Physics &Maths Roadmap – The Knowledge
Summary of an Honours Physics Degree - The Tip of the Iceberg!
Classical Physics
Newtonian Physics Newton’s 3 Laws F = ma
Optics Lenses Concave & Convex Lens formula power of a lens in diopters is the reciprocal of the focal length Fermat’s principle of least time Fresnel lens Poisson’s spot Cornu spiral
 One-slit and two-slit interference Mirrors
 Refractive index Snell’s Law
Laser Physics - population inversion carbon dioxide and helium-neon lasers Lamb dip
Mechanics Statics stress tensor Civil Engineering
 Dynamics Mechanical Engineering
 Centre of mass; moments and products of inertia
Hooke’s Law for a spring
 Simple pendulum = simple harmonic motion
Rigid body dynamics Euler angles Euler’s formulae solve for a spinning top nutation (nodding) & sleeping top
Curvature, binormal and torsion – Serret-Frenet equations
Ancient Astronomy – 12 signs of the Zodiac – 5 planets (= wandering stars) vs fixed stars Copernicus; Tycho Brahe; Kepler; Galileo
Modern Astronomy - telescopes – refractors & reflectors (Newtonian, Cassegrain; Schmidt & Makutzov Cassegrain)
Celestial Mechanics Kepler’s 3 Laws orbits are conic sections reduced mass
 u=1/r substitution  circle, ellipse, parabola, hyperbola.
Astrophysics – Hertzsprung-Russell (HR) diagram – main sequence OBAFGKMRNS – star types Cosmology – universe open, closed, or flat; gravitational lens; red shift (tired light?); Hubble constant; cosmological constant (Einstein’s mistake)
Continuum Mechanics Aeronautical engineering Joukowski aerofoils – conformal mapping hydrostatics Archimedes principle  Fluid Dynamics - Bernoulli; hydrodynamics; Navier-Stokes equations; Reynold’s number; sources and sinks; equation of continuity
 Euler or Lagrange method Polar decomposition theorem
 Method of images; flow past a cylinder
 Lagrangian & Hamiltonian – p’s and q’s – (generalised momenta and generalised coordinates/displacements)
Calculus of variations shape of a heavy rope or chain = catenary
Electricity & Magnetism Electrostatics & magnetostatics
 Electrodynamics & magnetohydrodynamics (MHD)
 Ferromagnetism; diamagnetism; paramagnetism; B-H hysteresis curve
Electromagnetism scalar & vector potential Maxwell’s equations – Lorentz gauge
 E/M spectrum – radio waves – microwaves – IF – light (ROY-G-BIV) – UV – X-rays
 Lienard-Wiechert potentials
 Biot-Savart law Helmholtz coils F = q(E + v x B) = Lorentz force law
 Maximum power transfer theorem - impedance matching – Z of free space – 300 ohms - antennae
 Electrical Engineering star-del – 3-phase
 Waveguides & Transmission lines
Laplace Transforms – Circuit analysis – Kirchoff’s 2 Laws (conservation of charge and conservation of energy)
 Two methods of circuit analysis ? & current loop Norton’s and Thevenin’s theorems current/voltage sources Poles and zeros in the s-plane; active and passive filters; passive components – resistors (Z=R); capacitors (Z=-j/ωC); inductors (Z=jωL) – Ohm’s law – V=IZ power factor cosφ
 Dirac delta function – limiting case of a normal function
 Heaviside unit step function – integral of Dirac delta function
Fourier Transforms – Time domain & Frequency domain – sine; square; triangular; sawtooth waveforms
Gravity & Electromagnetism Kaluza-Klein Theory
Modern Physics
Electronics Electronic engineering transistor invented 1948 by doping silicon with phosphorous, antimony or arsenic - producing P-type (deficiency of electrons = holes) and N-type (excess electrons); PN junction (diode) – depletion layer; PNP and NPN Bipolar junction transistors (BJT) hybrid parameters – hfe = gain = beta; t-equivalent parameters; small signal analysis; AC equivalent circuit.
 Design an amplifier from first principles
 Field effect transistors (FET) (N-channel; P-channel)
Diode clipper & Diode clamp; Oscillators – Hartree & Colpitz – logical contradiction
Power supply – fullwave/halfwave rectification – ripple voltage exponential charge/discharge – time constant τ = RC transformers transducers
Superheterodyne receiver block diagram: RF amp; local oscillator; mixer; IF amp; detector; AF amp
 Mixer non-linear response; sum & difference frequencies
 Noise Shot noise Thermal noise Flecker noise (1/f)
 Q of a tuned circuit  = L/R frequency response and resonance ω^2LC = 1
 Decibels
 Operational Amplifiers (LM741) - inverting/non-inverting input - positive & negative feedback
 555 timer Analogue to digital and digital to analogue conversion; flash encoders; dual slope converter
Solid state physics – 14 types of crystal (Bravais) lattice – BCC; FCC; etc; phonons
Special Relativity – Michelson-Morley experiment Galilean transformation – Lorentz transformation Minkowski space diagrams aberration of starlight; decay of ?particles; Thomas precession
Probability Quantum Mechanics Bohr model of the atom
Wavefunctions ψ(r,t) Heisenberg’s Uncertainty Principle - HUP
Time Dependent Schrödinger Equation (TDSE) - Separation of variables technique leads to -
Time Independent Schrödinger Equation (TISE) - Solve TISE for the Hydrogen atom – orbitals
Normalisation rule – integrate probability over all of space = 1
Matrix Mechanics – Pauli spin matrices
<Bra|c|ket> notation Hilbert Space Electron in a square potential well – quantum tunnelling
Thermodynamics 3 Laws Ideal gas law pV = nRT = NkT k = Boltzmann’s constant
 Entropy Absolute zero – heat reservoir – quasistatically - Enthalpy Heat capacity at constant volume; pressure
 ½ kT per degree of freedom; for a solid = 3R – black body radiation E = σT^4; σ = Stefan-Boltzmann constant
 Ideal gas law; van der Waals gas law; extensive and intensive quantities; TdS equations.
Statistics Statistical Physics Maxwell-Boltzmann Statistics (ultraviolet catastrophe) Wien’s displacement law  Maxwell’s demon counting states – converting sums (discrete) to integrals (continuous)
 Fermi-Dirac Statistics
 Bose-Einstein Statistics
Tensors of zeroth order = scalars
Tensors of first order = vectors Vector Analysis Div/grad/curl Gauss & Stokes laws
Tensors of second order = matrices Linear Algebra Tensor Calculus Cartesian tensors (other types?) dyads
 (x,y,z) ahg x (all hairy gorillas; have big feet; good for climbing)
 hbf y determinants; linear dependence and independence; Kramer’s rule
 gfc z Eigenvalues; eigenvectors
 General Relativity Einstein summation convention; Latin indices 0123 – Greek indices 1234 epsilon-del equality
 Coordinate transformations Covariant, contravariant & mixed tensors – raising and lowering indices
Waves – group velocity & phase velocity – formulae – plane waves
Solar cells – photoelectric effect – E = hν where h = Planck’s constant; ħ = h/2π
Ionosphere – sporadic E; spread F - Appleton-Hartree equation;
Energy integral (formula)
Experimental Physics
Systematic error Data Analysis Accuracy & Precision and the difference between the two.
Sampling: Aliasing; Nyquist frequency Error analysis – Gaussian distribution
First year experiments Black box Gyroscope Speed of light (formula)
Second year experiments Squash ball Undergraduate relativity experiment (paper by Dr. Higbie) Solar cells Bainbridge e/m X-ray crystallography Young’s modulus for glass Planck’s constant - Cosmology 1&2
Third year experiments Magnetron Radioactivity Nuclear magnetic resonance Half-lives of silver Steam engine  Speed of sound (exam);
Honours year project Bribie Island research station Gravity waves in the ionosphere 3 frequency experiment O & X waves computer program in FORTRAN
Honours year subjects Microwave spectroscopy – ammonia molecule
Honours year seminar EPR paradox – Bell’s theorem vs Quantum mechanics - hidden variables
Honours year essay Lightning
Methods of Mathematical Physics
Elementary transcendental functions – sine; cosine; tangent; secant; cosecant; cotangent; hyperbolic functions; x = tan (θ/2) substitution “trick”
The complex plane
Particular differential equations; Ricarti equation; separable equations; exact equations; Legendre’s equation; linear, first order equations; Conduction of Heat in Solids - Carslaw & Jaeger – Bessel’s equation and Bessel functions – existence and uniqueness of solutions.
Different coordinate systems: One chooses a coordinate system to match the symmetry of the physics problem being analysed;  div, grad, curl in these systems Cartesian coordinates (x,y,z) Cylindrical polar coordinates (r,θ,z) Spherical polar coordinates (r,θ,φ) Generalised curvilinear coordinates (h1,h2,h3)
 Del operator (3D) and d’Alembertian operator (4D)
Gauss’ and Stokes’ theorems – generalised to n dimensions
 Complementary function + particular integral
Laplace’s equation
 Poisson’s equation
 Green’s function solution by convolution
Pure Maths, Applied Maths and miscellaneous topics
Kurzweil integration; Greek alphabet – used so often in physics and maths – learn by heart; Peano’s axioms for the natural numbers; Complex numbers as ordered pairs of real numbers
Linear programming – Simplex method; Population dynamics – Lotka-Volterra equations; Traffic flow; Quaternion electromagnetism – Mr. McGregor; phonetic alphabet

