19.2 The case of E(3)

19.2 The case of E(3)

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

In the physical case of three spatial dimensions, the state space of the theory of a quantum free particle is again a Euclidean group representation, with the same relationship to the Schr¨odinger representation as in two spatial dimensions. The main diference is that the rotation group is now three dimensional and noncommutative, so instead of the single Lie algebra basis element $l$ we have three of them, satisfying Poisson bracket relations that are the Lie algebra relations of so(3)

$$\{l_1,l_2\}=l_3,\{l_2,l_3\}=l_1,\{l_3,l_1\}=l_2$$

The $p_j$ give the other three basis elements of the Lie algebra of $E(3)$ . They commute amongst themselves and the action of rotations on vectors provides the rest of the non-trivial Poisson bracket relations

$$\begin{array}{rl}\left\{l_1,p_2\right\}& =p_3,\left\{l_1,p_3\right\}=-p_2\\ \left\{l_2,p_1\right\}& =-p_3,\left\{l_2,p_3\right\}=p_1\\ \left\{l_3,p_1\right\}& =p_2,\left\{l_3,p_2\right\}=-p_1\end{array}$$

An isomorphism of this Lie algebra with a Lie algebra of matrices is given by

$$l_1\leftrightarrow \left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right)l_2\leftrightarrow \left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)l_3\leftrightarrow \left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$ $$p_1\leftrightarrow \left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)p_2\leftrightarrow \left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)p_3\leftrightarrow \left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right)$$

The $l_j$ are quadratic functions in the $q_j,p_j$ , given by the classical mechanical expression for the angular momentum

$$\mathbf{l}=\mathbf{q}\times \mathbf{p}$$

${}_{\mathrm{or}},$ in components

$$l_1=q_2{p}_3-q_3{p}_2,l_2=q_3{p}_1-q_1{p}_3,l_3=q_1{p}_2-q_2{p}_1$$

The Euclidean group $E(3)$ is a subgroup of the Jacobi group $G^J(3)$ in the same way as in two dimensions, and, just as in the $E(2)$ case, exponentiating the Schr¨odinger representation ${\Gamma }_S^{\prime }$

$${\Gamma }_S^{\prime }(l_1)=-i{L}_1=-i(Q_2{P}_3-Q_3{P}_2)=-\left(q_2\frac{\partial }{\partial q_3}-q_3\frac{\partial }{\partial q_2}\right)$$ $${\Gamma }_S^{\prime }(l_2)=-i{L}_2=-i(Q_3{P}_1-Q_1{P}_3)=-\left(q_3\frac{\partial }{\partial q_1}-q_1\frac{\partial }{\partial q_3}\right)$$ $${\Gamma }_S^{\prime }(l_3)=-i{L}_3=-i(Q_1{P}_2-Q_2{P}_1)=-\left(q_1\frac{\partial }{\partial q_2}-q_2\frac{\partial }{\partial q_1}\right)$$ $${\Gamma }_S^{\prime }(p_j)=-i{P}_j=-\frac{\partial }{\partial q_j}$$

provides a representation of $E(3)$

As in the $E(2)$ case, the above Lie algebra representation is just the infinitesimal version of the action of $E(3)$ on functions induced from its action on position space ${\mathbf{R}}^3$ . Given an element $g=(\mathbf{a},R)\in E(3)$ we have a unitary transformation on wavefunctions

$$u(\mathbf{a},R)\psi (\mathbf{q})=\psi (g^{-1}\cdot \mathbf{q})=\psi (R^{-1}(\mathbf{q}-\mathbf{a}))$$

Such group elements $g$ will be a product of a translation and a rotation, and treating these separately, the unitary transformations u are exponentials of the Lie algebra actions above, with

$$u(\mathbf{a},1)\psi (\mathbf{q})=e^{-i(a_1{P}_1+a_2{P}_2+a_3{P}_3)}\psi (\mathbf{q})=\psi (\mathbf{q}-\mathbf{a})$$

for a translation by $\mathbf{a},$ and

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

for $R(\varphi ,{\mathbf{e}}_j)$ a rotation about the j-axis by angle $\varphi .$

This representation of $E(3)$ on wavefunctions is reducible, since in terms of momentum eigenstates, rotations will only take eigenstates with one value of the momentum to those with another value of the same norm-squared. We can get an irreducible representation by using the Casimir operator ${\bar{P}}_1^2+P_2^2+P_3^2$ , which commutes with all elements in the Lie algebra of $E(3)$ . The Casimir operator will act on an irreducible representation as a scalar, and the representation will be characterized by that scalar. The Casimir operator is just 2m times the Hamiltonian

$$H=\frac{1}{2m}(P_1^2+P_2^2+P_3^2)$$

and so the constant characterizing an irreducible will be the energy 2mE. Our irreducible representation will be on the space of solutions of the time-independent Schr¨odinger equation

$$\frac{1}{2m}(P_1^2+P_2^2+P_3^2)\psi (\mathbf{q})=-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)\psi (\mathbf{q})=E\psi (\mathbf{q})$$

Using the Fourier transform

$$\psi (\mathbf{q})=\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}{e}^{i\mathbf{p}\cdot \mathbf{q}}\tilde{\psi }(\mathbf{p})d^3\mathbf{p}$$

the time-independent Schr¨odinger equation becomes

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

and we have distributional solutions

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

characterized by distributions ${\tilde{\psi }}_E(\mathbf{p})$ defined on the sphere $|\mathbf{p}{|}^2=2mE$

Such complex-valued distributions on the sphere of radius $\sqrt{2mE}$ provide a Fourier-transformed version ue of the irreducible representation of $E(3)$ . Here the action of the group $E(3)$ is by

$$\tilde{u}(\mathbf{a},1){\tilde{\psi }}_E(\mathbf{p})=e^{-i(\mathbf{a}\cdot \mathbf{p})}{\tilde{\psi }}_E(\mathbf{p})$$

for translations, by

$$\tilde{u}(0,R){\tilde{\psi }}_E(\mathbf{p})={\tilde{\psi }}_E(R^{-1}\mathbf{p})$$

for rotations, and by

$$\tilde{u}(\mathbf{a},R){\tilde{\psi }}_E(\mathbf{p})=\tilde{u}(\mathbf{a},1)\tilde{u}(0,R){\tilde{\psi }}_E(\mathbf{p})=e^{-i\mathbf{a}\cdot R^{-1}\mathbf{p}}{\tilde{\psi }}_E(R^{-1}\mathbf{p})$$

for a general element.

Previous section: 19.1 The quantum free particle and representations of E(2) · Next section: 19.3 Other representations of E(3) · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 210、211、212、213、214、215、216、217、218、219、220

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：96747283728210a94caf040995427342b95d9623aec43d018ba9712da0c11296

OCR 产物 SHA-256：96747283728210a94caf040995427342b95d9623aec43d018ba9712da0c11296