PHYS 30101 Quantum MechanicsPHYS 30101 Quantum Mechanics

Dr Sean Freeman Nuclear Physics Group

These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101

Lecture 15

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.

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

Recap: 4.3 Matrix representations in QM

We can describe any function as a linear combination of our chosen set of eigenfunctions (our “basis”)

Substitute in the eigenvalue equation for a general operator:

Gives:

Recap: 4.3 Matrix representations in QM

We can describe any function as a linear combination of our chosen set of eigenfunctions (our “basis”)

Substitute in the eigenvalue equation for a general operator:

Gives:

Multiply from left and integrate:

(We use )

And find:Exactly the rule for multiplying matrices!

Equation (1)

Pauli Spin Matrices:

4.3.2 Matix representations of Sx, Sy, Sz

Sx = ½ħ σx ; Sy = ½ħ σy ; Sz = ½ħ σz

Matrix representation: Eigenvectors of Sx, Sy, Sz

Eigenfunctions of spin-1/2 operators

4.3.3 Example: description of spin=1 polarised along the x-axis

In Dirac notation:

is

The Stern-Gerlach apparatus

The Stern-Gerlach apparatus

z

x

1 1/2 1/4 1/8

Unpolarised

Measure Sz

Select mz=+1/2 Measure Sx

Select mx=+1/2 Measure Sz

Select mz=+1/2

y

Successive measurements on spin-1/2 particles