Perturbation Theory Lecture 2 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

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)

Energy spectra of H-atom using Eq. 2

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.

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

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

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.

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

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

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

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

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

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

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.

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.

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)

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

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

Fine structure of H-atom

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

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

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

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.

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

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

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.

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.

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