 
Learning Physics and Maths at the Professional Level
In primary and secondary school, students are spoon-fed by teachers – a lesson reinforced by set homework.  As long as the student always understands the lesson, and does their homework, success usually follows.  However, at the tertiary level, subjects are no longer taught in the way that they are at primary and secondary level.  Students have to be weaned from their reliance on teachers - Teachers are replaced by lecturers, lessons become lectures (theory) and tutorials (practice).  Students have to learn how to feed themselves with the knowledge necessary to become a successful physicist or mathematician, or indeed to become successful in any field of higher education.
It’s very easy to be wise after the event, but I can look back on my studies and see where I could have done things better.
At high school, the mathematics courses I studied were based largely on the “Radcliffe and Dan” series of texts.  There were eight books in the series, which fitted well into eight semesters of grade 11 and 12 advanced maths (called Maths I and II – now known as Maths B and C).  The eight semesters were:
1. Preparatory Mathematics
2. Algebra and Calculus I
3. Geometry and Calculus II
4. Geometry and Calculus III
5. Matrices and Vectors
6. Mechanics
7. Probability and Statistics
8. Complex Numbers
In fact I did not study the last of these at high school – it was replaced with a unit on computers (which were nowhere near as ubiquitous then as they are now) - I say to my shame that I started university mathematics with no knowledge of complex numbers – a serious deficiency.
To become great you have to study the greats – in physics, there is no better place to start than “The Feynman Lectures on Physics – the definitive and extended edition” which you can purchase from Amazon for US$122.85 (plus p & h) – money well spent in my opinion.  Academics will build up over time a professional library in their chosen field of study – books are the academics’ “tools of trade” – you can never have too many books – for many years I have scoured the second hand and opportunity shops for used textbooks.  If the book you are after is a classic in its field, it may have been reprinted as a “Dover” edition.  Dover editions are very reasonably priced, particularly if purchased from a discount store like Amazon.  Schaum’s Outline series are also a low priced source of knowledge in most fields of study.  Apart from the classics, the older a textbook is, the cheaper it becomes – an outdated edition of a set text is worth next to nothing – even if the subject itself has changed little – but this is why one can find genuine bargains at the trash and treasure outlets.
Lecturers will tell you that “no textbook is perfect” – but then no set of lecture notes is perfect either!  In fact a textbook usually begins its life as a set of lecture notes, which after many years of refinement by the lecturer, evolves into a set textbook for that particular course.
From my own experience I have found that lecturers will sometimes bias a subject towards their own particular field of study – they do this with a view to channelling undergraduate students towards an honours or post-graduate project or even a Ph.D.  This can be a disadvantage to an undergraduate student, who requires a broad-based and solid foundation in a subject rather than a personal hobby-horse or “pet doctrine”.  (Yes, even rational and objective physicists and mathematicians can be guilty of subscribing to unorthodox theories.)  This is why sticking to a widely-accepted and set textbook is the best way to teach the foundations of an undergraduate subject.
So what would I suggest?  To ensure that your foundation is solid before you start university, I recommend working your way through Schaum’s Outline of College Physics.  Then for a first year undergraduate physics course there are quite a few texts that come to mind – Tipler is an obvious one, Serway another, Cutnell & Johnson, Halliday & Resnick, Ohanian.  As above, you will pay top dollar for the latest edition, but next to nothing for an older edition.
What is also required in first year physics is the mathematical basis for the later years of physics.  This means books like “Mathematics of Physics and Modern Engineering” by Sokolnikoff and Redheffer; “Mathematics for Engineers and Scientists” by Alan Jeffrey; and “Advanced Engineering Mathematics” by Kreyszig
If I remember anything about my degree, it is the names of some of the recommended texts: Elementary Classical Physics (vols.  1 & 2) by Weidner and Sells; Modern Physics by Tipler; Optics by Jenkins and White; Optics by Hecht and Zajac; Quantum Mechanics by Davydov; QM by Merzbacher; QM by Schiff; Thermal Physics by Morse; Special Relativity by French; Classical Mechanics by Goldstein; Mechanics by Chester; Classical Electricity and Magnetism by Panofsky and Phillips; Electricity and Magnetism by Bleaney and Bleaney; Analysis by Binmore; Principles of Mechanics by Synge and Griffiths.
Once you get a foothold into the literature, the bibliographies contained therein will lead you to further study of suitable texts.
Learning maths and physics – solving problems and getting the right answer is positive reinforcement, therefore the student enjoys the subject and is good at it.  However not solving problems or getting the wrong answer is negative reinforcement, therefore the student hates the subject and fails.

