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. ID: 583038
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Relativistic Quantum MechanicsLecture 3
Lectures on Quantum Field Theory by Ashok Das
Advanced Quantum Mechanics by
Relativistic Quantum Mechanics by Greiner
Quantum Field Theory by Mark
Klein Gordon EquationFor non-relativistic case, we know how to getSchrodinger equation from eq. using correspondingHermitian operators -----------(1) -------------(2)Slide3
For relativistic case, we start with relativistic energy momentum relationship -------(3)Using operator substitution and operating over scalar wave function ϕSlide4
Above equation is Klein Gordon equation.Putting and c in above equation will look as -------(5)Above eq will be invariant under Lorentz transformations.Slide5
operator is invariant
is assumed to be scalar and hence K.G.
Equation is invariantSlide6
Klein Gordon equation has plane wave solutions:
is Eigen function of energy-momentum operator
Thus, plane waves will be solution of K.G.
which gives positive and negative energy solutions.Slide8
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 ϕSlide9
--------(9)Before proceeding further, we recall continuityequation derived in non-relativistic quantum mechanics i.e. ----(10)WhereProbability density Current density -----(11)Slide10
(9) by and using natural system of units, we can write -----(12)Where -----(13) ------(14)Above eq gives current density and probabilitydensity from K.G. Eq.Slide11
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 positivedefinite. It is time independent and hence probabilityis conserved
(see chapter 3, QM by
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 ϕand .e.g. For (plane wave)Slide13
As E can be positive or negative and hence,
is discarded as quantum mechanical
wave equation for single relativistic particle.
As a quantized field theory, the Klein–Gordon
equation describes mesons.
scalar Klein–Gordon field describes
neutral mesons with spin 0.
field describes charged mesons with spin 0 and