Mon., Nov. 15:
#  hw#10 still due Friday...but see me if you have a real hardship related to the Math. Methods exam.
hw#11 will be assigned this Friday and due Wed. Nov. 24, but will be acceptable Mon. Nov. 29.
# Correction: In superfluid helium-3 the pairing mechanism is believed to involve the electron spins,
not the nuclear spins (except insofar as the fact that the nucleus is a fermion affects the symmetry
of the pair wavefunction).
# Note: When there is a supercurrent in a superconductor, the pair momentum, although nonzero,
is very small compared to the Fermi momentum. Therefore, while in principle one would expect
higher angular momentum components in the pair wavefunction (i.e. not just s-wave in a metallic
superconductor), this effect would be very small.
# Canonical quantization was discussed. See minutes from last Friday.
# Charge on a ring with magnetic flux: only the flux modulo the flux quantum Phi_0 is "felt" by
the charge. See minutes from last Friday.
# Otherwise free charge in a uniform magnetic field in z-direction:
motion in z-direction is free,  so classical trajectories are spirals.
Quantum mechanically we found using the Bohr-Sommerfeld semiclassical quantization condition
that the radius of the smallest orbit is r_0 = 3.63  10^-6 cm (B/1 Tesla)^-1/2.
If a charge is in a potential, e.g. the Coulomb potential of an atom, bound in an orbit much smaller
than r_0, then the magnetic field will only make a small correction. In this case part of the magnetic
Hamiltonian is much more important than the rest:

H = (p-(e/c)A)^2/2m = p^2/2m - (e/2mc)(p.A + A.p) + (e^2/2mc^2) A^2.

For a uniform magnetic field we can use the gauge A = B x r/2, for which H becomes

H = p^2/2m - (e/2mc) L.B +  (e^2B^2/8mc^2)(x^2 + y^2).

The term linear in B is of the form -mu.B, where mu = (e/2mc) L is the magnetic moment,
so gamma = e/2mc is the gyromagnetic ratio for orbital motion. This term is associated with
paramagnetism, since a system with a permanent dipole moment can lower its energy by
orienting the magnetic moment parallel to B. (If B is inhomogeneous, the system will
feel a force pulling it into the region where B is strongest.)

The term quadratic in B is always positive, and is associated with diamagnetism. (A system with
no permanent magnetic moment will have only positive energy in a magnetic field, hence will be
repelled from a region of strong field.)

# Let's compare orders of magnitude:
Set L = hbar, and x^2 + y^2 = r^2. Then, up to factors of pi and 2,
diamag. term / paramag. term = B r^2/Phi_0 ~ 10^-6 (Br^2/T-Angstroms^2).
For an atomic scale system at a B of one Tesla, the diamagnetic term is a million times
smaller than the paramagnetic term.
Comparing the paramagnetic term to the Coulomb energy in an atom on the other hand, we have
paramag term/ Rydberg =  B a^2/Phi_0, where a is the Bohr radius, which is again 10^-6 for
a one Tesla field. So the paramagnetic term is small for any laboratory fields. It is still important
however, mainly because it contributes to the Zeeman effect by the splitting of degenerate levels
according to the value of L_z.