Preparation for University Physics:  To ensure that your foundation is solid, you should work through Schaum’s College Physics and Schaum’s Mathematics for Physics Students before you start university.
Physics at a professional level.  When I learned physics, no physicist would be without a calculator.  Nowadays, no physicist should be without an algebraic manipulation program like Mathcad.  Proficiency – speed and accuracy at worked physics and maths problems is only achieved by practice.  Physics and maths examinations require speed and accuracy in problem solving.  Studying physics teaches one to think and to solve problems.  It is also a useful degree if you aspire to be an inventor.
Memories of University
One subject that stands out in my memory is Mr. McGregor’s MA375 (Applied Mathematics - Electromagnetic Theory - 1984).  Now Mr McGregor was, true to his name, a Scotsman, and somewhat of a maverick, as I was eventually to discover.  His approach to the subject of Electromagnetic Theory ruffled the feathers of some in the Physics Department, so much so that they wanted to make his subject incompatible with the corresponding Physics Department subject, PH337.
Now Mr. McGregor’s approach was unique in his use of "Quaternions", which were originally invented by the Irish mathematician, Hamilton.  These mysterious mathematical objects confounded me right from the start, but it is for this reason perhaps, that I remembered them.  I was eventually to learn that Quaternions, were in essence, merely a different notation to the "four-vectors" that mathematical physicists used.  A four-vector is a four-dimensional vector, three dimensions of space, and one dimension for time, while a Quaternion is also four dimensional, with a basis set (1,i,j,k).  Thus the correspondence is that (i,j,k) are for space, and the other dimension for time.
Mr. McGregor reformulated the laws of electromagnetic theory in this Quaternion notation.  Thoughts and phrases immediately spring to mind - "momergy", "the time on the SGIO clock", "boosts" and "turns", the distribution theory "hedgehog" (?!), and the mysterious "U4" thing ( = U2:U2) which involved such a horrendous calculation that he left it for us (the students) to do.
But this was not the only "bone of contention" that Mr. McGregor had with the establishment.  He proved, using "distribution theory", that if an electron is considered to be a point charge, then it does not radiate, in its orbit around the nucleus, which was the main stumbling block for the classical physicists’ description of the Bohr model of the atom.  (An orbiting electron is an accelerating charge, which according to classical theory, should radiate energy, and spiral into the nucleus.)  Mr. McGregor’s mathematics proved that an electron orbits the atom in a precessing elliptical orbit, and to repeat this proof was one of the examination questions.  This idea was dangerous because it struck at the heart of Quantum Mechanics, which stated that an electron could not be understood in the classical way of orbiting the nucleus like the planets orbit the sun.
Many years later, it dawned on me that an electron was, classically, not a point charge, but had a classical radius.  (see, for example, Classical Electricity and Magnetism by Panofsky and Phillips.)  I am unsure as to how this affected Mr. McGregor’s theory.  The problem with an electron being a point source is that its charge density (charge per unit volume) would be infinite!  What would be interesting to know, is how long it would take the electron to spiral into the nucleus, given the classical radius.
Another of Mr. McGregor’s insights was his rejection of the physicists’ notion of advanced and retarded potentials being possible solutions to the equations of electromagnetism - his idea was that advanced and retarded potentials were (I hope I get this right!) merely one solution with two different constants of integration - "End of Digression"
I bought a copy of Gravitation by Misner, Thorne and Wheeler - the most expensive and heavy book I had ever bought, and I brought it into our class.  Mr McGregor saw it and said, You won’t get much out of that!
I stumbled my way through the exam, and managed a "5", but to this day, more than thirty years later, the notes I have on the subject (which incidentally were originally recorded by Dr. Ross Paull) still, in the main, baffle me!
Back to Table of Contents

The Simple pendulum is not that simple!

