19 The Quantum Free Particle as a Representation of the Euclidean Group

The quantum theory of a free particle is intimately connected to the representation theory of the group of symmetries of space and time. This is well known for relativistic theories, where it is the representation theory of the Poincar´e group that is relevant, a topic that will be discussed in chapter 42. It is less well known that even in the non-relativistic case, the Euclidean group $E(3)$ of symmetries of space plays a similar role, with irreducible representations of $E(3)$ corresponding to free particle quantum theories for a fixed value of the energy. In this chapter we’ll examine this phenomenon, for both two and three spatial dimensions.

The Euclidean groups $E(2)$ and $E(3)$ in two and three dimensions act on phase space by a Hamiltonian group action. The corresponding moment maps (momenta $p_j$ for translations, angular momenta $l_j$ for rotations) Poisson-commute with the free particle Hamiltonian giving symmetries of the theory. The quantum free particle theory then provides a construction of unitary representations of the Euclidean group, with the space of states of a fixed energy giving an irreducible representation. The momentum operators $P_j$ give the infinitesimal action of translations on the state space, while angular momentum operators $L_j$ give the infinitesimal rotation action (there will be only one angular momentum operator $L$ in two dimensions since the dimension of $SO(2)$ is one, three in three dimensions since the dimension of SO(3) is three).

The Hamiltonian of the free particle is proportional to the operator $|\mathbf{P}{|}^2.$ This is a quadratic operator that commutes with the action of all the elements of the Lie algebra of the Euclidean group, and so is a Casimir operator playing an analogous role to that of the $SO(3)$ Casimir operator $|\bar{\mathbf{L}}{|}^2$ of section 8.4. Irreducible representations will be labeled by the eigenvalue of this operator, which in this case will be proportional to the energy. In the Schr¨odinger representation, where the $P_j$ are diferentiation operators, this will be a secondorder diferential operator, and the eigenvalue equation will be a second-order diferential equation (the time-independent Schr¨odinger equation).

Using the Fourier transform, the space of solutions of the Schr¨odinger equation of fixed energy becomes something much easier to analyze, the space of functions (or, more generally, distributions) on momentum space supported only on the subspace of momenta of a fixed length. In the case of $E(2)$ this is just a circle, whereas for $E(3)$ it is a sphere. In both cases, for each radius one gets an irreducible representation in this manner.

In the case of $E(3)$ other classes of irreducible representations can be constructed. This can be done by introducing multi-component wavefunctions, with a new action of the rotation group $SO(3)$ . A second Casimir operator is available in this case, and irreducible representations are eigenfunctions of this operator in the space of wavefunctions of fixed energy. The eigenvalues of this second Casimir operator turn out to be proportional to an integer, the “helicity” of the representation.

中文目录

Chapter contents

• 19.1 The quantum free particle and representations of E(2)
• 19.2 The case of E(3)
• 19.3 Other representations of E(3)
• 19.4 For further reading

Previous chapter: 18 Semi-direct Products

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