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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Slide1

Relativistic Quantum MechanicsLecture 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/

Slide2Klein Gordon EquationFor non-relativistic case, we know how to getSchrodinger equation from eq. using correspondingHermitian operators -----------(1) -------------(2)

Slide3For relativistic case, we start with relativistic energy momentum relationship -------(3)Using operator substitution and operating over scalar wave function ϕ

Slide4Considering

-------(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.

Slide5Note that

D’Alembertian

operator is invariant

Also

φ

is assumed to be scalar and hence K.G.

Equation is invariant

Slide6Klein Gordon equation has plane wave solutions:

Function

is Eigen function of energy-momentum operator

Eigen values

Slide7Thus, plane waves will be solution of K.G.

Eq

if

--------(6)

which gives positive and negative energy solutions.

Slide8Concept 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

Slide9

--------(9)Before proceeding further, we recall continuityequation derived in non-relativistic quantum mechanics i.e. ----(10)WhereProbability density Current density -----(11)

Slide10Multiplying

Eq

(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.

Slide11What 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

Zettili

)

Slide12But 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)

Slide13As 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.

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