Two Extensions of the WKB Approximation

Abstract

In this thesis, we study two extensions of the WKB approximation. (i) We show that the WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors, and (ii) we prove a general approximate quantization rule ∫𝑅𝐸 𝐿𝐸 𝑘0 𝑑𝑥 = (𝑁 + 12 )𝜋 for the bound states in the potential well of the 𝑛th-order Schrödinger equations 𝑒−𝑖𝜋𝑛/2∇ð‘›ð‘¥Ψ = [𝐸 − Δ(𝑥)]Ψ, where 𝑘0 = (𝐸 − Δ)1/𝑛 with 𝑁 ∈ â„•0. 𝑛 is an even natural number and 𝐿𝐸 and 𝑅𝐸 are the boundary points between the classically forbidden regions and the allowed region.

Firstly, we demonstrate that the WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap functions Δ(𝑥) in superconductors with the Bogoliubov quasiparticle energy 𝐸, the Fermi level 𝐸𝐹 , and the total momentum p at 𝐸𝐹 +𝐸. The Andreev bound states in the well of slowly varying |Δ(𝑥)| profile are studied, and the well may also be induced by the phase variation of Δ(𝑥) for massless Dirac fermions. By applying the WKB method to the Bogoliubov-de Gennes equation, we obtain two main results: (i) For 𝐸𝐹 ∼ 0, the bound states are determined by ∫𝑅𝐸 𝐿𝐸|p𝑑x| = (12 + 𝑛′)𝜋ℏ , where 𝑛′ ∈ â„•0, and 𝐿𝐸 and 𝑅𝐸 are the boundary points between the classically allowed region and forbidden regions, and (ii) for 𝐸𝐹 ≫ 𝐸 and |Δ(𝑥)|, the bound states are given by ∫𝑅𝐸
𝐿𝐸|p ± p𝐹 | 𝑑𝑥 = (12 + 𝑛′)𝜋ℏ with small 𝑝𝑦(𝑧). Empirical quantization conditions are provided for broader parameter regions. In addition to applying the traditional WKB method, we also develop a generalized WKB method to tackle semimetals with parabolic dispersion relationships. The applications of our results are discussed, for example, Dirac 𝜋-junctions or non-chiral Majorana wires, SNS junctions, the excitation threshold, and the tunneling rate in NSN junctions. In the 𝜋-junction, the Majorana zero modes correspond to the zero-point energy in the WKB formalism. This observation may provide insights into the Majorana bound state in a vortex and the Majorana fermions in high-energy physics.

Secondly, we prove a general approximate quantization rule ∫𝑅𝐸 𝐿𝐸 𝑘0 𝑑𝑥 = (𝑁 + 12 )𝜋 for the bound states in the potential well of the 𝑛th-order Schrödinger equations 𝑒−𝑖𝜋𝑛/2∇ð‘›ð‘¥Ψ = [𝐸 − Δ(𝑥)]Ψ, where 𝑘0 = (𝐸 − Δ)1/𝑛 with 𝑁 ∈ â„•0. The only hypothesis is that all expox nentially growing components are negligible, which is appropriate for not narrow wells. We discuss applications including the Schrödinger equation and Bogoliubov-de Gennes equation. For instance, the Bogoliubov-de Gennes equation of the semimetal with parabolic dispersion relationships is transformed into the mentioned differential equation with 𝑛 = 4. Therefore, our results greatly generalize the WKB approximation.

Overall, this thesis provides a formula to describe the Andreev bound states in nonuniform superconductors and expands the scope of the WKB approximation to a broader range of ordinary differential equations.