The Klein-Gordon Equation; Addendum

In 1925, shortly after Louis deBroglie asserted the wave-particle duality for matter and just a little short of a quarter century after Max Planck asserted the same duality for electromagnetic radiation, Erwin Schrödinger devised and presented an equation familiar to all students of modern physics. Following deBroglie’s description, Schrödinger assumed the existence of a wave associated with each and every particle of matter and described the wave with a state function (usually represented by the Greek letter psi) that encoded all relevant information about the particle. As the wave propagates, it carries the particle along with it, subject to the indeterminacy of Heisenberg’s principle. To gain a description of the state function, Schrödinger asserted that it must conform to the statement Hψ=Eψ, in which H and E represent, respectively, the Hamiltonian function associated with the particle and the particle’s total energy.

Schrödinger’s equation, in that raw form, appears to be a simple multiplication of the state function by two algebraic functions. In fact, H and E represent differential operators that alter the state function by extracting algebraic functions from it. If we ignore the potential energy for the nonce (thereby describing a particle free of any applied force), the Hamiltonian consists solely of the particle’s kinetic energy expressed as the square of the particle’s linear momentum divided by twice the particle’s mass. We have the basic differential operators representing linear momentum and total energy as

(Eq’ns 1)

so we write Schrödinger’s equation as

(Eq’n 2)

in which U represents the potential energy associated with the particle, if any.

That equation works very well in situations that, if enlarged, would conform to classical Newtonian mechanics; that is, alternatively, if Planck’s constant were to go to zero, Schrödinger’s quantum mechanics would become the Hamiltonian version of Newtonian mechanics. But in relativistic situations, clusters of events among which particles move at substantial fractions of the speed of light, Schrödinger’s equation will give us the wrong answers to the problems in which we apply it. We know that statement stands true to Reality because Schrödinger’s equation is not Lorentz invariant. It is second order in the spatial derivatives and first order in the temporal derivative, so if we apply the Lorentz Transformation to it, we get an equation containing additional terms that are first order in the spatial derivatives and second order in the temporal derivative: the equation has a different form in different frames of reference and that fact conflicts with the principle of Relativity. To satisfy that principle we need a fundamental equation of quantum mechanics that is either second order in the spatial and temporal derivatives or first order in both kinds of derivatives. In fact, we can have both.

For our fundamental equation we need an operator that will somehow extract the momentum and energy of the particle from the state function in some form and that is Lorentz invariant. We know that the dot product of any two four-vectors yields a value that remains unchanged when we apply the Lorentz Transformation to the four-vectors, so all we need to do is to multiply the momentum-energy four-vector associated with our particle by itself and we get

(Eq’n 3)

which is Lorentz invariant. Multiplying that equation by -c^{2} and
applying the result to the state function gives us the Klein-Gordon equation,

(Eq’n 4)

Substituting from Equations 1 and rearranging the equation slightly gives us

(Eq’n 5)

which looks just like the wave equation that we study in the electromagnetic theory.

We know how to solve that equation, so we have our state function as

(Eq’n 6)

in which we relate the wave number and the angular frequency to the particle’s linear momentum and energy through the statements p=Sk and E= Sω, in accordance with Planck’s theorem. But if we substitute Equation 6 into Equation 5 and apply the indicated differentiations, we get Equation 4 plus a cross term that involves multiplying the angular frequency by the rest mass-energy. That term should not be in our equation, but we can eliminate it easily. We need only modify Equation 6 to read

(Eq’n 7)

Substituting that solution into Equation 5 will give us Equation 4 because the differentiations give us two of the unwanted cross terms, which cancel each other out of the equation.

In making that connection we have tacitly attributed to the particle a negative mass-energy state; that is, we have assumed that the particle has available to it the possibility of becoming the antimatter equivalent of itself. In his 1949 paper "The Theory of Positrons", Richard Feynman asserted that such a possibility provides a clear understanding of the existence of antimatter in the Universe: he asserted that antimatter is simply matter traveling backward through time. Thus the Klein-Gordon equation obliges us to account for the existence of antimatter in much the way that Dirac’s equation does.

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