Translations and Rotations in the Euclidean Plane

Lecture 39

What is the significance of the operators

$$P_x = \frac{1}{i}~\frac{\partial}{\partial x}~~\qquad~~P_y = \frac{1}{i} ~\frac{\partial}{\partial y}\,,$$

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

$$(\nabla^2 +k^2)\psi = 0\,.$$

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

$$ds^2\equiv dx^2+dy^2=dr^2+r^2d\theta^2$$

as well as the Laplacian

$$\nabla^2\equiv\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\... ...ac{\partial}{\partial r} + \frac{1}{r^2}~\frac{\partial^2}{\partial\theta^2}\,.$$ (51)

There are three obvious such transformations:

(i)

translation along the $x$ -axis by an amount $a$ :

$$X_a\colon x\to x'=x+a$$

(ii)

translation along the $y$ -axis by an amount $b$ :

$$Y_b\colon y\to y'=y+b$$

(iii)

and also rotations around the origin by an angle $\gamma$ :

$$R_\gamma\colon\theta\to\theta '=\theta +\gamma\,.$$

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

$$dx^{\prime 2}+dy^{\prime 2} = dx^2+dy^2$$

$$dr^{\prime 2}+r^2d\theta^{\prime 2} = dr^2+r^2d\theta^2$$

or, in brief,

$$ds^{\prime 2} = ds^2$$

i.e., the distance $ds^2$ in the Euclidean plane is invariant under translations and rotations. It is also obvious that

$$\nabla^{'2}=\nabla^2 ~~.$$

Figure 5.2: The point transformation $x\to x+a$ induces a transformation which acts on functions according to the rule: $\psi (x,y)\to \psi (x-a,y)$ . Because of the minus sign, the transformed function is called the ``pull-back'' of $\psi (x,y)$ .

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

$$\psi (x,y)\to\psi (x- a,y) = \sum^\infty_0 \frac{(-a)^n}{n!}~ \frac{\partial^n\psi (x,y)}{\partial x^n}$$

$$\equiv e^{-a\frac{\partial}{\partial x}}\psi (x,y)$$

$$= e^{-iaP_x}\psi (x,y)\,.$$

See Figure 5.2. Thus, by exponentiating the operator $P_x=\frac{1}{i}~\frac{\partial} {\partial x}$ in a way which is identical to exponentiating a matrix, one obtains a linear operator which expresses a translation along the $x$ -axis. This operator

$$e^{-iaP_x} = 1-iaP_x+\frac{(-ia)^2}{2!} P^2_x +\cdots \equiv X_{a\ast}$$

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

$$X_{a\ast}\psi (x,y) = \psi (x- a,y)\,.$$

This translation transformation is evidently generated by the translation generator

$$P_x = \frac{1}{i}~\frac{\partial}{\partial x}\,.$$

The effect of the translation operator $X_{a\ast}$ is particularly simple when that operator is applied to an ``eigenvector'' of $P_x$ ,

$$P_x e^{i(k_xx+k_yy)}= k_x e^{i(k_xx +k_yy)}\,.$$

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

$$X_{a\ast}e^{i(k_xx+k_yy)} = e^{-iaP_x} e^{i(k_xx+k_yy)}$$

$$= \left[ 1-iak_x+\frac{(-ia)^2}{2!}k^2_x+\cdots\right] e^{i(k_xx+k_yy)}$$

$$= e^{-ik_xa} e^{i(k_xx+k_yy)}\,.$$

Thus, except for the phase factor $e^{-ik_xa}$ , 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 $x$ -axis. This result is the physical significance of the mathematical fact that a plane wave solution is an ``eigenvector'' of $P_x=\frac{1}{i}~\frac{\partial} {\partial x}$ . 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 $y$ -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. $P_x=\frac{1}{i}~\frac{\partial} {\partial x}$       ``$x$ -translation generator''
2. $P_y=\frac{1}{i}~\frac{\partial}{\partial y}$       ``$y$ -translation generator''
3. $L_\theta =\frac{1}{i}~\frac{\partial}{\partial\theta}$        ``rotation generator''

which generate the finite transformations

1. $X_{a\ast} = e^{-iaP_x}$             ``$x$ -translation by $a$ ''
2. $Y_{b\ast} = e^{-ibP_y}$             ``$y$ -translation by $b$ ''
3. $R_{\gamma\ast} = e^{-i\gamma L_\theta}$             ``$\theta$ -rotation by $\gamma$ ''

when they are applied to functions defined on the Euclidean plane. For example, the application of the rotation operator $R_{\gamma\ast}$ to $\psi (r,\theta )$ yields

$$R_{\gamma\ast}\psi (r,\theta ) = \psi (r,\theta -\gamma )\,.$$