Relativistic Quantum Mechanics Lecture 3 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl Relativistic Quantum Mechanics by Greiner Quantum Field Theory by Mark Srednicki http://www.physicspages.com/2015/11/12/klein-gordon-equation/ http://www.quantumfieldtheory.info/

Klein Gordon Equation For non-relativistic case, we know how to get Schrodinger equation from eq. using corresponding Hermitian operators -----------(1) -------------(2)

For relativistic case, we start with relativistic energy momentum relationship -------(3) Using operator substitution and operating over scalar wave function ϕ

Considering -------(4) Above equation is Klein Gordon equation. Putting and c in above equation will look as -------(5) Above eq will be invariant under Lorentz transformations.

Note that D’Alembertian operator is invariant Also φ is assumed to be scalar and hence K.G. Equation is invariant

Klein Gordon equation has plane wave solutions: Function is Eigen function of energy-momentum operator Eigen values

Thus, plane waves will be solution of K.G. Eq if --------(6) which gives positive and negative energy solutions.

Concept of probability density and current density in Klein Gordon equation: Writing K.G. Eq. and its complex conjugate -------(7) ------(8) Multiplying (7) from left by ϕ* and (8) by ϕ

--------(9) Before proceeding further, we recall continuity equation derived in non-relativistic quantum mechanics i.e. ----(10) Where Probability density Current density -----(11)

Multiplying Eq (9) by and using natural system of units, we can write -----(12) Where -----(13) ------(14) Above eq gives current density and probability density from K.G. Eq.

Note the difference in probability density from S.E. What will be form of J and ρ in Eqs (13) and (14), if we do not use natural system of units? Note the difference in probability density from S.E. And K.G. eq. i.e. Eq. (11) and (14) In case of S.E. probability density is always positive definite. It is time independent and hence probability is conserved (see chapter 3, QM by Zettili )

But eq. (14) (ρ from KG eq) can take negative values as well. This is because KG eq. is 2nd order in time derivative and we can use any initial value of ϕ and . e.g. For (plane wave)

As E can be positive or negative and hence, ρ also. Klein Gordon eq is discarded as quantum mechanical wave equation for single relativistic particle. As a quantized field theory, the Klein–Gordon equation describes mesons. The hermitian scalar Klein–Gordon field describes neutral mesons with spin 0. The non-hermitian pseudoscalar Klein–Gordon field describes charged mesons with spin 0 and their antiparticles.