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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
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Normalisation method Defining ----(1) We write solution as (2) Where, α and β are normalization constants.
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are normalized as ----(3) which is for same spin components. For different spin components it vanish.
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We now calculate -----(4)
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Negative energy solutions
(5) Also -----(6)
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Wave function (adjoint spinor)
---(7) e.g. ----(8) ----(9)
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Using (8) -----(10)
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Similarly, using (9) (11) For relativistic normalization, we will not have normalization condition -----(12) Probability density transform like time component of a four vector
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For relativistic covariant normalization, we need
---(13) In rest frame Independent free particle wave function With above normalization condition (eq 13), (14)
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Using (4), (5) and (13) (15) (16)
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Normalized +Ve and –Ve energy solutions are
----(17) Also ---(18) Which is Lorentz scalar.
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Positive and negative energy solutions are orthogonal
= 0. (19)
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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)
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From this, we have -----(22) (23)
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Also ---(24) Which is again scalar.
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