PHYS 30101 Quantum MechanicsPHYS 30101 Quantum Mechanics

Dr Gavin Smith Nuclear Physics Group

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

Lecture 14

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. Spin

(algebra almost identical to orbital angular momentum algebra – except we can’t write down explicit analogues of spherical harmonics for spin eigenfunctions)

Commutation relations (plus two others by

cyclic permutation of x,y,z)

By convention we choose to work with eigenfunctions of S2 and Sz which we label α and β

So, the eigenvalue equations are:

Any general spin-1/2 wavefunction χ can be written as a linear combination of the complete set of our chosen eigenfunction set:

χ = a α+ b β

The coefficients a and b give the weighting and relative phases of the α and β eigenstates.

Normalization: a2 + b2 = 1

The wavefunction χ could be, for example, that of a spin-1/2 particle polarised in the x-direction (an eigenstate of Sx)

We now find the coefficients a, b for this state as an example

(there’s only two eigenfunctions in the set)

χ = a α+ b β

Eigenfunctions and eigenvalues of Sx, Sy, Sz described in this way:

RECAP: 4.2 Dirac notation

Dirac

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:

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)

Matrix representation:

Eigenvectors of Sx, Sy, Sz

Eigenfunctions of spin operators (from lecture 13)