19.1 The quantum free particle and representations of E(2)

19.1 The quantum free particle and representations of E(2)

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · Lie algebra representation · unitary representation · irreducible representation · Lie bracket

We’ll begin for simplicity with the case of two spatial dimensions. Recall from chapter 18 that the Euclidean group $E(d)$ is a subgroup of the Jacobi group $G^J(d)=H_{2d+1}⋊Sp(2d,\mathbf{R})$ . For the case $d=2$ , the translations ${\mathbf{R}}^2$ are a subgroup of the Heisenberg group $H_5$ (translations in $q_1,q_2)$ and the rotations are a subgroup $SO(2)\subset Sp(4,\mathbf{R})$ (simultaneous rotations of $q_1,q_2$ and $p_1,p_2)$ . The Lie algebra of $E(2)$ is a sub-Lie algebra of the Lie algebra ${\mathfrak{g}}^J(2)$ of polynomials in $q_1,q_2,p_1,p_2$ of degree at most two.

More specifically, a basis for the Lie algebra of $E(2)$ is given by the functions

$$l=q_1{p}_2-q_2{p}_1,p_1,p_2$$

on the $d=2$ phase space $M={\mathbf{R}}^4$ , where $l$ is a basis for the Lie algebra $\mathfrak{so}(2)$ of rotations, $p_1,p_2$ a basis for the Lie algebra ${\mathbf{R}}^2$ of translations. The non-zero Lie bracket relations are given by the Poisson brackets

$$\{l,p_1\}=p_2,\{l,p_2\}=-p_1$$

which are the infinitesimal version of the rotation action of $SO(2)$ on ${\mathbf{R}}^2$ . There is an isomorphism of this Lie algebra with a matrix Lie algebra of 3 by 3 matrices given by

$$l\leftrightarrow \left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),p_1\leftrightarrow \left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),p_2\leftrightarrow \left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right)$$

Since we have realized the Lie algebra of $E(2)$ as a sub-Lie algebra of the Jacobi Lie algebra ${\mathfrak{g}}^J(2)$ , quantization via the Schr¨odinger representation ${\Gamma }_S^{\prime }$ provides a unitary Lie algebra representation on the state space $\mathcal{H}$ of functions of the position variables $q_1,q_2$ . This will be given by the operators

$${\Gamma }_S^{\prime }(p_1)=-i{P}_1=-\frac{\partial }{\partial q_1},{\Gamma }_S^{\prime }(p_2)=-i{P}_2=-\frac{\partial }{\partial q_2}\text{\hspace{1em}(19.1)}$$

and

$${\Gamma }_S^{\prime }(l)=-iL=-i(Q_1{P}_2-Q_2{P}_1)=-\left(q_1\frac{\partial }{\partial q_2}-q_2\frac{\partial }{\partial q_1}\right)\text{\hspace{1em}(19.2)}$$

The Hamiltonian operator for the free particle is

$$H=\frac{1}{2m}(P_1^2+P_2^2)=-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}\right)$$

and solutions to the Schr¨odinger equation can be found by solving the eigenvalue equation

$$H\psi (q_1,q_2)=-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}\right)\psi (q_1,q_2)=E\psi (q_1,q_2)$$

The operators $L,P_1,P_2$ commute with $H$ and so provide a representation of the Lie algebra of $E(2)$ on the space of wavefunctions of energy $E.$

This construction of irreducible representations of $E(2)$ is similar in spirit to the construction of irreducible representations of $SO(3)$ in section $8.4$ . There the Casimir operator $L^2$ commuted with the $SO(3)$ action, and gave a diferential operator on functions on the sphere whose eigenfunctions were spaces of dimension $2l+1$ with eigenvalue $l(l+1)$ , for l non-negative and integral. For $E(2)$ the quadratic function $p_1^2+p_2^2$ Poisson commutes with $l,p_1,p_2$ . After quantization,

$$|\mathbf{P}{|}^2=P_1^2+P_2^2$$

is a second-order diferential operator which commutes with $L,P_1,P_2$ This operator has infinite dimensional eigenspaces that each carry an irreducible representation of $E(2)$ ). They are characterized by a non-negative eigenvalue that has physical interpretation as 2mE where $m,E$ are the mass and energy of a free quantum particle moving in two spatial dimensions.

