1. 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

2. 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 .

3. 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)

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

5. 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)

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

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

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

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

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

11. 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.

12. 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

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

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

15. 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

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

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

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

19. 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 .

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

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

22. 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

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

24. 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)

25. 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