Wed. Nov. 17:
# Explained paramagnetism and diamagnetism. Showed demonstration:
Copper Sulfate is paramagnetic, attracted to strong field region of magnet,
while Bismuth is diamagnetic, repelled from strong field.
# Paramagnetism, being from permanent dipole moments, is subject to thermal fluctuations.
Boltmann factor exp(- E/kT) =~ 1 - E/kT for temperatures high compared with E.
(The Bohr magneton has the value 0.67 K/T (i.e. Kelvin/Tesla), so at e.g. one Tesla
the magnetic dipole energy is very small compared to kT until the temperature
gets near 1K.) In this case, the magnetic response will be inversely proportional to temperature.
# Magnetic moments: We found the paramagnetic term -(e/2mc)L.B, so the orbital magnetic moment
of the charge is (e/2mc)L, so the orbital gyromagnetic ratio is (e/2mc). This can also be understood in
a classical model: mu = (1/2c)\int d^3r r x j. For a current loop of area A this is just IA/c,
which for a circle of radius r and orbit of frequency w is
(ew/2pi)(pi r^2)/c = (e/2c) w r^2 = (e/2mc) L, where L = mwr^2 is the angular momentum.
So we get the same gyromagnetic ratio. Note r drops out of the gyromagnetic ratio. Thus,
for an extended body rotating rigidly, if the charge to mass ratio of each ring of fixed radius
is the same, the gyromagnetic ration of the whole body will also be e/2mc. For an electron,
the orbital magnetic moment when L=hbar is thus minus
the Bohr magneton: mu_B = |e|hbar/2m_e c = 5.8 x 10^-5 eV/T = 0.67 K/T.
# It was discovered from spectroscopy that electrons also have spin, and that the spin gyromagnetic
ratio is g(e/2mc), with g=2. Thus the spin magnetic moment g(e/2mc)S = ehbar/2mc = -mu_B is also minus the
Bohr magneton (the g-factor cancels the 2 in the denominator of S=hbar/2).
The extra factor of 2 could not be understood from a picture of a spinning
ball of charge and mass with fixed charge to mass density ratio. The Dirac equation predicts precisely
this factor of 2. An experiment by Nafe, Nelson and Rabi (1947), using the molecular beam magnetic
resonance method discussed earlier in the semster, indicated that g deviates from 2 by something around
a fourth of a percent. Better experiments pinned it down to 2(1 + 1.19 10^-3), which was soon explained
by Schwinger based on a QED (quantum electrodynamics) calculation, which yielded
g = 2(1 + alpha/2pi + O(alpha^2)) = 2(1 + 1.16 10^-3 + O(alpha^2)),
where alpha = e^2/hbar c =~ 1/137 is the fine structure constant.
Experiments have now refined this down to g = 2.002 319 304 718 (564), and theory has accounted
for the first 7 digits, after which it has not been calculated (I think).
# The total magnetic moment of an electron is thus
mu_tot = mu_orb + mu_spin = (e/2mc)[L + g S] =~ (e/2mc)[L + 2 S].
# The spin magnetic moment of the proton is mu_p = 2.79 mu_N, where mu_N = ehbar/2m_p c is the nuclear magneton .
This is some 2000 times samller than the Bohr magneton, since m_p/m_e =~ 2000. Recall why it is inversely
proportional to the mass: for a given angular momentum, the ``angular velocity" of  a mass m is inversely
proportional to m, and it is the angular velocity, not the angular momentum, that determines the current and therefore
the magnetic moment.  The reason for the factor of 2.79 is that the proton has internal structure, and the
ratio of charge to mass density is not uniform.
# The spin magnetic moment of the neutron is mu_n = -1.91 mu_N. Although it is neutral, the neutron has internal
currents which produce a magnetic moment.  (The neutron is composed of (charged) quarks and  (neutral)
gluons. Although the number of quarks is not definite, it is primarily one up quark with charge 2/3 and two
down quarks each with charge -1/3 in units of e_0. The magnetic moments of the proton and neutron are in principle
calculable in QCD (quantum chromodynamcis), however this calculation is not feasible at present.)
# The deuteron is an s-wave, spin triplet bound state of proton and neutron. The magnetic moment of the deuteron
thus comes from the aligned proton and neutron spins, and is approximately given by mu_p + mu_n = 0.88 mu_N.
Experimentally, it is given by mu_d = 0.86 mu_N.

Fri. Nov. 19:
GAUGE INVARIANCE:
# The two sourceless Maxwell equations, div B=0 and curl E = (1/c) d_t B  (where d_t = partial wrt t),
imply that E and B are derivable from a scalar potential phi and a vector potential A:

E = -grad phi - (1/c)d_t A,             B = curl A,

Conversely, if potentials define E and B in this way, they automatically satisfy the sourceless Maxwell eqns.
Potentials (A', phi') will define the same E and B fields as (A, phi) if they are related by a gauge transformation:

A' = A + grad g ,    phi' = phi - (1/c)d_t g,

where the gauge parameter g(x,t) is an arbitrary function of space and time.
# To simplify notation here, let us adopt units with hbar=c=1.
# The Hamiltonian H = [p - eA]^2/2m + e phi is not gauge invariant, but the physics is. That is,
there exists a unitary transformation U such that

(i d_t - H') U = U  (i d_t - H) ,     namely, U = exp(i eg).

In other words, the gauge transformation of the potentials is accompanied by a transformation of the wave function,
to psi' = exp(ieg) psi. Then if psi satisfies the Schrodinger eqn with H, psi' will satisfy it with H'.
# The probability density psi*psi is unchanged by a gauge transformation. The conserved current in the
presence of a background magnetic field is modified in such a way that it is gauge invariant:

j = Re(psi* v psi), with v = (p - eA)/m.