From our discussion of the free particle in chapter 11 we see that, in momentum space, solutions of the Schr¨odinger equation are given by

$$\tilde{\psi }(\mathbf{p},t)=e^{-i\frac{1}{2m}|\mathbf{p}{|}^{2}t}\tilde{\psi }(\mathbf{p},0)$$

and are parametrized by distributions

$$\tilde{\psi }(\mathbf{p},0)\equiv \tilde{\psi }(\mathbf{p})$$

on ${\mathbf{R}}^2$ . These will have well-defined momentum ${\mathbf{p}}_0$ when

$$\tilde{\psi }(\mathbf{p})=\delta (\mathbf{p}-{\mathbf{p}}_0)$$

The position space wavefunctions can be recovered from the Fourier inversion formula

$$\psi (\mathbf{q},t)=\frac{1}{2\pi }{\int }_{{\mathbf{R}}^2}{e}^{i\mathbf{p}\cdot \mathbf{q}}\tilde{\psi }(\mathbf{p},t)d^2\mathbf{p}$$

Since, in the momentum space representation, the momentum operator is the multiplication operator

$$\mathbf{P}\tilde{\psi }(\mathbf{p})=\mathbf{p}\tilde{\psi }(\mathbf{p})$$

an eigenfunction for the Hamiltonian with eigenvalue E will satisfy

$$\left(\frac{|\mathbf{p}{|}^2}{2m}-E\right)\tilde{\psi }(\mathbf{p})=0$$

$\tilde{\psi }(\mathbf{p})$ can only be non-zero if $\begin{array}{r}E=\frac{|\mathbf{p}{|}^2}{2m}\end{array}$ , so free particle solutions of energy E will thus be parametrized by distributions that are supported on the circle

$$|\mathbf{p}{|}^2=2mE$$

Figure 19.1: Parametrizing free particle solutions of Schr¨odinger’s equation via distributions supported on a circle in momentum space.

Going to polar coordinates $\mathbf{p}=(p\cos \theta ,p\sin \theta )$ , such solutions are given by distributions $\dot{\psi }(\mathbf{p})$ of the form

$$\tilde{\psi }(\mathbf{p})={\tilde{\psi }}_E(\theta )\delta (p^2-2mE)$$

depending on two variables $\theta ,p.$ . To put this delta-function in a more useful form, recall the discussion leading to equation 11.9 and note that for $p\approx \sqrt{2mE}$ one has the linear approximation

$$p^2-2mE\approx 2\sqrt{2mE}(p-\sqrt{2mE})$$

so one has the equality of distributions

$$\delta (p^2-2mE)=\frac{1}{2\sqrt{2mE}}\delta (p-\sqrt{2mE})$$

In the one dimensional case (see equation 11.10) we found that the space of solutions of energy E was parametrized by two complex numbers, corresponding to the two possible momenta $\pm \sqrt{2mE}$ . In this two dimensional case, the space of such solutions will be infinite dimensional, parametrized by distributions ${\tilde{\psi }}_E(\mathbf{\theta })$ on the circle.

It is this space of distributions ${\tilde{\psi }}_E(\mathbf{\theta })$ on the circle of radius $\sqrt{2mE}$ that will provide an infinite dimensional representation of the group $E(2)$ , one that turns out to be irreducible, although we will not show that here. The position space wavefunction corresponding to ${\tilde{\psi }}_E(\mathbf{\theta })$ will be

$$\begin{array}{rl}\psi (\mathbf{q})& =\frac{1}{2\pi }\iint e^{i\mathbf{p}\cdot \mathbf{q}}{\tilde{\psi }}_E(\theta )\delta (p^2-2mE)pdpd\theta \\ & =\frac{1}{2\pi }\iint e^{i\mathbf{p}\cdot \mathbf{q}}{\tilde{\psi }}_E(\theta )\frac{1}{2\sqrt{2mE}}\delta (p-\sqrt{2mE})pdpd\theta \\ & =\frac{1}{4\pi }{\int }_0^{2\pi }{e}^{i\sqrt{2mE}(q_1\cos \theta +q_2\sin \theta )}{\tilde{\psi }}_E(\theta )d\theta \end{array}$$

Functions ${\tilde{\psi }}_E(\mathbf{\theta })$ with simple behavior in $\theta$ will correspond to wavefunctions with more complicated behavior in position space. For instance, taking $\begin{array}{r}{\tilde{\psi }}_E(\theta )=\end{array}$ $e^{-in\theta }$ one finds that the wavefunction along the $q_2$ direction is given by

$$\begin{array}{rl}\psi (0,q)& =\frac{1}{4\pi }{\int }_0^{2\pi }{e}^{i\sqrt{2mE}(q\sin \theta )}{e}^{-in\theta }d\theta \\ & =\frac{1}{2}{J}_n(\sqrt{2mE}q)\end{array}$$

where $J_n$ is the n’th Bessel function.