Theoretical physics is essentially about formulating a problem using the known laws of physics, and then solving this problem using the physicist’s bag of “mathematical tricks”.

In high school physics you probably studied the simple pendulum, and derived the mathematical formula for the period of oscillation.  During the derivation of this formula, one makes the approximation sin θ = θ.  This is only true for small angles (small swings) when θ is measured in radians.  If this approximation is not made, there is no solution in terms of the so-called elementary transcendental functions (meaning sine (sin), cosine (cos), and tangent (tan)).  Many problems in physics are like this – no exact solution exists, and an approximate or numerical solution is required.

________________________________________
Maths and Physics Revision: I like maths and physics – even my first love academically speaking.  I took physics as far as an honours degree.  One goal is to revise my degree.  I believe the internet should make tertiary education cheaper, if not completely free.  I could teach you everything I can remember about my degree, but I can’t give you a certificate for it.   

Science
Conservation versus Dissipation: In physics, conservative forces are good forces; dissipative forces waste energy (e.g. friction)
When solving physics problems, one solves the general case first – using letters to represent numbers.  Then the last step is to plug the numbers into the equation and calculate the numerical answer.
Physics is the most exact of all the sciences – yet it is not as exact as some think!

How many of today’s electronic gadgets will work in eternity, where (I think) the speed of light is infinite?  They may work better, or not at all – I don’t know.
Time travel is impossible because time itself is imaginary – the past and the future only exist in our imagination – the only time that is real is the present.  However this will change in the age to come – the fourth (imaginary) dimension, time, is replaced by (real) eternity, which can be travelled through back and forth like space!  Mathematicians will be pleased with this!  Quaternions will accurately describe the four dimensions!

Is God a mathematician?  I believe God’s maths would put our maths to shame!

For as the heavens are higher than the earth,
So are My ways higher than your ways,
And My thoughts than your thoughts.  Isaiah 55:9

Our maths begins with Euclidean geometry, which we all learn at school, and contains within it the idea that the ancient Greeks believed, that the Earth is flat! (Euclidean geometry is the geometry of the infinite flat plane, and the word geometry means geo (the earth) + metry (to measure)!)  If instead of an infinite flat plane, we start with a sphere – with one pole the origin, and the other pole the point at infinity, would we end up with a new version of maths?  For every shape we can imagine, would we discover a different version of maths?

Some say the constants must be constant, or else the Global Positioning System (GPS) would go haywire!  However if the constants were constant, the GPS would not have to be continually re-synchronised with atomic clocks on earth?
The laws of electricity and magnetism only apply up to certain limits – an electric or magnetic field cannot be increased indefinitely – something has to break down.  Science knows that with high power lasers, one enters a field of study called non-linear optics – the response of the medium to the high power E/M field changes from linear to non-linear.  Similarly with the physics of flying saucers – a strong enough E/M field causes space-time itself to break down, possibly creating a black hole.
I want to teach you everything I can remember about my honours physics degree (for free!).  After all, it cost me nothing in terms of filthy lucre, but it did cost me 12 years of diligent school work, and the best part of 4 years at university.  In those days, university was free if your marks at high school were good enough. After all those hours sitting in lecture theatres absorbing knowledge, some of it is still there. I can cast my mind back to those days, listening to my lecturers, and some things do come back to me.  Physics is often seen as an intimidating subject – and it can be, particularly if you have no-one to explain it to you.  My results would have been better if my lecturers were more accessible and available to answer my questions.  Young undergraduates and postgraduates really need mentors who will take them under their wings and guide the progress of their careers.
STEM – Science, Technology, Engineering & Maths

All science is either physics or stamp collecting – Ernest Rutherford

It is good to see that the Government is starting to realise the importance of the STEM subjects.  Sadly for me it’s about 35 years too late!  However it’s not too late for those starting out in these fields, so if I can impart some of this knowledge to the next generation then hopefully some good will come of my experience.

My Degree:
Pure maths: MP171;MP172;MP173;MP174
Statistics: MS171
Applied Maths: MA170; MA172; MA173; MA271;MA272;MA273;MA275;MA375
Computer Science: CS100

Everything else was physics:
Astronomy: PH124;PH224;PH324
First Year: PH130;PH131;PH170;PH171
Second Year: PH231;PH232;PH236;PH240;PH253;PH270;PH271;PH272;PH273
Third Year: PH331; PH332; PH333; PH334; PH337; PH350; PH356; PH358; PH365; PH372; PH375
Honours Year: PH490: The whole year was classed as a single subject.

