Relativistic Quantum Mechanics Lecture 6 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.quantumfieldtheory.info/

Normalisation method Defining ----(1) We write solution as ----------(2) Where, α and β are normalization constants.

are normalized as ----(3) which is for same spin components. For different spin components it vanish.

We now calculate -----(4)

Negative energy solutions ---------(5) Also -----(6)

Wave function (adjoint spinor) ---(7) e.g. ----(8) ----(9)

Using (8) -----(10)

Similarly, using (9) -------(11) For relativistic normalization, we will not have normalization condition -----(12) Probability density transform like time component of a four vector

For relativistic covariant normalization, we need ---(13) In rest frame Independent free particle wave function With above normalization condition (eq 13), ----------(14)

Using (4), (5) and (13) --------(15) --------(16)

Normalized +Ve and –Ve energy solutions are ----(17) Also ---(18) Which is Lorentz scalar.

Positive and negative energy solutions are orthogonal = 0. ----------(19)

Note that --------(20) Normalization discussed above is for massive particle Only. Alternative, normalization condition which work Well for massive and mass-less particles is ------(21)

From this, we have -----(22) --------(23)

Also ---(24) Which is again scalar.