Equations 19.1 and 19.2 give the representation of the Lie algebra of $E(2)$ on wavefunctions $\psi (\mathbf{q})$ . The representation of this Lie algebra on the ${\tilde{\psi }}_E(\mathbf{\theta })$ is given by the Fourier transform, and we’ll denote this by ${\tilde{\Gamma }}_S^{\prime }$ . Using the formula for the Fourier transform we find that

$${\tilde{\Gamma }}_S^{\prime }(p_1)=-\frac{\tilde{\partial }}{\partial q_1}=-i{p}_1=-i\sqrt{2mE}\cos \theta$$ $${\tilde{\Gamma }}_S^{\prime }(p_2)=-\frac{\tilde{\partial }}{\partial q_2}=-i{p}_2=-i\sqrt{2mE}\sin \theta$$

are multiplication operators and, taking the Fourier transform of 19.2 gives the diferentiation operator

$$\begin{array}{rl}{\tilde{\Gamma }}_S^{\prime }(l)& =-\left(p_1\frac{\partial }{\partial p_2}-p_2\frac{\partial }{\partial p_1}\right)\\ & =-\frac{\partial }{\partial \theta }\end{array}$$

(use integration by parts to show $\begin{array}{r}{q}_j=i\frac{\partial }{\partial p_j}\end{array}$ and thus the first equality, then the chain rule for functions $f(p_1(\theta ),p_2(\theta ))$ for the second).

This construction of a representation of $E(2)$ starting with the Schr¨odinger representation gives the same result as starting with the action of $E(2)$ on configuration space, and taking the induced action on functions on ${\mathbf{R}}^2$ (the wavefunctions). To see this, note that $E(2)$ has elements $(\mathbf{a},R(\varphi ))$ which can be written as a product $(\mathbf{a},R(\varphi ))=(\mathbf{a},1)(0,R(\varphi ))$ or, in terms of matrices

$$\left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & a_1\\ \sin \varphi & \cos \varphi & a_2\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{ccc}1& 0& a_1\\ 0& 1& a_2\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right)$$

The group has a unitary representation

$$(\mathbf{a},R(\varphi ))\to u(\mathbf{a},R(\varphi ))$$

on the position space wavefunctions $\psi (\mathbf{q})$ , given by the induced action on functions from the action of $E(2)$ on position space $\overline{{\mathbf{R}}^2}$

$$\begin{array}{rl}u(\mathbf{a},R(\varphi ))\psi (\mathbf{q})& =\psi ((\mathbf{a},R(\varphi ){)}^{-1}\cdot \mathbf{q})\\ & ==\psi ((-R(-\varphi )\mathbf{a},R(-\varphi ))\cdot \mathbf{q})\\ & =\psi (R(-\varphi )(\mathbf{q}-\mathbf{a}))\end{array}$$

This representation of $E(2)$ is the same as the exponentiated version of the Schr¨odinger representation ${\Gamma }_S^{\prime }$ of the Jacobi Lie algebra ${\mathfrak{g}}^J(2)$ , restricted to the Lie algebra of $E(2)$ . This can be seen by considering the action of translations as the exponential of the Lie algebra representation operators ${\Gamma }_S^{\prime }(p_j)=-i{P}_j$

$$u(\mathbf{a},1)\psi (\mathbf{q})=e^{-i(a_1{P}_1+a_2{P}_2)}\psi (\mathbf{q})=\psi (\mathbf{q}-\mathbf{a})$$

and the action of rotations as the exponential of the ${\Gamma }_S^{\prime }(l)=-iL$

$$u(0,R(\varphi ))\psi (\mathbf{q})=e^{-i\varphi L}\psi (\mathbf{q})=\psi (R(-\varphi )\mathbf{q})$$

One also has a Fourier-transformed version $\tilde{u}$ of this representation, with translations now acting by multiplication operators on the ${\psi }_E$

$$\tilde{u}(\mathbf{a},1){\tilde{\psi }}_E(\theta )=e^{-i(\mathbf{a}\cdot \mathbf{p})}{\tilde{\psi }}_E(\theta )=e^{-i\sqrt{2mE}(a_1\cos \theta +a_2\sin \theta )}{\tilde{\psi }}_E(\theta )\text{\hspace{1em}(19.3)}$$

and rotations acting by rotating the circle in momentum space

$$\tilde{u}(0,R(\varphi )){\tilde{\psi }}_E(\theta )={\tilde{\psi }}_E(\theta -\varphi )\text{\hspace{1em}(19.4)}$$

Although we won’t prove it here, the representations constructed in this way provide essentially all the unitary irreducible representations of $E(2)$ , parametrized by a real number $E>0$ . The only other ones are those on which the translations act trivially, corresponding to $E=0$ , with $SO(2)$ acting as an irreducible representation. We have seen that such $SO(2)$ representations are one dimensional, and characterized by an integer, the weight. We thus get another class of $E(2)$ irreducible representations, labeled by an integer, but they are just one dimensional representations on $C$.

Chapter contents · Next section: 19.2 The case of E(3) · 中文版本

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