Advanced Quantum Mechanics II PHYS 40202

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To be used in future

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A.2 Coordinate systems

We are normally only interested in orthogonal coordinate systems, where the surfaces of constant coordinate values (isosurfaces) intersect each other along

. There are a large number of these, linked to certain 2D mappings of the (complex) plane. We shall mainly restrict ourselves to three.

A.2.1 Cartesian coordinates↓↓

Cartesian coordinates are the normal rectangular coordinates

and

you are quite used to. In these coordinates we have

and

A.2.2 Spherical coordinates↓↓

Defined as

We have

. Use

We have

and

A.2.3 Cylindrical coordinates↓↓

Defined as

We have

. Use

We have

and

A.2.4 Guide to other choices

There are a substantial set of other orthogonal coordinates that have use in certain cases; most of these are the three dimensional analogue of 2D coordinates obtained by a conformal map, extended as either cylindrical or polar coordinates. The simplest of these are discussed in many books (parabolic, spheroidal, ellipsoidal, toroidal, bispherical, bipolar, conical, ....). In engineering this choice of coordinate system is sometimes referred to as “field theory”.

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