Download Lecture 12

PHYS 30101 Quantum Mechanics
Lecture 12
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
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