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: www.man.ac.uk/dalton/phys30101

Lecture 11

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.

RECAP: 3. Angular Momentum

L = R x P (I’m omitting “hats” but remember they’re there)

Thus

Lx = Y Pz – Z Py and two similar by cyclic change of x, y, z

We used those to show

[ Lx, Ly] = i ħ Lz and two similar by cyclic change of x, y, z

Since the operators for the components of angular momentum do not commute, there is NO set of common eigenfunctions for any of the pairs of operators.

Thus a state of definite eigenvalue Lz can not have definite values for either Lx or Ly.

Add this to your notes:

Today:

Using [ Lx, Ly] = i ħ Lz and two similar by cyclic change of x, y, z

We will show

[ L2, Lx] = [ L2, Ly] = [ L2, Lz] = 0

Thus there exists a common set of eigenfunctions of L2 and Lx

And there exists a common set of eigenfunctions of L2 and Ly

And there exists a common set of eigenfunctions of L2 and Lz

By convention we usually work with the last set of eigenfunctions.

NOTE: we can always describe a state which is an eigenfunction of, say, Ly by a linear combination of the Lz eigenfunctions.

Also Today:

3.1 Angular momentum operators in spherical polar coordinates

Using

And the unit vector relationship

We will show

And we won’t show but will be prepared to accept that:

Continuing:

3.1 (continued) Eigenfunctions and eigenvalues of L2 and Lz – the Spherical Harmonics

3.2 Finding eigenfunctions and eigenvalues in 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

Spherical HarmonicsRepresentation (dark and light regions have opposite sign) and explicit expressions.

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