Advanced Quantum Mechanics Schwabl · Problem 6.5

The Coulomb Potential

Physics — Quantum Mechanics · quadratic-dirac-equation-landau-levels

Problem

Consider the quadratic form of the Dirac equation from Problem 6.4 with the fields E=E0 (1,0,0)\mathbf{E} = E_0\,(1,0,0) and B=B (0,0,1)\mathbf{B} = B\,(0,0,1), where it is assumed that E0/Bc≤1E_0/Bc \leq 1. Choose the gauge A=B (0,x,0)\mathbf{A} = B\,(0,x,0) and solve the equation with the ansatz ψ(x)=e−iEt/ℏ ei(kyy+kzz)φ(x)Φ ,\psi(x) = \mathrm{e}^{-\mathrm{i}Et/\hbar}\,\mathrm{e}^{\mathrm{i}(k_y y + k_z z)}\varphi(x)\Phi\,, where Φ\Phi is a four-spinor that is independent of time and space coordinates. Calculate the energy eigenvalues for an electron. Show that the solution agrees with that obtained from Problem 5.3 when one considers the nonrelativistic limit, i.e., E0/Bc≪1E_0/Bc \ll 1. Hint: Given the above ansatz for ψ\psi, one obtains the following form for the quadratic Dirac equation: [K(x,∂x)1+M] φ(x)Φ=0 ,[K(x,\partial_x)\mathbb{1} + M]\,\varphi(x)\Phi = 0\,, where K(x,∂x)K(x,\partial_x) is an operator that contains constant contributions, ∂x\partial_x and xx. The matrix MM is independent of ∂x\partial_x and xx; it has the property M2∝1M^2 \propto \mathbb{1}. This suggests that the bispinor Φ\Phi has the form Φ=(1+λM)Φ0\Phi = (\mathbb{1} + \lambda M)\Phi_0. Determine λ\lambda and the eigenvalues of MM. With these eigenvalues, the matrix differential equation reverts into an ordinary differential equation of the oscillator type.

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