# The gauge parameter g(x,t) is fixed just once, but since the charge q appears in the Hamiltonian, the appropriate
U = exp(iqg) for a given particle type depends on q.
# The coupling to electromagnetic fields makes quantum theory invariant under local phase transformations of the
wave function. In a sense, local phase symmetry implies the existence of electromagnetism. This symmetry in fact
implies conservation of charge, and it fixes the form of the coupling between the electromagetic potentials and the
charged particles. The dynamics of the electromagnetic field itself---i.e. the Maxwell equations with source terms---
is supplemental information. These equations however are the only equations consistent with Lorentz invariance
and gauge invariance that are second  order in derivatives of the potentials. This is easy to see with Lorentz-covariant
formalism, but I will not go into it here.
 # The notion of electromagnetic gauge covariance is the tip of an iceberg. One way to glimpse the iceberg is to
write (p_i - eA_i) = -i (d_i  - ie A_i) =: -i D_i,  and  (id_t - ephi) = i (d_t  + ie phi) =: iD_t.
The operators D_i and D_t are gauge-covariant derivatives, in the sense that D'_i psi' = U D_i psi, and
similarly for D_t, i.e. D U = U D. The gauge transformation of D is cancelled by the derivative of U.
The Schrodinger equation in tems of the gauge covariant derivatives is iD_t psi = -(1/2m) D_iD_i psi.
A differential eqaution constructed with gauge covariant derivatives will transofrm covariantly under gauge
transformations, so if satisfied in one gauge it will be satisfied in all gauges, i.e. it will be gauge invariant.
# In general relativity the role of the gauge transformations is played by general coordinate transformations,
and the equations are expressed in terms of a derivative operator that is is covariant with respect to these,
so that the equations are valid for all coordinate systems.
# In 1954 Yang and Mills invented a generalization of electromagnetic gauge invariance using matrix-valued
potentials and gauge transformations. Their motivation was that various symmetries among particles had been
suspected, so it seemed natural to group together certain particles into multiplets, which we can
think of as multicomponent wavefunctions Psi.  Yang and Mills were trying to construct a theory that would be
symmetric under local unitary transformations, Psi -> Psi' = U Psi, where now U is a unitary matrix which is
an element of some group. A special case of this is electromagnetism, where U is the 1x1 matrix exp(ieg).
The phases comprise the group called U(1).  The group U(n) is the n x n unitary matrices, while SU(n) is the
subgroup of those with unit determinant. The matrices do not commute, i.e. the group is non-abelian, which
requires a generalization of the gauge transformation law. Suppose we want the derivative D = d - ieA to be
gauge-covariant in the sense that D' U = U D. Then

(d - ieA') U =  U d - ieA' U + (dU) = U(d - i eA) = U d - ie U A, which implies that
-ieA' U + (dU) = -i eU A, so
A' = U A U^-1  - (i/e)  (dU) U^-1.

If U=exp(ieg) this reduces to the usual electromagnetic gauge transformation A' = A + e dg,
but for non-Abelian groups it is different. In particular, the gauge potential A must be matrix-valued.
It can be seen that the matrix i dU U^-1 is hermitian if U is unitary, so A should be hermitian. If U has
unit determinant, then in addition A should be trace-free. In addition to the gauge transformation rule,
Yang & Mills worked out the analog of Maxwell's equations, which turned out to necessarily be
a non-linear equation. The particles corresponding to the potential fields A are called gauge bosons, and are
analogous to the photon, but due to the non-linearity they interact directly with each other, unlike photons.
# Initially Yang-Mills theory looked intriguing, but problematic, since the gauge particles were, like
the photon, necessarily massless, since a mass term in their equation would violate gauge invariance,
and no extra massless particles were seen in nature.
Nevertheless the idea looked very nice, and eventually it turned out to be correct.
The standard model of particle physics has gauge group SU(3) x SU(2) x U(1).
The SU(3) is the QCD color gauge group. Since there are 8 independent 3 x 3 tracefree hermitian matrices,
there are 8 gauge fields, called gluons. The SU(2) x U(1) part is the electroweak group. The electromagnetic
U(1) is a combination of a U(1) subgroup of SU(2) and the explicit U(1) factor. The remaining three generators
are the charged W+ and W- and the neutral Z boson. The gluons are in a sense massless, however
due to the dynamical binding mechanism, called color confinement, gluons do not propagate freely.
The W and Z bosons on the other hand do propagate freely (until they decay), but they are massive.
This is possible, without violating gauge invariance, thanks to the phenomenon of
spontaneous symmetry breaking: one of the fields to which they couple, the Higgs field, has a vacuum
expectation value, which gives them an effective mass.