Download Lecture 15

PHYS 30101 Quantum Mechanics
Lecture 15
Dr Sean Freeman
Nuclear Physics Group
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.
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:
Equation (1)
Gives:
Multiply from
left and integrate:
)
(We use
And find:
Exactly the rule for
multiplying matrices!
4.3.2 Matix representations of Sx, Sy, Sz
Sx = ½ħ σx
;
Pauli Spin Matrices:
Sy = ½ħ σy
;
Sz = ½ħ σz
Eigenfunctions of spin-1/2 operators
Matrix representation: Eigenvectors of Sx, Sy, Sz
4.3.3 Example: description of spin=1 polarised along the x-axis
is
In Dirac notation:
The Stern-Gerlach
apparatus
The Stern-Gerlach apparatus
z
y
x
Successive measurements on spin-1/2 particles
1
Unpolarised
1/2
1/4
Select
mz=+1/2
Measure Sz
Select
mx=+1/2
Measure Sx
1/8
Select
mz=+1/2
Measure Sz