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The magnetic vector potential, Klein-Gordon equation and Klein’s paradox in relativistic quantum mechanics

David L. Schaffer

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Thesis (M.S.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Physics

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Matrix Operations

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Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1) where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that (Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2) One possible interpretation of this equation is that we should make the usual substitutions pˆ→ ~ i ∇ and Hˆ → −~ i ∂ ∂t , which leads to[( −~ i ∂ ∂t − eφ )2 − ( ~ i ∇− eA )2 c2 ] ψ = m2c4ψ . (11.3) This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.

Taylor & Francis eBooks•Aug 9, 2006•0 citations
A detailed consideration of the Klein-Gordon equation in relativistic quantum mechanics is presented in order to offer more clarity than many standard approaches. The equation is frequently employed in the research literature, even though problems have often been raised regarding its second-order nature, the status of its negative-energy solutions and the formulation of particle density and flux. Most of these problems can be avoided by dismissing the negative-energy solutions. An application of the equation to a broad wave-packet shows that a small amendment to the usual relativistic formalism can be helpful to demonstrate continuity with the non-relativistic case, although difficulties remain when the proposed quantum state has a broad relativistic energy distribution.
We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits. 1
Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1) where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that (Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2) One possible interpretation of this equation is that we should make the usual substitutions pˆ→ ~ i ∇ and Hˆ → −~ i ∂ ∂t , which leads to[( −~ i ∂ ∂t − eφ )2 − ( ~ i ∇− eA )2 c2 ] ψ = m2c4ψ . (11.3) This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.