.
5. Approximations. Use Taylor expansions for small changes, e.g. sin x ' x, cos x '
4. Doppler effect. For waves in a medium. For electromagnetic waves (photons).
1 − x2 /2, (1 + x)α ' 1 + αx, ln(1 + x) ' x for x  1.
5. Diffraction: narrow slits. Double slits and many slits (i.e. diffraction grating).
6. Differential equations in physics. Use of small volume elements in deriving
You should be able to derive the condition for constructive interference: nλ =
a differential equation describing the physics of the problem. (e.g., in mechanics,
d sin θ, and to sketch the diffraction patterns for the two cases as a function of the
consider the forces applied on the element and the acceleration that arises from
angle of diffraction θ.
these applied forces.)
6. Diffraction: single slit/finite diffraction grating. Effect of finite slit width
for double slit. Effect of finite number of slits for diffraction grating. Implication
2 Mechanics for theoretical resolution of optical instruments.
1. Newton’s laws. Make sure you draw on a diagram all the physical forces on the 7. Refraction. Explain refraction as a consequence of change in velocity. Snell’s
body, and then relate them to the resulting acceleration. Don’t mix up F with law.
ma if you want to make sure the signs come out correctly.

2. Inertial frames. “Fictitious” inertial forces in non-inertial frames. 4 Quantum Mechanics

3. Rotation. Centrifugal force (or centripetal acceleration) as an inertial force.
1. DeBroglie relations. p = h̄k and E = h̄ω. What do we mean by this?
Angular momentum and angular velocity: L = Iω. Derivation of moments of
inertia, I, for rods, discs and spheres.

1 2
 2. Heisenberg uncertainty principle: momentum and position. ∆p ∆x ≥ Equipotentials and the method of images: charge near metallic sheet, metallic
h̄/2. Explain the meaning of ∆p and ∆x. Relationship to single-slit diffraction. sphere.
Implication for the kinetic energy of confined particles.
6. Force on charge. F = q(E+v×B), e.g., cyclotron motion in a uniform magnetic
3. Heisenberg uncertainty principle: energy and time. ∆E ∆t ≥ h̄/2. Impli- field.
cation for linewidths in atomic spectra.
7. Dielectric. Permittivity  = r 0 and electric polarizability.
4. Observables & operators. Observables are represented by Hermitian operators.
Each operator has a set of eigenfunctions and eigenvalues. When this observable 8. EM wave in vacuo. c2 = 1/µ0 0 . Polarisation: linear/circular. In free space,
is measured, the outcome has to be one of these eigenvalues. The wavefunction E ⊥ B and both are normal to direction of propagation.
for the system becomes the eigenfunction corresponding to this eigenvalue (“wave- 9. Poynting vector. S = E × B/µ0 gives energy density current (energy flowing
function collapse”). through unit area per unit time). Radiation pressure: p = 2S/c for a perfectly
5. Operators for momentum, kinetic energy and total energy. p̂ = −ih̄ ∂/∂x. reflecting mirror.
What are the eigenfunctions of the momentum operator? Hamiltonian operator : 10. Metals: Ohm’s Law. Conductivity σ and resistivity ρ. Relation between re-
Ĥ = p̂2 /2m + V (r). Eigenstates and energy levels. sistivity and resistance R. Ohm’s law : J = σE where J is the current density
6. Schrödinger equations. Time-dependent and time-independent forms. Derive (charge flowing through unit area per unit time).
the time-independent form from the time-dependent form. 11. EM wave in dielectric. Refraction: Snell’s law, total internal reflection. Brew-
7. 1D scattering. Transmission and reflection at a potential step or a potential ster angle.
barrier. Classically forbidden region. Tunnelling through a barrier: dependence 12. EM wave in conductors. Skin depth: absorption of EM wave in a conductor.
on barrier width.
13. EM wave in waveguides. TE and TM modes.
8. 1D potential wells. Bound states in a infinitely deep potential well and in a
finite potential well. Compare the eigenstates and energy levels in these two cases.
Bound states in a double well: tunnelling and energy level splitting. 6 Thermodynamics and Statistical Physics
9. Simple harmonic motion. Quantum oscillator in a parabolic well: En = 1. Basic concepts. Heat and reservoirs. Entropy. Reversibility. State variables.
h̄ω0 (n + 1/2). Sketch wavefunctions for the different energy levels.
2. Laws of thermodynamics. 1st law: energy conservation dU = dQ − dW .
(Define U , Q and W carefully to get signs right.) 2nd law: dS ≥ 0.
5 Electromagnetism
3. Fundamental equation. dU = T dS − pdV .
1. Maxwell’s equations. You are expected to remember these four equations.
4. Equations of State. Ideal gas: pV = N kT . Van der Waals gas.
2. Gauss’ law. Electric field due to static charges. Applications: point charge, line
charge, charged plane, charged sphere (inside and outside). 5. Simple processes. Work done, heat exchanged, entropy change during adiabatic,
isothermal, free expansions.
3. Ampère’s law. Magnetic field due to steady currents. Applications: field around
a current-carrying wire, inside a long solenoid. 6. Engines. Carnot cycle. Definition of efficiency η. Efficiency of Carnot engine.

