Goos-H{ä}nchen Shift for Relativistic Particles Based on Dirac's Equation

Technical Summary: Goos-Hänchen Shift for Relativistic Particles Based on Dirac's Equation

Problem Statement
The Goos-Hänchen (GH) shift is a well-established optical phenomenon where a finite-width beam undergoes a lateral displacement parallel to the interface during total internal reflection. While extensively studied in optics and non-relativistic quantum mechanics (e.g., for neutrons and electrons in graphene), a generalization to relativistic quantum mechanics has remained incomplete. The primary obstacle is the existence of the Klein paradox, where reflection coefficients for Dirac fermions incident on high potential barriers can exceed unity, rendering standard scattering calculations and subsequent GH shift derivations ambiguous or meaningless. This paper addresses the calculation of the GH shift for relativistic Dirac fermions obliquely incident on a three-dimensional infinite potential barrier, specifically resolving the issues posed by the Klein paradox.

Methodology
The authors employ a theoretical framework based on the Dirac equation for a free particle interacting with a step potential $V(x)$.

1. Resolution of the Klein Paradox: Following the approach proposed by Wang (2020), the authors resolve the ambiguity of reflection coefficients by selecting specific energy solutions based on the incident and transmitted regions. The incident wavefunction is treated as a positive-energy solution, while the transmitted wavefunction in the high-potential region ($V > E$) is treated as a negative-energy solution. This selection ensures that the sum of reflection and transmission coefficients equals unity ($R+T=1$), effectively avoiding the paradox.
2. Scattering Setup: The study considers a Dirac particle with momentum $\vec{p} = (p_x, 0, p_z)$ incident on a potential barrier defined by $V(x) = V$ for $x > 0$ and $0$ for $x \leq 0$. The wavefunctions for the incident, reflected, and transmitted regions are constructed using the general solutions of the Dirac equation, incorporating spinor components and continuity conditions at the interface ($x=0$).
3. GH Shift Calculation: To calculate the GH shift, the authors model the incident particle as a finite-width beam rather than a plane wave, analogous to the Artmann method in optics. They utilize an angular spectrum analysis, expressing the wavefunction as a superposition of plane waves with a distribution of transverse momenta. By applying a reflection operator and analyzing the phase shift $\theta$ of the reflection coefficient in the regime of total reflection (where the transmitted momentum component becomes imaginary), they derive the lateral shift $\Delta z$ using the stationary phase method: $\Delta z = -\partial \theta / \partial k_z$.

Key Results
The paper derives an analytical expression for the relativistic quantum GH shift ($\Delta z$) as a function of the incident energy $E$, potential barrier height $V$, and incident angle $\phi$.

• Analytical Formula: The derived shift is given by:
  $$\Delta z = \frac{c\hbar(E^2 - m^2c^4 - 2EV + p^2c^2 \cos 2\phi) \tan \phi}{[-(E^2 + m^2c^4) + p^2c^2 \cos 2\phi] \sqrt{m^2c^4 - (E - V)^2 + p^2c^2 \sin^2 \phi}}$$
• Negative GH Shift: A significant finding is that the relativistic GH shift can be negative. This contrasts with non-relativistic cases where the shift is typically positive. The conditions for a negative shift depend on the relationship between $E$, $V$, and $\phi$. Specifically, when $V < E$, a negative shift occurs if the potential satisfies $V < (E - m^2c^4/E)\cos^2\phi$.
• Behavioral Trends:
  • The shift is zero at normal incidence ($\phi = 0$).
  • As the incident angle approaches grazing incidence ($\phi \to \pi/2$), the shift diverges to infinity.
  • When $V \geq E$, the shift remains positive.

Significance and Claims
The authors claim that their work provides a consistent theoretical treatment of the GH shift for relativistic particles by successfully navigating the Klein paradox through the rational selection of energy solutions.

• Theoretical Distinction: The paper highlights a fundamental difference between relativistic and non-relativistic quantum mechanics: the possibility of a negative GH shift in the relativistic regime.
• Practical Implications: The authors suggest that their theory has practical significance for the design of neutron waveguides.
• Observability: Acknowledging that the magnitude of the shift $\Delta z$ is extremely small and difficult to observe directly, the authors propose that the associated phase shift $\delta$ (where $\delta = p_{z0}\Delta z / \hbar$) may serve as a more viable experimental signature to validate the theory.
• Limitations: The study is limited to the case of an infinite potential barrier in the absence of electromagnetic fields. The authors note that investigating the GH shift in the presence of electromagnetic fields requires further work due to the increased complexity of the Dirac equation.

In conclusion, the paper establishes a framework for calculating relativistic GH shifts, demonstrating that under specific relativistic conditions, the lateral shift can reverse direction, a phenomenon not present in non-relativistic analogs.