1. 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

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

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

4. We now calculate
-----(4)

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

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

7. Using (8)
-----(10)

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

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

10. Using (4), (5) and (13)
(15)
(16)

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

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

13. 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)

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

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