Autumn · project 5 of 20

The Klein paradox

Topic: Relativistic quantum mechanics

Goal: Study the scattering of a relativistic fermion on a potential step within the Dirac equation, understand the origin of the so-called Klein paradox, and show that it disappears when the current density and the spectrum of states are interpreted correctly.

We consider the one-dimensional scattering of a fermion of mass \(m\) on the potential step \[V(x)=V_0\,\theta(x)\] within the single-particle Dirac equation, without second quantization.

  1. The Dirac equation in 1+1 dimensions

    Consider the stationary Dirac equation \[\left[ -i\gamma^0\gamma^1\partial_x + \gamma^0 m + V(x) \right]\psi(x)=E\psi(x).\]

    1. Choose an explicit representation of the \(\gamma\) matrices in 1+1 dimensions.

    2. Write the equation in matrix form and find the plane-wave solutions in the regions \(x<0\) and \(x>0\).

  2. Spectrum and wave vectors

    For the regions \[\text{I: } x<0, \quad V=0, \qquad \text{II: } x>0, \quad V=V_0,\] find the corresponding wave vectors \(k_1\), \(k_2\).

    1. Show that for \[E<V_0-m\] the wave vector \(k_2\) remains real.

    2. Explain why this regime does not exist in nonrelativistic theory.

  3. Boundary conditions

    Write the full solution as \[\psi_I = \psi_{\text{in}} + R\,\psi_{\text{ref}}, \qquad \psi_{II}=T\,\psi_{\text{tr}},\] and impose continuity of the wave function at \(x=0\).

    1. Find the reflection coefficient \(R\) and the transmission coefficient \(T\).

    2. Check that \(R+T\neq1\) with the naive definition of probability.

  4. Current density

    Define the Dirac current density \[j^\mu=\bar{\psi}\gamma^\mu\psi, \qquad j^x=\psi^\dagger\alpha\psi.\]

    1. Compute the current for the incident, reflected and transmitted waves.

    2. Show that current conservation, \[j_{\text{in}}=j_{\text{ref}}+j_{\text{tr}},\] holds exactly.

    3. Explain why \(|R|>1\) is nevertheless possible.

  5. Group velocity and interpretation

    1. Compute the group velocity \[v_g=\frac{dE}{dk}\] in region II.

    2. Show that the sign of the group velocity coincides with the sign of the current.

    3. Explain why the solution in the region \(x>0\) corresponds to states with negative energy relative to the local vacuum.

  6. Physical interpretation

    1. Explain how the paradox disappears when the solutions are interpreted as antiparticle states.

    2. Discuss why the language of particles and antiparticles appears already at the level of the single-particle Dirac equation.

    3. Compare with the nonrelativistic case (the Schrödinger equation).

  7. Numerical analysis

    Using python:

    1. Plot \(R(E)\) and \(T(E)\) for different values of \(V_0\).

    2. Mark the region \(E<V_0-m\).

    3. Plot the currents \(j_{\text{in}}\), \(j_{\text{ref}}\), \(j_{\text{tr}}\) and check their balance.

  8. Discussion

    1. Where exactly does the “paradox” arise if the interpretation is wrong?

    2. Why is the Klein paradox not a violation of causality?

    3. Which parts of this analysis require second quantization, and which do not?

  9. References

    • B. R. Holstein, Klein’s paradox, Am. J. Phys. 66, 507 (1998).

    • P. A. M. Dirac, The Principles of Quantum Mechanics.

    • W. Greiner, Relativistic Quantum Mechanics.

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