Relativistic Quantum Mechanics Lecture 10 Books Recommended: Relativistic Quantum Mechanics and Field Theory by Franz Gross Quantum Mechanics by Bransden and Joachain Advanced Quantum Mechanics by Schwabl Relativistic Quantum Mechanics by Greiner Quantum Mechanics II second course in Quantum Theory By Robin H Landau

Dirac Eq. in Central potential Dirac eq. in presence of spherical symmetric potential -------(1) In above we have four coupled equations Using spherical symmetry property can be reduced to two coupled differential equations

We use symmetry properties of the problem to reduce above equation two coupled eqns. Symmetries of Motion orbital angular momentum is ------(2) and spin angular momentum -----(3) L and S are not constant of motion because they do not commute with Hamiltonian --------- --------(4)

Total angular momentum is a constant of motion i.e., [J, H] = 0 Also is constant of motion. Parity operator defined by is also constant of motion -----------------(5) Conserved quantities

Consider solution of Dirac eq in the two component form i.e. ------(6) F and G have opposite parity and parity of overall state Is determined by upper component -----(7) Total angular momentum has form -----(8)

Upper and lower components of solutions can be expanded in terms of generalized spherical harmonics . Above harmonica are constructed from vector addition of spatial harmonic and spin half states States are represented as ------(9) Where f and g are function of radial coordinate only.

Above states are eigenstates of angular momentum and parity -----(10) We can write raising and lowering operators as -----(11)

states are written in terms of CG coefficients ---(12) For each j, the vaue of l will be j + ½ and j-1/2. Parity of will depend upon whether l is even or odd. We define new quantum no k to define the states ------(13)

are expressed in terms of k. In terms of k CG coefficients will be -----(14) ---------(15)

We now write the solutions in the form ------(16) We will now use the identities -----(17) Exercise: Prove eq (17). See QFT by Franz Gross

Reduction of Dirac equation to two equations: Assuming solution Dirac eq can be written as (using 1st Eq of 17) --(18) Using (9) and (17), we can write ----(19)

Rearranging (19) ---(20)

Hydrogen like atoms Potential will be -------(21) Using the functions --------(22) Eqns. In (20) can be written as ----(23)

When -----(24) Which shows that the bound states solutions will take the form ------(25)

We scale equations using -----(26) Using above Eq. (23) become ---(27) Where -------(28)

Eq. (27) are solved using power series method -----(29) Using (29) in (27) and equating (n-1)th power of ρ, -------(30)

Same eq will be obtained for ν when n= 0 -----(31) For non-trivial solution of above eq, we have ----(32)

Negative values will not be considered to avoid singularity. Thus ------(33) From Eq. (30), eliminating An-1 and Bn-1, we get -----(34) Using (34) in (30), we obtain recursion relation ----(35) Above eq will be used to find the Eigen value eq.

-------(36) Which is unacceptable solution and therefore, series must terminate. For some integer N ------(37) Using (28), ---------(38)

Solving for E, we get ------(39) Eliminating ν ---------(40) Which is exact sol of Dirac Eq for hydrogen atom.

Energy level scheme We define Using above and also k in terms of j, we get Which give energy of Dirac particle bound by Coulomb potential.

Dirac wave function where