The eigenvalue problem of one-dimensional Dirac operator
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The eigenvalue problem of one‑dimensional Dirac operator Jacek Karwowski1 · Artur Ishkhanyan2,3 · Andrzej Poszwa4 Received: 23 June 2020 / Accepted: 6 October 2020 © The Author(s) 2020
Abstract The properties of the eigenvalue problem of the one-dimensional Dirac operator are discussed in terms of the mutual relations between vector, scalar and pseudo-scalar contributions to the potential. Relations to the exact solubility are analyzed. Keywords One-dimensional Dirac equation · Vector potential · Scalar potential · Pseudo-scalar potential · Supersymmetry · Effective mass · Schrödinger equation · Lévy-Leblond equation · Bound states · Non-relativistic limit · Pauli approximation
1 Introduction
𝛹 (x, t) =
We are concerned with the simplest quantum system: a single particle in a one-dimensional space. Its time evolution in the Schrödinger picture and in the coordinate representation is governed by the evolution equation: [ ] 𝜕 iℏ − 𝖧(x) 𝛹 (x, t) = 0. (1) 𝜕t Solutions of the eigenvalue problem of the Hamiltonian
𝖧 𝜓E (x) = E 𝜓E (x)
(2)
give the stationary state energies and wave functions. From here solution of (1) may be obtained as
* Jacek Karwowski [email protected] Artur Ishkhanyan [email protected] Andrzej Poszwa [email protected] 1
Institute of Physics, Nicolaus Copernicus University, 87‑100 Toruń, Poland
2
Russian-Armenian University, 0051 Yerevan, Armenia
3
Institute for Physical Research, NAS of Armenia, 0203 Ashtarak, Armenia
4
Faculty of Mathematics and Computer Science, University of Warmia and Mazury, 10‑710 Olsztyn, Poland
∑ C(E) e−iEt∕ℏ 𝜓E (x) dE. ∫E
(3)
We have chosen to describe a single fermion using the Lorentz-covariant quantum theory. The space-time is twodimensional, and physical quantities may be described by scalars, pseudo-scalars and two vectors composed of time and space components. The relativistic dispersion relation for a free particle
𝖤2 = c2 𝗉2 + m2 c4 ,
𝖤 = 𝖧0 ,
𝗉=
ℏ d , i dx
(4)
implies that the free-particle Hamiltonian 𝖧0 is a Hermitian square root of the second-order differential operator in the rhs of this relation, i.e. it is the one-dimensional free Dirac operator [1]. By applying the classical procedure of Dirac, reduced to one dimension, we get
𝛼 𝗉 + 𝛽 mc2 , 𝖧0 = c𝛼
(5)
with 𝛼 𝛽 + 𝛽 𝛼 = 0 , 𝛼 2 = 𝛽 2 = 1 and x ∈ ⟨−∞, ∞⟩1. Since 𝛼 and 𝛽 have to be Hermitian, their eigenvalues can only 𝛼 ) = Tr(𝛽𝛽 ) = 0 and, consebe ±1 . It is easy to see that Tr(𝛼 quently, the dimension of these matrices has to be even. The Pauli matrices
1
The momentum operator is Hermitian if x ∈ ⟨−∞, ∞⟩ , but it is non-Hermitian on a semi-axis. By properly choosing boundary conditions one can construct a Hermitian extensions of 𝗉 for x in a finite interval x ∈ ⟨ −L, L ⟩ [2, 3].
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(
) ( ) 10 01 𝜎0 = , 𝜎1 = , 01 10 ( ) ( ) 0 −i 1 0 𝜎2 = , 𝜎3 = , i 0 0 −1 form a complete set of 2 × 2 matrices. One can readily check that any two of three anti-commuting matrices fulfill
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