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PHYS 30101 Quantum Mechanics
Lecture 17
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
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
4.4 Measurement of a spin component
Magnetic moments (a) due to orbital angular momentum
r
e
Electron in orbit produces a
magnetic field (like bar magnet)
and therefore has a magnetic
dipole moment:
μl = gl l μB
μl = gl l μB
μB = eħ/2me
(the Bohr magneton)
gl = -1 (gyromagnetic ratio or g-factor)
l= 1, 2, 3…
4.4 Measurement of a spin component
Magnetic moments (a) due to orbital angular momentum
Electron in orbit produces a
magnetic field (like bar magnet)
and therefore has a magnetic
dipole moment:
r
e
μl = gl l μB
μl = gl l μB
μB = eħ/2me
(the Bohr magneton)
gl = -1 (gyromagnetic ratio or g-factor)
l= 1, 2, 3…
(b) Due to intrinsic spin
μs = gs s μB gs = -2 exactly from Dirac theory
gs = -2.0023192 (inc. QED corrections)
μs = gs s μB
s=½
thus
μs = - μB is a good approximation
Nuclear magnetic moments
(10-3 smaller than atomic moments)
Proton s = ½
μp = g s s μN
Nuclear magneton μN
Neutron s = ½
= eħ/2mp
gs = +5.58
(not a Dirac point
particle but must have
substructure (quarks)
μn = g s s μN
gs = - 3.83
(also not a Dirac point particle.
Quark model wavefunctions can
explain moment)
The Stern-Gerlach
apparatus


Rae,
Eq. (5.30)
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
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.
Coupling two angular momenta
When M (= m1 + m2) is a constant of motion, m1 and m2 are
not well defined
S
L
We shall try and follow this convention:
Capitals J, L, S indicate angular momentum vectors
with magnitudes that can be expressed in units of
ħ:
L2 = l ( l + 1) ħ2
Lower case j, l, s indicate quantum numbers that
are integer or half-integer:
l = 0, 1, 2, 3…
s = 1/2
j = 1/2 , 3/2, 5/2
Lower case vectors j, l, s indicate vectors whose
components along a quantization axis are integer or halfinteger values (ie not in units of ħ).