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arXiv:1907.05842

On the correspondence principle for the Klein-Gordon and Dirac Equations

Kevin Hernández, Sergio E. Aguilar-Gutierrez, J. Bernal

Published Jul 12, 2019• 1 citation
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Abstract

We investigate the asymptotic behavior of the solutions to the Klein-Gordon and Dirac equations using the local spatial averaging approach to Bohr's correspondence principle in the large principal quantum number regime. The procedure is applied in two basic problems in $1+1$-dimensions, the relativistic quantum oscillator and the relativistic particle in a box. In the harmonic oscillator cases, we find that the corresponding probability densities reduce to their respective classical single-particle distributions plus a series of terms suppressed by powers of the $\hbar$ constant, while particle in a box cases show a different structure for the quantum corrections.

DOI
10.48550/arxiv.1907.05842

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•Dec 31, 2019•0 citations
We investigate the asymptotic behavior of the solutions to the Klein-Gordon and Dirac equations using the local spatial averaging approach to Bohr's correspondence principle in the large principal quantum number regime. The procedure is applied in two basic problems in 1+11+11+1-dimensions, the relativistic quantum oscillator and the relativistic particle in a box. In the harmonic oscillator cases, we find that the corresponding probability densities reduce to their respective classical single-particle distributions plus a series of terms suppressed by powers of the ℏ\hbarℏ constant, while particle in a box cases show a different structure for the quantum corrections.
In this paper we apply the Bohr's correspondence principle to analize the asymptotic behavior of the Klein-Gordon and Dirac probability densities. It is found that in the non-relativistic limit, the densities reduce to their respective classical single-particle probability distributions plus a series of quantum corrections. The procedure is applied in two basic problems, the relativistic quantum oscillator and the relativistic particle in a box. The particle and antiparticle solutions are found to yield the same classical distribution plus quantum correction terms for the proposed limit. In the quantum oscillator case, a κ\kappaκ parameter modifies the probability distribution. Its origin is briefly discussed in terms of energy.
In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr’s correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum number. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator and the particle in a box, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck’s constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.
Abstract The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantised energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein–Gordon equation in the Feshbach–Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.
In the quantum field,due to Bohr correspondence principle,quantum mechanics go to classical mechanics in large quantum number.Based upon Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle,quantum-classical correspondence of particles in magnetic is studied.Applying Heisenberg correspondence principle in relativistic realm,the exact wave functions of Klein-Gordon equation are obtained for particles in magnetic with quantum matrix element in a rectangular coordinate system of new representation.It turns out that in the classical limit matrix elements of quantum operators reduces to the classical solutions and with classical approach,the coordinate is cyclical variation with time,the orbit of particles is a circle,and the movement is uniform circular motion in uniform magnetic field.
Present work is devoted to studying the spatio-temporal of the Dirac and the Klein Gordon (KG) particles confined in a one-dimensional box. We discuss the quantum carpets and revivals time for each particle. Moreover, we explain that the relativistic phenomena not only decrease the density of the corresponding carpets but also affect the revivals time of each particle within a box. Further, we impose different limiting conditions on the obtained Dirac energy solution and explain the wellknown slight-relativistic and non-relativistic quantum carpets, respectively. In addition, we extend our study to explain the quantum carpets of the KG particle at different energy states and compare its revivals time with the Dirac particle.
The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.