1. Lecture 18 3D Cartesian Systems
Only 4 lectures left + 1 revision session
Come to see me before the end of term
I’ve put more sample questions and answers in Phils Problems
Past exam papers
Complete solution from last lecture
Have a look at homework 2 (due in on 12/12/08)
Remember Phils Problems and your notes = everything

2. Schrödinger’s equation
Introduction to PDEs
In many physical situations we encounter quantities which depend on two or more variables, for example the displacement of a string varies with space and time: y(x, t). Handing such functions mathematically involves partial differentiation and partial differential equations (PDEs).
Wave equation
Elastic waves, sound waves, electromagnetic waves, etc.
Schrödinger’s equation
Quantum mechanics
Diffusion equation
Heat flow, chemical diffusion, etc.
Laplace’s equation
Electromagnetism, gravitation, hydrodynamics, heat flow.
Poisson’s equation
As (4) in regions containing mass, charge, sources of heat, etc.

3. 3D Coordinate Systems 1. PDEs in 3D Cartesian Coordinates
We can describe all space using coordinates (x, y, z), each one ranging from -∞ to +∞.
1. PDEs in 3D Cartesian Coordinates
Consider the wave equation. In one dimensional space we had
In 3D equation becomes
which may be written in shorthand as ,
Let us look for a solution of the form ,
i.e. we substitute and separate the variables, as done in 1D ….

4. 3D Coordinate Systems (*)
Substituting these into PDE then dividing both sides by
gives …
(*)
As before for 1D case we need solutions that can be zero more than once to fulfil boundary conditions, so we choose each term to equal a negative constant to ensure we get LHO style solutions.
Let
Comparing with (*) the defined constants, w, kx, ky, kz are related by
Each of the ODEs above has the normal harmonic solutions, which we can write in terms of sines and cosines below. .
X(x) ~
Y(y) ~
Z(z) ~
T(t) ~

5. 3D Coordinate Systems X(x) ~ Y(y) ~ Z(z) ~ T(t) ~
Giving special solutions of the form
Or sometimes it is more convenient to use complex exponentials,
Then depending on the boundary conditions we can get special solutions such as:
where
and
As we might have expected, these solutions are plane waves with wavevector k (which is also the direction of travel of the wave).

6. 3D Coordinate Systems
A general solution can then be written as a sum over all special solutions. By applying boundary conditions we can then determine which terms contribute and the allowed values of kx, ky, kz as before in 1D examples.
For example, suppose we have a box with dimensions L1, L2, L3 in the x, y, z directions respectively and know that Y must vanish at the walls and that it is zero at t = 0. Then the special solutions after these boundary conditions have been applied will be:
where
So each special solution, or ‘mode’ will be characterized by three integers, n1, n2, n3.
And this mode will have angular frequency
A common question is to deduce how many different modes (i.e. unique combinations of integers n1, n2, n3) exist in a given frequency range w to w + dw ? e.g. Planck’s Law for blackbody radiation.

7. Just integrating over x gives
3D Coordinate Systems
2. Integrals in 3D Cartesian Coordinates
We have dV = dx dy dz, and must perform a triple integral over x, y and z. Normally we will only work in Cartesians if the region over which we are to integrate is cuboid.
Example 1 : Find the 3D Fourier transform, if
and
and
The integral is just the product of three 1D integrals, and is thus easily evaluated:
Just integrating over x gives
Mistake in notes
This is therefore a product of three sinc functions, i.e.

8. Corrections to notes- sorry!!
Homework 2
Lecture 17