PHYS 30101 Quantum MechanicsPHYS 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 12

Syllabus

1. Basics of quantum mechanics (QM) Postulate, operators, eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent Schrödinger equation, probabilistic interpretation, compatibility of observables, the uncertainty principle.

2. 1-D QM Bound states, potential barriers, tunnelling phenomena.

3. Orbital angular momentum Commutation relations, eigenvalues of Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical harmonics Yl,m.

4. Spin Noncommutativity of spin operators, ladder operators, Dirac notation, Pauli spin matrices, the Stern-Gerlach experiment.

5. Addition of angular momentum Total angular momentum operators, eigenvalues and eigenfunctions of Jz and J2.

6. The hydrogen atom revisited Spin-orbit coupling, fine structure, Zeeman effect.

7. Perturbation theory First-order perturbation theory for energy levels.

8. Conceptual problems The EPR paradox, Bell’s inequalities.

Last lecture we found operators for L2 and Lz in spherical polar coordinates:

Eignefunctions could be found by separation of variables:

The eigenfunctions are called Spherical Harmonics. The eigenvalues are:

TODAY:

3.2 Finding eigenfunctions and eigenvalues is a more abstract way using the ladder operators.

3.3 We show states of definite eigenvalue Lz have axial symmetry.

3.4 Coefficients connected to the ladder operators

4. Spin – intrinsic spin= ½ħ angular momentum of electron, proton and neutron.

Using the ladder operators we will show:And find the commutation relations:

So starting with the eigenvalue equation

φ = eigenfunction, β =eigenvalue (=mħ)

we can find other eigenfunctions with eignevalues one unit of ħ different from β i.e. (m+1)ħ and (m-1)ħ

Possible orientations of the l=2 angular momentum vector when the z-component has a definite value.

4. Spin

4.1 Commutators, ladder operators, eigenfunctions, eigenvalues

4.2 Dirac notation (simple shorthand – useful for “spin” space)

4.3 Matrix representations in QM; Pauli spin matrices

4.4 Measurement of angular momentum components: the Stern-Gerlach apparatus