 ### 7. Relativistic Systems in One Dimension

The fundamental equation of relativistic quantum physics is the Dirac equation. It combines the special theory of relativity with quantum mechanics. In this chapter, we restrict our attention to free particles moving in one space dimension, leaving the more general case to the next chapter. In one dimension, relativistic particles are described by wave functions with two components, which are comparatively easy to visualize and, nevertheless, show many features of higher-dimensional solutions that have puzzled a whole generation of physicists.

Dirac originally derived his equation by linearizing the classical relativistic energy-momentum relation using matrix-valued coefficients (Section 7.2). A direct generalization of the nonrelativistic interpretation of wave functions in terms of position probability densities leads to a natural Hilbert space for the Dirac equation and to a tentative one-particle interpretation (Section 7.3). In Section 7.4, we find plane-wave solutions of the Dirac equation and form (approximately) localized wave packets as "continuous superpositions" of plane waves. It turns out that the Dirac operator (the energy observable of a free particle) has a spectrum consisting of positive and negative energies. Correspondingly, the Hilbert space can be decomposed as a direct sum of subspaces with positive and negative energy (Section 7.5).

Surprisingly, the kinematics of wave packets according to the Dirac equation turns out to be a rich and interesting subject (Section 7.6). The presence of a limiting velocity (the velocity of light) leads to characteristic distortions of wave packets in position space. Moreover, a wave packet with negative energy moves in a direction opposite to its average momentum. A very peculiar effect called Zitterbewegung - an oscillating motion of the average position - occurs under certain conditions for superpositions of positive and negative energies. This and other interference effects are discussed in Section 7.7}. In Section 7.8, we describe wave packets in energy space and in velocity space. These are representations where the energy or the (classical) velocity are diagonal. The energy and velocity distributions are important for an analysis and understanding of a wave packet's time evolution.

The free-particle Dirac equation is relativistically invariant. The group of Poincaré transformations has a unitary representation in the Hilbert space. We discuss the Lorentz contraction of wave packets and find interesting interference effects for a velocity transformation of superpositions involving positive and negative energies (Section 7.9).

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• 7.1. Introduction 324
• 7.2. The Free Dirac Equation 325
• 7.2.1. The relativistic energy-momentum relation 325
• 7.2.2. The Dirac operator 326
• 7.3. Dirac Spinors and State Space 327
• 7.3.1. A Hilbert space for the Dirac equation 327
• 7.3.2. The standard interpretation 328
• 7.3.3. The relativistic momentum space 330
• 7.3.4. A solution of the Dirac equation 331
• 7.4. Plane Waves and Wave Packets 332
• 7.4.1. Diagonalization in momentum space 332
• 7.4.2. Eigenvectors and plane-wave solutions 334
• 7.4.3. Building wave packets 337
• 7.5. Subspaces with Positive and Negative Energies 339
• 7.6. Kinematics of Wave Packets 343
• 7.6.1. The limiting velocity 343
• 7.6.2. Negative-energy wave packets 346
• 7.7. Zitterbewegung 347
• 7.7.1. The standard position operator 347
• 7.7.2. Velocity of the standard position 349
• 7.7.3. The classical velocity operator 350
• 7.7.4. Time evolution of the standard velocity 351
• 7.7.5. Time evolution of the standard position 352
• 7.7.6. Superpositions and Zitterbewegung 353
• 7.7.7. Superluminal motion? 357
• 7.8. Special Topic: Energy Representation and Velocity Space 359
• 7.8.1. Variable substitutions 360
• 7.8.2. Energy representation 362
• 7.8.3. Velocity representation 364
• 7.9. Relativistic Invariance 365
• 7.9.1. Lorentz transformations 365
• 7.9.2. Transformation of the Dirac equation 367
• 7.9.3. Representation of Lorentz boosts 369
• 7.9.4. Invariance of the free-particle Dirac equation 370
• 7.9.5. Unitary implementation of Lorentz boosts 371