Klein Paradox

Introduction

The Klein Paradox, proposed by Swedish physicist Oskar Klein, is a counterintuitive phenomenon in quantum mechanics and quantum field theory, particularly in the context of relativistic quantum mechanics described by the Dirac equation. The paradox refers to the penetration of high-energy particles through high potential barriers, contrary to the predictions of non-relativistic quantum mechanics.

Basics of the Klein Paradox

The Klein Paradox arises in the study of relativistic quantum mechanics. It concerns the behavior of a fermion (for example, an electron) impinging on a potential step that is higher than the energy of the fermion. In non-relativistic quantum mechanics, such a scenario would result in the fermion being reflected by the potential barrier. However, in the context of the Dirac equation, a fully relativistic treatment predicts that the fermion can tunnel through the barrier with a probability greater than 1, which seems paradoxical.

The Dirac Equation

The Dirac equation, a relativistic wave equation describing fermions, is fundamental to understanding the Klein Paradox. The equation in natural units () reads:

where is the wave function of the fermion, is the fermion mass, are the gamma matrices, and is the four-gradient.

Explanation of the Klein Paradox

When a fermion encounters a potential barrier of height , the Klein-Gordon equation (square of the Dirac equation) gives two solutions, corresponding to positive and negative energy states. If is greater than the fermion’s rest energy (), the negative energy solution becomes relevant, and it is possible for a fermion to tunnel through the barrier by transitioning to a negative energy state. This phenomenon, termed “Klein tunneling,” results in perfect transmission through the barrier, which is the so-called Klein Paradox.

The Role of Quantum Field Theory

The resolution to the Klein Paradox comes from the framework of quantum field theory, which allows for the creation and annihilation of particles. In this context, the “negative energy states” correspond to the creation of particle-antiparticle pairs. Therefore, what seems like a paradox in the framework of single-particle quantum mechanics is explained within the broader context of quantum field theory.

Conclusion

The Klein Paradox underscores the limitations of single-particle quantum mechanics when dealing with relativistic particles and high-energy phenomena. Its resolution through quantum field theory emphasizes the necessity of the latter in the complete understanding of quantum mechanical phenomena.

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