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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
and spin angular momentum
------(2)
-----(3)
L and S are not constant of motion because they do
not commute with Hamiltonian
--------- --------(4)
Total angular momentum
motion i.e., [J, H] = 0
Also
is a constant of
is constant of motion.
Parity operator defined by
of motion
is also constant
-----------------(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
or odd.
will depend upon whether l is even
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