Cambridge IB Chemistry course with some IB Physics blended in.
Ionic ModelIn the lattice, ions experience the force from neighbours, next-neighbours, next-next-neighbours, …, forming a conditionally convergent series.
The sum is (or rather from experimental data is deduced to be) Madelung const...
This is a only a summary of knowledge.
Fourier transformThere are many conventions as in where to put the normalisation 2\pi factor.
Forward Fourier transform (Fourier analysis)
\tilde{f}(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,dx
Inverse Fourier transform (Fourier synthesis)
f(x)=\frac{1}...
FunctionalsJust like functions take variables as arguments, functionals take functions as arguments.
In the following sections, the functional of the form
G[y]=\int_\alpha^\beta f(y,y' ; x) \, dxis considered.
Functional derivativesDefinition:
\begin{align}
\delta G &= G[y+\delta y]-G[y]\\
&=\...
Key concepts
Linear operator
Homogeneous:
equation: no forcing, or, a multiple of a solution y, \alpha y is also a solution.
problem: in addition, \alpha y also satisfies B.C.
Boundary conditions
Dirichlet: the function value on the boundary is defined
Neumann: the normal derivative value on t...
MatricesVector spaceDefinition
A set of elements on which vector addition and scalar multiplication are defined
Closed under these operations
Includes identity \mathbf{0} for vector addition
Note:
2D spaces \mathbb{R}^2 \neq \mathbb{C}, since \mathbb{C} has a single point of \infty.
SpanThe s...
Contour IntegrationIdea: A powerful method of doing integrals by using the properties of complex functions.
Analytic functions
To be differentiable at a point, a function must satisfy Cauchy-Riemann.
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \,\frac{\partial v}{\partial x} ...
Overall pictureScale
Type
Energy scale / \pu{cm-1}
EM wave
Electronic
10000
UV/VIS, X-ray
Vibration
1000
IR
Rotation
1
Microwave
Requirements
Rotation (Microwave) Spectra
Need permanent dipole
x, y or z transforms as totally symmetric representation
Infrared
Need change in di...
Irrelevant content.
Notes for Cambridge IB Introduction to QM course. Password access required.
