Relativistic Quantum Mechanics Lecture 5 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/ https://www2.warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf

Solution of Dirac Equation Dirac eq in momentum representation -----(1) In coordinate representation ---(2) Since the matrices are 4 by 4, the solution Will also be a column matrix having 4 components .

Multi-component wave function suggest the spin Angular momentum. It means Dirac equation must describe particle with spin. For this we need to find the solution. Let plane wave solution is --------(3)

With ------(4) Putting above back into Dirac eq -------(5) Where ---------(6)

Considering motion along –z direction -----(7) Thus, ------(8) Using matrix form of gamma matrices ( Note –ve sign with 2nd term in Eq (8) is missing) -----(9)

Set of 4 equations have solution if -----(10) which give ---------------------------(11)

Thus solution exist only for energies -----(12) Each of the value is doubly degenerate which may be due to spin structure.

Particle at rest Consider plane wave solution ----(13) Here -----(14) For particle at rest p = 0, --------(15)

Spatial derivative also vanishes when p = 0 ----------(16) From Dirac equation -----------(17)

Expanding gamma matrices -------(18) Which is Eigen value equation. Four independent solutions exist. Two with E = m and Two with E = -m

Solutions are ---(19) First two solutions are for particle with positive Energy with spin up and spin down respectively. Third and fourth are for particle with negative Energy and again with spin up and spin down.

For complete solution with ψ, e-iEt factor will also Appear, with E = m for u1 and u2 and E = -m for u3 and u4 Spin ½ particle with spin-up Spin ½ particle with spin-down Spin ½ antiparticle with spin-up Spin ½ antiparticle with spin-down

If function is Eigen-function of spin matrix Sz, then -----(20) Which is for particle with spin up.

Solution for particle in motion Plane wave solution ------(21) Using this, Dirac Eq -----(22)

We use following two component form We can write -----(23) We use following two component form for 4-component spinor (also known as bispinor) --------(24) For upper two components For lower two components

From Eq (22), we can write now ---------(25) Above eq lead to following coupled equations -----(26)

From 2nd relation in Eq (26), we have Using above Eq. in 1st relation of (26), we get Which is relativistic energy momentum relationship.

Note that -----(27) Now from Dirac Eqs., (28), we have --------(28)

We consider first solution --------(29) Using (28) and (29) in (24), we can write ------(30) With p = 0, above Eqns. reduces to free particle Solution with E> 0 .

Now we use ------(31) and this give ----(32) which is for E<0.

Thus, we write -----(33) Exercise: Discuss the non-relativistic limit of above Solutions.

Feynman Stuckelberg interpretation for –Ve energy solution Convention -----(34) Spin flips for antiparticles i.e. Santiparticle = -Sparticle In above v1 is for spin up and v2 is for spin down

Interpretation: -Ve energy particle travelling backward in time or +Ve energy antiparticle moving forward in time.

The general solutions u1, u2, v1 and v2 are not Eigen-states of spin matrix. This is because -----(35) If ----(36) i.e. If particle is moving along z-direction only --------(37)

Solutions in (37) are eigenstates of Sz Positive and negative energy solutions u1, u2, u3, u4 Positive energy solutions u1, u2, v3, v4 Solution v3 and v4 satisfy Dirac equation since sign of reverses