Download L z

PHYS 30101 Quantum Mechanics
Lecture 11
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: www.man.ac.uk/dalton/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.
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
Add this to your notes:
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.
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 Harmonics
Representation (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.