First Year University: (1982)
Semester One:
Computer Science:
CS100 INTRODUCTION TO PROGRAMMING (Dr. Lister)
Applied Mathematics:
MA170 INTRODUCTION TO APPLIED MATHS 1H (Prof. A. F. Pillow & Dr. Ken Smith)
Pure Mathematics:
MP171 ANALYSIS 1AH (Prof. Rudolf Vyborny)
MP173 FOUNDATIONS OF MATHS 1H (Prof. C.S. Davis)
Statistics:
MS171 PROBABILITY THEORY 1AH
Physics:
PH130 MECHANICAL, GENERAL & THERMAL PHYSICS 1H (Dr. Mainstone)
PH170 EXPERIMENTAL PHYSICS (Dr. G. Tuck)
Semester Two:
Applied Mathematics:
MA172 DYNAMICS 1H
MA173 APPLIED DIFFERENTIAL EQUATIONS 1H (Dr. Bracken)
Pure Mathematics:
MP172 ANALYSIS 1BH (Prof. Rudolf Vyborny)
MP174 ALGEBRA 1H (Dr. Ann Street)
Physics:
PH124 ASTRONOMY 1 (Dr. Ross)
PH131 ELECTRICITY, MAGNETISM & OPTICS 1H (Dr. Crouchley)
PH171 EXPERIMENTAL PHYSICS: ELECTRICAL (Dr. Tuck)
Second Year University: (1983)
Semester One:
Applied Mathematics:
MA271 DYNAMICS 2H (Dr. V.G. Hart)
MA273 VECTOR ANALYSIS & CONTINUUM MECHANICS 2H (Prof. A. F. Pillow)
Physics
PH231 THERMODYNAMICS (Dr. Crouchley)
PH236 RELATIVITY & CLASSICAL MECHANICS 2H (Dr. Frost & Dr. Dalton)
PH240 PHYSICAL OPTICS 2H (Prof. J. David Whitehead)
PH270 INTRODUCTORY ELECTRONICS (Dr. Peter Munro)
PH273 EXPERIMENTAL OPTICS (Dr. Jack Higbie)
Semester Two:
Applied Mathematics
MA272 WAVES, DIFFUSION & POTENTIAL THEORY 2H (Dr. Tony Watts)
MA275 APPLIED DIFFERENTIAL EQUATIONS 2H (Dr. Ken Smith)
Physics:
PH224 ASTRONOMY 2 (Dr. Ross)
PH232 QUANTUM PHYSICS 2H (Dr. Jack Higbie)
PH253 CIRCUITS & ELECTROMAGNETISM (Dr. Metchnik)
PH271 APPLIED ELECTRONICS (Dr. Munro)
PH272 EXPERIMENTAL MODERN PHYSICS (Dr. Higbie)
Third Year University: (1984)
Semester One:
Applied Mathematics:
MA375 ELECTROMAGNETIC THEORY 3H (Mr. MacGregor)
Physics:
PH324 ASTRONOMY 3-STELLAR EVOLUTION & COSMOLOGY (Dr. O'Mara)
PH331 STATISTICAL MECHANICS (Dr. Dearden)
PH332 QUANTUM PHYSICS (Dr. Brian Dalton)
PH333 CIRCUIT THEORY 3H (Dr. Hajkowicz)
PH365 EXPERIMENTAL DATA ANALYSIS (Dr. Keith Jones)
PH375 EXPERIMENTAL PHYSICS 3AC (Dr. Heckenberg & Dr. Lasich)
Semester Two:
Physics:
PH334 RELATIVITY 3H (Dr. Frost)
PH337 ELECTROMAGNETIC THEORY (Prof. Ralph Parsons)
PH350 LINEAR SYSTEMS (Prof. Whitehead?)
PH356 SOLID STATE PHYSICS (Dr. Lucas)
PH358 LASER PHYSICS (Dr. Norm Heckenberg)
PH372 ELECTRONICS IN PHYSICAL MEASUREMENT (Dr. Keith Jones)
Fourth Year University: (1985)
Physics:
PH490 Physics Honours
Data Analysis IVH (Dr. Hastie)
Ionospheric Physics IVH (Dr. Keith Jones)
Quantum Mechanics IVH (Dr. O'Mara)
Physical Oceanography IVH (Dr. Mainstone)
General Relativity IVH (Dr. O'Mara)
Solid State Physics IVH (Dr. Lucas)
Microwave Spectroscopy IVH (Prof. Ralph Parsons)
Electromagnetic Theory IVH (Prof. Ralph Parsons)

Essay: "An Introduction to Lightning and its Causes"
Seminar: "Bell's Theorem versus Quantum Mechanics" on Friday, September 13th, 1985.
Honours Project: "Movements of the Ionosphere at Different Heights"

Bibliography
Here is a list of some of the text books that I came across during my studies.  Most of these books are still available; some are classics in their field of study; some are available in new editions; others are out of print but should be available second hand.

Maths:
• Calculus by Spivak + Solutions Manual for same (MP171 & MP172)
• Analysis by Binmore (MP171 & MP172)
• Mathematical Models by Haberman (MA170)
• Mechanics by Chester (MA172)
• Differential Equations by Wylie (MA173)
• Principles of Mechanics by Synge & Griffith (MA271)
• Fluid Dynamics by Batchelor (MA273)

Physics:
• The Feynman Lectures on Physics vols 1,2 & 3
• Elementary Classical Physics vol. 1 & 2 by Weidner & Sells (PH130 & PH131)
• Physics by Halliday and Resnick (PH130 & PH131)
• Astronomy by Abell (PH124 & PH224)
• Thermal Physics by Morse (PH231)
• Thermal Physics by Kittel & Kromer (PH331)
• Optics by Jenkins & White (PH273)
• Optics by Hecht & Zajac (PH273)
• Elementary Modern Physics by Weidner & Sells
• Modern Physics by Tipler + Solutions Manual for same (PH232)
• Quantum Mechanics by Davydov (PH332)
• Quantum Mechanics by Schiff (PH332)
• Quantum Mechanics by Merzbacher (PH332)
• Special Relativity by French (PH236)
• Classical Mechanics by Goldstein (PH236)
• Tensor Calculus by Spain
• Gravitation by Misner, Thorn & Wheeler
• Electricity and Magnetism by R.C. Cross (PH131)
• Electromagnetism by Bleaney & Bleaney (PH337)
• Electromagnetic Theory by Panofsky and Phillips (PH337)
• The Art of Electronics by Horowitz and Hill + Laboratory Manual for same (PH270 & PH271)
• Basic Electronics by Brophy (PH270 & PH271)
• Electronics for Physical Scientists by Havill & Walton (PH270 & PH271)

Useful Reference Books:
• Mathematical Handbook by Schaum's Outline Series
• Mathematical Handbook by Korn & Korn
• Tables and Formulae by Abramowitz and Stegun
• Theoretical Physics by Joos

Some Comments:
My goal is to impart as much of this knowledge as I can in a logical, rational format, including some of the pearls of wisdom about maths and physics that I picked up along my journey.  Maths and physics is like another language, so if I teach you the basics of this language then you’ll be able to read the literature for yourself and understand its meaning.

