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MATHEMATICAL METHODS IN PHYSICS & ASTRONOMY
2003–2004
Colin Wilkin
(cw@hep.ucl.ac.uk)
The frst year 1B21 course and the second year 2B21 course should together
provide all the necessary mathematical techniques For the understanding oF the com
pulsory (core) courses in the whole Physics and Astronomy and related degree pro
grammes. The two second year courses which rely heavily on the mathematical
Formalism developed in 2B21 are Quantum Mechanics and Electromagnetic theory.
Since Electromagnetism is in the second semester, the course will start with the
things needed For Quantum Mechanics,
viz
matrices and eigenvalues, partial di±er
ential equations, and the solutions oF the Legendre di±erential equation. By imposing
boundary conditions on this, we will derive quantisation oF angular momentum about
the same time as you will see it in the Quantum Physics course. The properties oF
the Legendre polynomials and spherical harmonics derived here should help you un
derstand some oF the Quantum phenomena. Though you will have met the gradient
di±erential operator in the 1B21 course, Electromagnetism requires that you be able
to apply also the divergence and curl operators, as well as use the divergence and
Stokes’ theorem which are introduced at the end oF the course. The ²ourier analysis
is also oF use in second year Electromagnetism, but it will required in many other
courses such as modern optics or signal processing.
Even iF it is not true, the assumption has to be made that you have understood
and can reproduce most oF the frst year 1B21 material, including the Notes on
Integration. You will have to remember the contents oF the ²ormulae Sheet because
it will not be issued in second year examinations. It can thereFore do no harm at all
iF you spent some time going through last year’s notes on the subject.
There are about 34 lectures in total in the 2B21 course. The methodology is
rather similar to 1B21 but, because there are no associated problem solving classes,
there are about 9 discussion classes where I will go over examples. ²ollowing sug
gestions From students in past years, most oF these examples will be chosen by me so
that they ft well with the course and the problem sheets. That seemed to work quite
well in the past, but it is not intended to stop students raising their own questions.
The discussion classes will normally be in the second slot on Thursday aFternoons,
except For the frst week when I shall lecture For two hours instead — it is hard to
discuss problems beFore getting well into the course.
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 Spring '09
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