1. Perturbation Theory Lecture 2 continue Books Recommended:
Quantum Mechanics, concept and applications by
Nouredine Zetili
Introduction to Quantum Mechanics by D.J. Griffiths
Cohen Tanudouji, Quantum Mechanics II
Introductory Quantum Mechanics, Rechard L. Liboff

2. Fine Structure of Hydrogen Atom
Hamiltonian for H-atom
------(1)
Above eq. have KE and PE interaction between
Proton and electron.
Energy of nth state
---(2)

3. Energy spectra of H-atom using Eq. 2

4. Eq. (2) is not whole story about H-atom.
This energy have some coorection and that lead to
Fine Structure of Hydrogen Atom.
This correction is small by a factor of α2 and can
be treated perturbation theory.
Smaller then this is Lamb Shift.
Smaller than Lamb
Shift is Hyerfine
slitting.

5. Fine structure include :
Relativistic corrections and spin orbit coupling.
Relativistic Corrections:
KE term in Eq. (1) is (classical exression)
----(3)
In operator form
(4)

6. In relativistic theory KE will be
---(5)
In terms of relativistic momentum KE will be
----(6)
Where now p is
(7)

7. Expanding Eq. (6) in non-relativistic limit
-----(8)
Small relativistic correction to classical KE is given by
----(9)
Motive is to find correction due to above
Hamiltonian using perturbation theory

8. Hydrogen atom is degenerate system but can use
Non-degenerate perturbation theory.
Recall the theorem
In present case L2 and Lz are such operators. These
Commute with p4 and their eigenstates can be
treated as good states.

9. Applying 1st order perturbation to eq (8)
---(10)
From Schrodinger eq (for unperturbed state)
----(11)
Using (11) in (10)
------(12)

10. For H-atom
Using above in (12)
-----(13)
Where
----(14)
a is Bohr radius.

11. Using (14) in (13)
------(15)
Using
In above, we get
----(16)
Note that relativistic correction is smaller than En by
a factor En/mc2 ~ 2*10-5

12. Spin Orbit Interactions
It arise because of interaction of electron
Spin magnetic moment with the magnetic
Field due to orbital motion of proton.

13. Hamiltonian corresponding to spin-orbit interaction
----(17)
Where μ is Spin magnetic moment of electron.
Magnetic field due to proton:
Where
Orbital angular momentum of electron

14. Thus,
----(18)
Also, electron spin magnetic moment will be
-----(19)
Eq. (17) become
------(20)

15. Eq. (20) is need to correct as electron is accelerating
aroud nucleus and this lead to Thomas precession .
This lead to a factor ½ in Eq (20) and finally the
Spin orbit interaction Hamiltonian will be
---(21)
Now we need to find the corrections to energy
due to above interaction term.

16. Note the following
Thus, Hamiltonian (Eq.(21)) do not commute with
L and S.
Now, note these
Thus, Hamiltonian (21) commute with L2, S2 and J.

17. Eigenstates of Lz and Sz are not good states to use
in perturbation theory.
However, Eigenstates of L2, S2, J2 and Jz are good.
----(22)
Eigen values of L.S
----(23)

18. Also
----(24)
Using (23) and (24), expectation value of (21) is,
----(25)
Which is correction due to spin-orbit interaction.

19. Adding (16) and (25), we get total correction due to
fine structure
------(26)
Total energy of H-atom including above corrections
----(27)

20. Fine structure of H-atom

21. Fine structure formula by solving Dirac equation
Exercise: Obtain equation (27) from above.

22. Zeeman Effect: Phenomenon of splitting of energy
levels of atoms in presence of external magnetic
Field.
For single electron (H atom)
---(1)
Where
--(2)

23. Using (2) in (1)
----(3)
Splitting depend upon strength of external MF
compared to internal field.
Estimate for internal field (see last section):

24. If , fine structure dominated
And we have Weak Zeeman effect. In this
case eq (1) will have small contribution
And can be treated erturbatively
If , we have strong Zeeman Effect.

25. Week field Zeeman Effect:
First order correction gives
---(4)
Recall n, l, j and mj are good quantum numbers.
Also
---(5)

26. In above, expression in bracket is called
Lande g-factor gJ

27. Considering M.F. Along Z-axis, we have
--(6)
Where
Total energy is now sum of contribution from
Fine structure and eq (6) resulting from weak
External field.

28. For example: For ground state
n = 1, l = 0, s = ½, j = ½ , mj = ±½, and gJ = 2
Thus, Energy will be
Which shows energy level
will split corresponding to
mj = ±½.

29. Strong Zeeman Effect
Unperturbed energy
First order correction due to fine structure