Physicists use letters (both lower case and upper case) of our alphabet and even the Greek alphabet* to stand for quantities that can be measured.  One is free to choose any letter to stand for anything, however some letters are commonly used to mean certain things.  For example at school we learn that x often stands for an unknown quantity.  To locate a point in four dimensional space-time (4-D) the values of the three space dimensions and time could be written as a group of four (x, y, z, t) (a four-vector).  Angles are often given small Greek letters.  The three angles of a triangle could be given the Greek letters α (alpha), β (beta), and γ (gamma). The mass of the earth could be given a capital M and the mass of an orbiting satellite a lower case m.  The radius of the earth could be given a capital R and the radius of the satellite a lower case r.  To avoid confusion, if you are referring to electric field and energy, both of which are commonly denoted by a letter E (though energy is a scalar and electric field is a vector), then you might choose to use a script E for energy to distinguish it from the electric field E.

*It is therefore a useful exercise to learn the Greek alphabet off by heart.

Some Equations from High School

Newton’s law is F = ma where F is force, m is mass and a is acceleration.  (The bold type means that this is a vector equation – see below.)
Einstein’s formula is E = mc2 where E is energy, m is mass (as above) and c2 is the speed of light squared. E, m and c are all scalars.
Ohm’s law is V = I R where V is voltage, I is current and R is resistance.
In physics, when you multiply two quantities, you usually leave out the multiplication sign, and multiplication is assumed.
So V = I x R is written V = I R.

Coordinate systems are used to locate a point in space relative to some origin given the symbol O.
The most commonly used coordinate system is the Cartesian system (x, y, z) named after Rene Descartes who thought of it.  This means that x is the distance from the origin along the X axis, (length), y is the distance from the origin along the Y axis, (breadth), and z is the distance from the origin along the Z axis, (height). Other coordinate systems are chosen depending on the symmetry of the experiment under analysis.  If an experiment has cylindrical symmetry then cylindrical polar coordinates (r, θ, z) are used.  Spherical polar coordinates (r, θ, φ) are used for experiments with spherical symmetry.  Choosing the appropriate coordinate system for the experiment means that the maths will be simpler and easier to solve exactly (if it can be solved exactly).

Number Systems (some pure maths)
Natural numbers are (1, 2, 3, 4, ...) (… means to infinity, for which the symbol is ∞ )
Add zero and we get the whole numbers (0, 1, 2, 3, 4, …)
Add negative numbers and we get the integers (…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …)
Add all the numbers that can be expressed as fractions (vulgar fractions as we called them in primary school), and we get the rational numbers.  (Too many to list, but numbers like ⅓, ⅚, ⅞ etc.)
Add all the numbers that can’t be expressed as fractions, (irrational numbers), like √n when n is a natural number but not a perfect square (√2, √3, √5, etc), as well as transcendental numbers like e and π (which I admit I don’t understand very well!) and we get the real numbers.
(There is a saying amongst mathematicians that God invented the natural numbers, and everything else was invented by man.)

Scalars, Vectors, Matrices, and Tensors
A scalar has magnitude but no direction (like mass m).
A vector has magnitude and direction (like force F and acceleration a).  Vectors must be distinguished from scalars in some way.  With today’s computers, it is very easy to denote a vector by using bold type, as force F and acceleration a above.  The most commonly used way to denote a vector is with a tilde (~) below the letter. In N dimensions, a vector has N components.  (So in three dimensional space (3-D), vectors have three components, and in four dimensional space-time (4-D) vectors have four components (four-vectors).)
A matrix relates one vector to another.  Although some mathematicians don’t like this definition, simply put if you divide two vectors you get a matrix.  In N dimensions, a square matrix has N rows by N columns or N2 components. (So in three dimensional space (3-D), matrices have three rows and three columns or nine components, and in four dimensional space-time (4-D) matrices have four rows and four columns or sixteen components.)

Matrices are usually written with two subscripts, such as Aij (Latin subscripts) or Aμν (Greek subscripts).  The first subscript denotes the row and the second subscript denotes the column.

In N dimensions,
A scalar is a tensor of order 0, with N0 = 1 component.
A vector is a tensor of order 1, with N1 = N components.
A matrix is a tensor of order 2, with N2 components.
Tensors of higher order n have Nn components.
So the order of a tensor is the number of subscripts it has.

An example of a tensor of order 3 is the permutation tensor, given the symbol εijk which is defined as follows:

The Kronecker delta is the symbol δij which is defined as:

A very useful identity involving the permutation tensor is the ε-δ identity (called the epsilon-del identity),

εikmεpsm = δipδks – δisδkp 

To further explain this equation, we are using the Einstein summation convention.  This means that when a subscript is repeated, (in this case the subscript m) it means that we sum over all the values that m can have.
 
The epsilon-del identity is something that is tedious to prove (there are 64 possible combinations), but you only have to prove it once in your career!

Another useful theorem (does it have a name?) is that any arbitrary N2 matrix can be expressed as the sum of a symmetric matrix and an anti-symmetric matrix.  This is self-evident, because

Aij = ½(Aij + Aji) + ½(Aij – Aji)

The contents of the first bracket is symmetric (if you swap i and j the bracket is unchanged) and the contents of the second bracket is anti-symmetric (if you swap i and j you obtain the negative of the bracket).

 

 

Statistical Physics and Thermodynamics
Statistical physics uses the laws of large numbers (statistics) to mathematically model systems with large numbers of particles (such as gases). Sums can be turned into integrals.

There are three forms of statistics studied by physicists:
Classical particles obey Maxwell-Boltzmann statistics.
Bosons obey Bose-Einstein statistics.
Fermions obey Fermi-Dirac statistics.

