Next: Symmetries of the Helmholtz Up: The Helmholtz Equation Previous: Complete Set of Commuting   Contents   Index

## Translations and Rotations in the Euclidean Plane

Lecture 39

What is the significance of the operators

and what are they good for? The answer is that they express the translation invariance of the Euclidean plane and that they generate the rectilinear translations of the wave system governed by the Helmholtz equation

Let us see what this means and why this is so.

The Euclidean plane is characterized by various symmetry transformations which leave invariant the distance

as well as the Laplacian

 (51)

There are three obvious such transformations:
(i)
translation along the -axis by an amount :

(ii)
translation along the -axis by an amount :

(iii)
and also rotations around the origin by an angle :

These are point transformations. Even though a transformation takes each point of the Euclidean plane into another, the distance between a pair of points before the transformation is the same as the distance after this pair has been transformed to a new location. This is expressed by the equality

or, in brief,

i.e., the distance in the Euclidean plane is invariant under translations and rotations. It is also obvious that

One can also apply any one of the three symmetry transformation (i)-(iii) to a function, say , and obtain a new function. The rule is

See Figure 5.2. Thus, by exponentiating the operator in a way which is identical to exponentiating a matrix, one obtains a linear operator which expresses a translation along the -axis. This operator

is, therefore, called a translation operator. It translates a wave pattern, a solution to the Helmholtz equation from one location to another, i.e.,

This translation transformation is evidently generated by the translation generator

The effect of the translation operator is particularly simple when that operator is applied to an eigenvector'' of ,

In that case, one obtains a power series in the eigenvalue ,

Thus, except for the phase factor , the plane wave remaines unchanged. It is a translation eigenfunction. In other words, a plane wave is invariant (i.e. gets changed only by a constant phase factor) under translation along the -axis. This result is the physical significance of the mathematical fact that a plane wave solution is an eigenvector'' of . It expresses the physical fact that a plane wave is a translation invariant solution of the Helmholtz equation.

Analogous considerations lead to the definition of translations along the -axis and rotations around the origin. Thus, corresponding to the three point transformations (i), (ii), and (iii) earlier in this section, one has the three generators

1.        -translation generator''
2.        -translation generator''
3.        rotation generator''
which generate the finite transformations
1.              -translation by ''
2.              -translation by ''
3.              -rotation by ''
when they are applied to functions defined on the Euclidean plane. For example, the application of the rotation operator to yields

Next: Symmetries of the Helmholtz Up: The Helmholtz Equation Previous: Complete Set of Commuting   Contents   Index
Ulrich Gerlach 2010-12-09