4. Faraday’s law. Electric field around a loop due to time-varying magnetic flux 7. Free energies. Free energies as functions of state variables. Internal energy
through the loop. Application U (S, V ), Helmholtz F (T, V ) = U −T S, Gibbs G(T, p) = U −T S +pV . Expressions
for dF and dG. Maxwell relations.
5. Electrostatic potential. Deriving electric field from the potential and vice
versa: E = −∇V . Application: electostatic potential energy of a charged sphere. 8. Classical statistical physics. Statistical definition of entropy: microstates and
macrostates. Boltzmann distribution for classical particles.

3 4
 9. Quantum statistics. Identical particles: bosons and fermions. Pauli exclu- 7. Optical properties of semiconductors. p-n junction as a light-emitting diode
sion principle and the Fermi energy. Bose-Einstein and Fermi-Dirac distributions. or light detector. Relation between the band gap and the photon energy.
Density of states. Application: black-body radiation, heat capacity.

9 Particle and Nuclear Physics

7 Atomic Physics
1. Fundamental particles. Particles and antiparticles.
1. Spectral lines. Use of spectral lines to understand atomic energy levels. Broad-
ening of spectral lines: intrinsic, Doppler, collisions. 2. Nuclear decay. Alpha, beta, gamma decays.

2. Hydrogen atom. Principal quantum number n and angular momentum l. En- 3. Nuclear reactions. Fission and fusion: basic concepts of mass defect.
ergy levels: s, p, d, f ,. . .. 4. Nuclear stability. Semi-empirical mass formula: origin of the terms and their
3. Vector model. LS coupling: total angular momentum: J. influence of the stability of the nucleus. Nuclear shell model.

4. Selection rules. Concept.

10 Probability and Statistics
5. Perturbation theory. Application of first-order perturbation theory in atomic
physics. 1. Probability distributions. Discrete distributions: probabilities Pi with nor-
P
malisation
R i Pi = 1. Continuous distributions: probability densities p(x) with
6. Zeeman effect. p(x)dx = 1. The sum/integration is taken over all possible values of the random
variable x.
7. Molecules. Bonding. Rotational modes and infra-red spectroscopy.
2. Expectation values. For random variable x, the expectation value of any func-
8. Lasers. Spontaneous and stimulated emission. Population inversion. P R
tion f (x) is hf (x)i = i f (xi )Pi for discrete distributions, f (x)p(x)dx for con-
tinuous distributions.
8 Electrons in Solids/Solid State Physics 3. Mean. µ = hxi =
P
xi Pi for discrete distributions,
R
xp(x)dx for continuous
i

1. Free electron model. Counting states in k-space. Density of states g(E). Fermi distributions.
level EF & Fermi wavevector kF . Metals: dependence of physical properties on 4. Variance. var x = h(x − µ)2 i = hx2 i − hxi2 . Standard deviation: σ = (var x)1/2 .
g(EF ), e.g. electronic heat capacity.
5. Normal/Gaussian distribution. p(x) = (2πσ 2 )−1/2 exp[−(x−µ)2 /2σ 2 ], e.g. for
2. Electron bands in solids. Qualitative discussion of the formation of electrons the result of an experimental measurement of a continuous variable with expected
bands from localised orbitals. Particles and holes. Effective mass. value µ and error σ.
3. Bloch’s theorem. Electrons in a periodic potential. Reciprocal space: Brillouin 6. Binomial distribution. Probability of M success out of N trials with probability
zone. p for success in an individual trial. Mean and variance.
4. Insulators, semiconductors and metals. Explanation in terms of the Fermi 7. Poisson distribution. P (n) = e−µ µn /n! for a rare event to occur n times if it
level relative to the position of electron bands. Importance of band gaps. occurs on average µ times. Variance = mean. e.g. number of nuclear decays in a
5. Semiconductors. Intrinic carrier density by thermal properties. Extrinsic carri- given interval of time among a large collection of radioactive nuclei.
ers: n-type and p-type doping. 8. Independent events/measurements. The mean/variance of the sum is the
P
6. p-n junction. Physics at the interface: shifted electron bands, depletion layer. sum of the means/variances. So, suppose X = N j=1 xj is the sum of N outcomes

Application as diode: current-voltage characteristic, forward and reverse bias, for the same random variable x, then hXi = N hxi and var X = N var x.
breakdown.

5 6
 P
9. Central Limit Theorem. For large N , the sum X = N j=1 xj has a Normal
distribution with mean N µ and variance N σ 2 if x has mean µ and variance σ 2 ,
irrespective of probability distribution for x.

10. Estimating the mean from data. For N independent measurements of x, best
P
unbiased estimate of µ is x̄ = N −1 N
j=1 xj . Fractional error ∝ 1/N
1/2
.

11. Estimating the variance from data. Unbiased estimate of variance is [N/(N −
P
1)][N −1 N 2 2
j=1 xj − x̄ ].