The ideal gas law: PV=nRT=NkT (The first form is taught in high school where n is the number of moles and R is the ideal gas constant; in the second form N is the number of particles and k is Boltzmann’s constant.)  There are other gas laws, such as the van der Waals equation of state, which takes into account the van der Waals forces between atoms.
Thermodynamics is tricky because describing a gas involves three variable quantities: P (pressure), V (volume), and T (temperature).  So pressure can be considered a function of volume and temperature; or volume a function of pressure and temperature; or temperature a function of pressure and volume.
In thermodynamics there are intensive quantities and extensive quantities.  In the ideal gas, pressure and temperature are intensive quantities while volume is an extensive quantity.  This means that when you half an extensive quantity (volume), the intensive quantities (pressure and temperature) remain the same.

Circuit Analysis - Some basic electrical engineering
In Australia the electricity that powers our homes (the grid) is 240V at 50Hz. V stands for volts, Hz stands for hertz (once called cycles per second).  This is alternating current (AC) to distinguish it from direct current (DC). Most automobiles have a 12V DC electrical system.
Alternating current circuits (sinusoidal waveform) at a given frequency f where ω = 2πf were analysed using two methods: the voltage drop method and the current loop method.  At this frequency the three types of passive components (resistors R, inductors L & capacitors C) have complex impedances given by: R, jωL, and 1/jωC (= -j/ωC) where j=√-1 (physicists usually use i while engineers use j for √-1).  Complex numbers are nothing to be afraid of.  They are just a shorthand way of doing maths in two dimensions*.  In MP173 we were taught that complex numbers can be thought of as an ordered pair of real numbers.  Voltage (V) is measured along the X-axis and current (I) along the Y-axis, and then the complex impedance (Z) is a (two dimensional) vector, and Ohm’s law becomes:

V = I Z

For a resistor, Z = R
For an inductor, Z = jωL
& for a capacitor, Z = 1/jωC = -j/ωC  (the second equality is found by multiplying top and bottom by j and use j2 = -1)

*(In other words complex numbers are two dimensional.  There are also hypercomplex numbers or quaternions which are four dimensional, which I came across in MA375.  Even eight dimensional octonians have been invented by mathematicians, however as Captain Mannering said to Corporal Jones, I think you’re delving into the realms of fantasy there Jones!  My personal view is that I confine my thoughts to the four dimensions that we are all familiar with, namely three dimensions of space (length, width and height) and one dimension of time, although sometimes I do think about a fifth dimension.  I suppose this makes me more of an applied mathematician than a pure mathematician!)

The most important equation of complex numbers is Euler’s formula:
eiθ = cos θ + i sin θ
Or if you are an engineer,
ejθ = cos θ + j sin θ

We can multiply this equation by r, the distance from the origin O and obtain
r eiθ = r (cos θ + i sin θ) = r cos θ + i r sin θ
x = r cos θ is the real part along the X-axis and y = r sin θ is the imaginary part along the Y-axis.
So we can write a two dimensional (2-D) complex number as an ordered pair (x, y) = (r cos θ, r sin θ)

Complex numbers, like real numbers, can be added, subtracted, multiplied and divided.  The power of Euler’s formula is in the ease of multiplying two complex numbers:

Problems with Classical Physics
The Ultraviolet Catastrophe

Quantum Mechanics
TDSE – the Time Dependent Schrödinger Equation is solved for the hydrogen atom, and the solution turns out to be the orbitals that are the basis of the periodic table which is the foundation of chemistry.  It is quite remarkable that this equation can be solved, and that the solution is a well-known mathematical function.  From the solution comes four quantum numbers:

The principal quantum number n
The spin quantum number l
The magnetic quantum number m

By the process of separation of variables, the time dependency can be removed, leading to the TISE – the Time Independent Schrödinger Equation.

Measurement and Error Analysis is a very important area of (experimental) physics, for the reason that (to a physicist) a measurement without an estimation of error is meaningless.
If I were to measure the diagonal of this computer screen, I would say it is 60cm ± 3mm (this symbol ± means plus or minus).  This could also be expressed as a fraction, 3mm/600mm, 1 part in 200 = 0.005, or as a percentage 0.5%.

In an actual physics experiment, you would measure the same quantity many times (say 10 times).  You would then do a statistical analysis of these 10 measurements, and express the final result as the average (mean) plus or minus the standard deviation as the error in your result. 

