1. Relativistic Quantum Mechanics
Lecture 9
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. Non-Relativistic Limit of Dirac Particle
Positive energy solution of Dirac eq is
(1)
Negative sol
---(2)

3. In (1) ----------(3) In (2) ----------(4)
Major component when v<<c
Minor Component when v<<c

4. +Ve energy sol satisfy
-----(5)
Thus, two component eq
-----(6)

5. 2nd Eq. In (6) gives
-----(7)
Using (7) 1st eq of (6) gives
----(8)

6. (9)
Using (9) in (8)
-----(10)
which is non-relativistic form of Dirac Eq.

7. Electron in external magnetic field
We use minimal coupling scheme
(1)
Using coordinate representation
(2)
Also, defining
-----(3)

8. Let electron interacting with M.F.
---(4)
Dirac eq for +Ve energy sol take the form
Or (5)
(6)

9. 2nd Eq in (6) take the form
--(7)
Using (7) in (1st eq of (6), we get
(8)

10. Using identity
-----(9)
Right side of (8) become
---(10)

11. Using (10) in (8), --(11) From (7) ---(12)
Which is Schrodinger equation for a charged electron with a minimal coupling to an external vector field along with a magnetic dipole interaction with the external magnetic field.

12. Magnetic moment operator associated with
electron is identified as
Which suggested gyro magnetic ratio
g = 2
We know
Since for two component electron
and thus

13. Quantum electrodynamics adds very small
corrections to the value g = 2 and also the
experimental value has very small difference from
theoretically predicted.
Particles with internal structures have value of g
very different from 2. We say there anomalous
contribution to magnetic moment
e.g.
where