1. PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) j.billowes@manchester.ac.uk These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 Lecture 21

2. 6. The hydrogen atom revisited - Reminder of eigenfunctions, eigenvalues and quantum numbers n, l, m l of hydrogen atom. 6.1 Spin-orbit coupling and the fine structure. 6.2 Zeeman effect for single electron atoms in (a) a weak magnetic field (b) a strong magnetic field 6.3 Spin in magnetic field: QM and classical descriptions

3. Plan: Include coupling of orbital and spin angular momenta in Hamiltonian for hydrogen atom L S

4. The shift in energy of a state is the eigenvalue of the spin-orbit Hamiltonian: l=1, s=1/2 j=3/2 (4 states) j=1/2 (2 states) Aħ 2 /2 -Aħ 2 The energy centroid is unchanged: 4 X A/2 = 2 X A m l =+1, 0, -1 m s =+1/2, -1/2 (6 states) -μ.B

5. 6.2(a) Weak-field Zeeman effect L S L and S remain coupled to J. Classically J precesses slowly around field B, keeping J z = M a constant B 6.2(b) Strong-field Zeeman For electron, B is much greater than the field it ”sees” due to its orbital motion. S and L independently precess around B keeping m s and m l constants of motion B mlml msms S L

6. Landé g-factor

7. The state l=1, s=1/2 j=3/2 j=1/2 Aħ 2 /2 -Aħ 2 g J = -4/3 g J = -2/3 (Spin-orbit splitting)

8. (-1,+1/2) (+1,-1/2) Strong field Weak field Zeeman structure for l = 1, s = 1/2 orbital

9. For single-electron atom in an external magnetic field B applied along z-axis the full Hamiltonian is H = H 0 + H so + H mag P 2 /2m + V A/2(J 2 -L 2 -S 2 ) -µ B /ħ (g l L z + g s S z ) B In strong external magnetic field this term is much greater than the spin-orbit interaction H so (which we now ignore). H mag does not commute with J so eigenfunctions are no longer x

10. 6. The hydrogen atom revisited - Reminder of eigenfunctions, eigenvalues and quantum numbers n, l, m l of hydrogen atom. 6.1 Spin-orbit coupling and the fine structure. 6.2 Zeeman effect for single electron atoms in (a) a weak magnetic field (b) a strong magnetic field 6.3 Spin in magnetic field: QM and classical descriptions

11. 6.3 Classical and QM description of precession B B L θ dLdL dφdφ

12. 6.3 QM description of classical precession Consider a state with angular momentum L in an external field B H mag = -g l μ B l z B Eigenstates are |l, m z > These are stationary states. L points with equal probability everywhere on the surface of the cone. Time evolution of a state (section 1.6(a)) is | Ψ(t) > = e -iEt/ħ |l, m z > The phase factor cannot be directly measured. No precession observed. B

13. Matrix representation: Eigenvectors of S x, S y, S z Eigenfunctions of spin-1/2 operators (from lecture 13) 4.3.3 Example: description of spin=1 polarised along the x-axis In Dirac notation: is Now consider a state with angular momentum L polarised along the x -axis at t=0 in a magnetic field B applied along the z -axis B x

14. Energies of |l, m z > states are E = +ħω, 0, -ħω where ω = -g l μ B B / ħ (which actually equals the classical Larmor precession frequency) Time evolution of the initial state is We are now able to observe interference between the different phase factors – these are the “quantum beats” discussed in section 1.6(b) of this course. E = -μ.B