Accuracy and Precision mean different things to a physicist.  The common way to explain this is with a dart board.  If you play darts like me, then the darts would be all over the place, and nowhere near the bullseye – neither precise nor accurate. If all your darts hit the dart board in a random fashion but in a loose cluster centred around the bullseye, then you would be accurate but not precise.  If you were aiming for the bullseye, but all your darts scored triple 20, then you would be precise but not accurate.  If all your darts hit the bullseye, then you would be both accurate and precise. 
How Well Do You Remember Your High School Physics?
People ask, "What use is physics?", without having an understanding of what physics is all about.  Engineering is applied physics.  A motor car, a microwave oven, this computer, and a television are but a few of the thousands of inventions that all work by known physical laws - the laws of physics.  Here are a few physics problems that I hope will bring some meaning and relevance to the physics you may (or may not) have learned in high school. 
Physics is needed to answer the following problems:
1. Calculate the average speed of a sprinter who can complete the 100m sprint in 10 seconds,
(a) in metres per second,
(b) in kilometres per hour.
2. The "four-minute mile" was one of the milestones (no pun intended) of athletics.  The old measure of the mile was 1760 yards.  A yard was 36 inches, and an inch was 2.54 cm.  Calculate the average speed of the first person to break the four-minute mile,
(a) in miles per hour,
(b) then in metres per second,
© and then in kilometres per hour.
3. A fast bowler’s deliveries have been clocked at 140km/h.  Assuming a cricket pitch of 20 metres in length, what time does this give the batsman to react to such a delivery? In cricketing terms, what is meant by "an extra metre of pace"? (It used to be an extra "yard" of pace.)
4. A dragster can complete the metric quarter mile (400metres) in 6 seconds from a stationary start.  Calculate the average acceleration, and the top speed, assuming the average acceleration over 400 metres.  Assume the dragster’s mass is 1000kg.  What is the average force required to accelerate the dragster over the 400 metres? At top speed, how much kinetic energy does the dragster possess? What is the net power developed in the engine (after losses), in kW, and then in brake horse power? (1 horsepower = 745.7 Watts) (Ans.  Just less than 2000 horsepower)
5. How long would it take a stone dropped from a 400 metre cliff to reach the bottom of the cliff? (Acceleration due to gravity = 9.8metres per second per second.) Compare this with the dragster of question 4.
6. The original definition of a "metre" was that it was one ten-millionth of the distance along the earth’s surface from the equator to the pole (north or south).
(a) From this definition, calculate the Earth’s radius. 
(b) From the answer to (a), calculate the Earth’s surface area. 
© From the answer to (b), and the assumption that the Earth’s surface is two-thirds water, calculate the surface area of the world’s oceans and seas. 
(d) A chunk of ice breaks off the Antarctic.  Its size is 200 km long by 50 km wide by 500 metres deep.  Calculate the volume of water it contains, assuming that ice reduces in volume by one-tenth when it melts into water. 
(e) Using answers (c) & (d) above, and the assumption that to begin with, the ice was completely above water, what would be the increase in the earth’s ocean level, if all the water contained in the iceberg melted into the world’s oceans?
Answers:
A: 13mm
B: 13cm
C: 13m
D: 13km
(f) Do I sell my waterfront property (If I owned one!)? More importantly, would I be able to find a buyer?
7. A trajectory problem: A hammer thrower wants (of course) to throw the hammer as far as possible.  Calculate the range of the hammer, as a function of the angle to the horizontal, that it is thrown (assuming no wind resistance).  In the range of angles from zero to ninety degrees, prove using differential calculus that at 45 degrees the range is a maximum, and calculate this range.
8. Why is an accident at 100 km/h so much more dangerous than one at 60 km/h? (Because kinetic energy increases as the square of the velocity, so a car at 100 km/h has 100x100/(60x60), or 2.77 times the energy of the car at 60km/h.  Even though the speed has only increased by 66.7%, the energy has increased by 177%, and it is the energy that must be dissipated in the collision at impact (by greater forces through greater distances, causing greater damage either in the vehicle or in the passengers).  This tells us why people rarely survive plane accidents, because the kinetic energy of  aircraft is so much greater.  In a school zone, if the speed is reduced by one third from 60km/h to 40km/h, the energy decreases by (36-16)/36 x 100%, or 56%, more than one-half.
9. The insolation from the sun is given as 1300 watts per square metre.  A solar panel measures 305mm x 305mm, is rated at 4 watts, and costs $87.95.  Calculate the efficiency of the solar panel, in percent.  If electricity costs 12.5 cents per kilowatt-hour, how many hours of continuous solar power is required before the panel pays for itself? Even at 8 hours of strong sunlight per day, how many days (or years) is this?
10. An electric jug is rated at 240Volts, 10 Amps.  How long will it take to boil a litre of water, from room temperature of 25 degrees Celsius? (Assuming boiling point is 100 degrees Celsius).  How many kilowatt-hours of electricity is this? What is the cost, assuming 12.5 cents per kilowatt-hour? (Specific heat capacity of water is 4.2 kilojoules per kilogram per degree) Ans.  About one cent.
11. Calculate from first principles the conversion factor between psi (Pounds per square inch), and kPa (kilopascals).  (1 kg = 2.2 pounds; 1 inch = 2.54 cms).  Ans: 1psi = 6.90 kPa.  (Use g = 9.81m/(s^2)) (N.B.  psi is actually pounds force per square inch).
12. If you are at the equator, and there is a full moon directly overhead, is the time closest to:
1. 6.00 am
2. 12 noon
3. 6.00 pm
4. 12 midnight
13. If you are at the equator, and there is a full moon on the eastern horizon, is the time closest to:
1. 6.00 am
2. 12 noon
3. 6.00 pm
4. 12 midnight
14. If you are at the equator, and there is a full moon on the western horizon, is the time closest to:
1. 6.00 am
2. 12 noon
3. 6.00 pm
4. 12 midnight
15. A satellite orbits the earth at a height of 200 km above the earth’s surface.  At what speed must it be travelling? If it hits a meteoroid the size of a grain of sand (1mm x 1mm x 1mm) at this relative velocity, what energy is dissipated in the collision? Compare this energy with that of an average rifle bullet.
16. A helicopter problem: (Does a helicopter work by pushing air down?) A helicopter has n rotor blades, measuring l long by w wide, inclined at an angle q to the horizontal.  If the helicopter’s mass is m, how fast must the blades rotate (in RPM) to effect lift-off? (This is much harder - more like a first year university problem!) One way to tackle this problem is to consider an element of one rotor blade length dl, moving at speed v = r x omega.  Imagine a packet of air hitting the rotor blade and bouncing off it at a certain angle, imparting a certain amount of momentum.  Then integrate over the length of the rotor blade, multiplying by the number of blades.
17. An Olympic size swimming pool slopes evenly from one metre deep at the shallow end to two metres deep at the deep end, and is fifty metres long by twenty metres wide.  Calculate the volume of water it contains.
A Couple of Maths Problems:
18. It is a well known property of numbers that they are divisible evenly by nine iff (iff means "if and only if") the sum of their digits is divisible by nine.  (eg.  63 is divisible by 9 because 6+3 = 9).  Can you think of an easy general proof of this property?
19. For clock watchers: At precisely what times do the minute and hour hands exactly co-incide? Is there any other time besides 12.00.00 that all three hands (hour, minute and second) exactly co-incide? Prove your result!
20. Thinking caps on for this one: We all know that 2 + 2 = 4 and 2 x 2 = 4, but when (in the decimal system) does 2 + 2 ≠ 2 x 2? (Hint: I’m more physicist than mathematician!)