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 http://www.quantumfieldtheory.info/

Non-Relativistic Limit of Dirac Particle Positive energy solution of Dirac eq is -------(1) Negative sol ---(2)

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

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

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

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

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

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

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

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

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.

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

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