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1 Comment on ’Another form of the Klein-Gordon equation’

Andreas Aste

Published Jan 1, 2001• 1 citation
OpenAlex

Abstract

It is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. PACS code: 11.10.-z Keywords: Klein-Gordon equation, relativistic field theory. 2 The alternative formulation of the Klein-Gordon equation [1, 2] proposed by M.D. Kostin reads [3] i¯h ∂φ ∂t = +mc2φ + c ( ˆ⃗p ⃗ ψ) (1) i¯h ∂ ⃗ ψ ∂t = −mc2 ⃗ ψ + c ˆ ⃗pφ, (2) where φ(⃗r,t) and ⃗ ψ(⃗r,t) are ’scalar ’ and ’vector ’ probability amplitudes, respectively, and ˆ ⃗p = −i¯h ⃗ ∇. Defining the probability density P = φ ∗ φ + ( ⃗ ψ ∗ ⃗ ψ) (3) and the probability current density one readily derives the probability conservation equation ⃗S = c(φ ∗ ⃗ ψ + φ ⃗ ψ

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It is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. PACS code: 11.10.-z Keywords: Klein-Gordon equation, relativistic field theory. 2 The alternative formulation of the Klein-Gordon equation [1, 2] proposed by M.D. Kostin reads [3] i¯h ∂φ ∂t = +mc2φ + c ( ˆ⃗p ⃗ ψ) (1) i¯h ∂ ⃗ ψ ∂t = −mc2 ⃗ ψ + c ˆ ⃗pφ, (2) where φ(⃗r,t) and ⃗ ψ(⃗r,t) are ’scalar ’ and ’vector ’ probability amplitudes, respectively, and ˆ ⃗p = −i¯h ⃗ ∇. Defining the probability density P = φ ∗ φ + ( ⃗ ψ ∗ ⃗ ψ) (3) and the probability current density one readily derives the probability conservation equation ⃗S = c(φ ∗ ⃗ ψ + φ ⃗ ψ
In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
In this paper, we shall derive a new conserved current for Klein-Gordon equation. The first component of this current is non-negative and reduces to ∣ϕ∣2|\phi|^2∣ϕ∣2 in the non-relativistic limit. Therefore, it can be interpreted as a suitable probability density for spinless particles. In addition, this current is time-like and so prevents faster than light particle propagation. We will see the probability density has a considerable deviation from ∣ϕ∣2|\phi|^2∣ϕ∣2 providing the uncertainty in momentum is much greater than m0cm_0cm0​c.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.