Mon., 11/1:
Prof. Xiangdong Ji lectured on:
# Reduction of 2 body problem to 1 body problem with reduced mass.
# Decomposition of total angular momentum into orbital of the center
of mass and
relative to the center of mass: J_tot = J_cm + J_rel.
Each of J_cm and J_rel satisfy the angular momentum commutation relations
by themselves,
and they commute with each other.
# The spin of the proton, for example, refers to the total angular
momentum relative to the center of mass.
It is composed of many contributions, both orbital and spin of the
quarks, and of the gluons.
Really, one has a quark-gluon field state, without any definite numnber
of constituent particles.
# Identical particles come in two varieties: bosons and fermions, for
which the many particle state is
respectively symmetric and antisymmetric.
Wed., 11/3
# IDENTICAL PARTICLES:
Recall composite systems in quantum mechanics: Hilbert space of AB
is H_AB = H_A x H_B,
where x stands for tensor product. If A and B are copies of a single
particle, then we can meaningfully interchange the factors with a permutation
operator: P_12 |a>|b> := |b>|a>. P_12 is unitary.
For n particles there are n! permutations, which can all be expressed
as combinations of interchanges P_ij. Particles are indistinguishable
or identical if all observables are unchanged under permutation
of the particles.
That is, P* O P = O for every observable O and every
permutation P.
# If particles are identical then the state P|v> is indistinguishable
from the state |v> by any observable. Logically it is possible that nevertheless
P|v> is not the same as |v>, however we have found that
in nature this never happens.
(The possibility that P|v> and |v> are not the same state goes
under the name of parastatistics.
Research into this possibility was pioneered by Prof. O.W. Greenberg
(with A. Messiah),
who is in our department.)
Thus P|v> = u |v> for some phase factor u.
If P is an interchange P_ij, then since (P_ij)^2=1, u must be +1 or
-1.
Moreover, since any permutation can be composed of interchanges, u
must always be +1 or -1.
Moreover, u must be the same for all interchanges. This follows since
P_12 P_23 P_12 = P_13,
so u_12 u_23 u_12 = u_13, so u_23 = u_13, etc.
If u is +1 for the interchanges it is +1 for any permutation.
Identical particles satisfying this condition are called bosons.
If on the other hand u is -1 for interchanges, it is the sign of the
permutation (-1)^P for any permutation P, which is +1 or -1 if
the permutation is even or odd (i.e. an even or odd number of nterchanges
respectively). Such particles are called fermions.
# Fermions obey the Pauli exclusion principle: no two
can be in the same quantum state.
This follows from the antisymmetry under exchange P_ij v(...i...j...)
= - v( ...j...i...), so v(...i...i...) = 0.
This affects the counting of states for fermions, i.e. the statistics.
One says fermions obey Fermi-Dirac statistics, and bosons obey Bose-Einstein
statistics.
# The possibility of particles with u neither +1 nor -1 is called anyons.
They can only exist if H_AB is not H_A x H_B. This can happen for
non-elementary excitations in condensed matter systems confined to two
spatial dimensions. Perhaps more on this later.
# Spin-statistics connection: particles with integer spin are
bosons, while those with half-integer spin are fermions. This can be derived
from relativistic quantum mechanics. It is odd that relativity should be
needed, and in fact I think it is not.
Thus for example electrons, protons, and neutrons are fermions, while
photons (spin-1) are bosons.
# A system composed of an even/odd number of fermions and
any number of bosons is a boson/fermion
(and has half/whole integer spin). For example, a hydrogen atom
is a boson, a helium-4 atom is a boson, while a helium-3 atom is a fermion.
Fri., 11/5:
# 4He and 3He behave very differently
at low temperature: 4He a superfluid at 2.2K, 3He a superfluid at around
2mK.
# Homonuclear diatomic molecules: e.g. 12C or 16O are spin-0 bosons.
Nuclear wavefunction
is therefore symmetric under pair interchange. Center of mass
part is automatically symmetric,
so relative part must also be. Pair interchange reverses the relative
coordinate r1-r2, so the
relative wave function must be even under parity, so only even angular
momenta are allowed.
This affects the rotational spectra. If nuclei have spin then it is
the combined spin and space
state that must be symmetric or antisymmetric for bosons or fermions
respectively. For example,
the hydrogen molecule.
# Particle with space and spin-1/2 degrees of freedom: Hilbert sapce
= H_space x H_spin.
States: |v> = \int dx |x>(v_u(x) |u> + v_d(x) |d>), where |u>
and |d> are spin up and down states.
Sometimes think of as a two-component wavefunction (v_u(x), v_d(x)).
# Two spins-1/2: {uu, ud, du, dd}= {uu, (ud +du)/sqrt[2], dd} + {(ud
-du)/sqrt[2]}.
The first three are symmetric under interchange, the last antisymmetric.
The three transform
into each other under rotations, and comprise a spin-1 representation
of the rotation group,
while the last is invariant, so carries a spin-0 rep. The three are
called the triplet rep., while
the one is the singlet. (We know the three form a spin-1 rep
since there are three of them,
but we can also see it explicitly: the largest J_z = S_1z + S_2z is
hbar, on uu:
(S_1z + S_2z)|uu> = hbar |uu>. The other states are obtained by applying
the lowering op. J_-.)
# Two-proton states: |v> = \int dx1 dx2 |x1x2> (sum_{m=-1,0,1}
v_m (x1,x2) |1m> + v_s(x1,x2) |00>).
The wave function v_s(x1,x2) must be symmetric under interchange,
since |00> is antisymmetric,
so only even angular momenta about the center of mass are allowed,
while v_m(x1,x2) must
be anti-symmetric, so only odd relative angular momenta are allowed.
The singlet state of the
molecule is called para-hydrogen, while the triplet state is
called ortho-hydrogen. There are
three times as many of the latter as the former.
# The nuclear spins interact only very weakly with each other or with
electrons, so a gas
of H_2 molecules prepared as the ground state parahydrogen (with l=0)
will, after being
warmed up to room temp, maintain its para form for weeks.
# Degenerate Fermi gas: fermions packed as densely as allowed
by the exclusion principle.
N fermions in a box of volume L^3 have density n = N/L^3. What is the
relation between
this and the maximum wavevector? Allowed wavevectors with periodic
boundary
conditions are 2pi/L(n_x,n_y,n_z). (The bc's will not affect bulk properties
if the are enough
particles.) Thus the volume in k-space per state is (2pi/L)^3,
so the number of states in a sphere
of k-space up to a radius k_F, allowing for two spin states of a spin-1/2
particle,
is 2 x 4pi (L/2pi)^3 \int_0^kF dk k^2 = (k_F L)^3/3pi^2. If the
N particles fill the states, up to the
states with |k|=k_F, the Fermi wavevector, the density is therefore
n =k_F^3/3pi^2, or
k_F = (3pi^2 n)^1/3. Note that apart from the 3pi^2, this follows from
dimensional analysis:
n^1/3 is the only inverse length of relevance. The volume per particle
is on the order of the
Fermi wavelength 2pi/k_F.
# A degenerate Fermi gas has a degeneracy pressure, since if
the gas is squeezed in space,
k_F goes up, so the kinetic energy goes up. This is what holds up a
white dwarf star (with
electron degeneracy) or a neutron star (with neutron degeneracy).
# So far we discussed a homogeneous Fermi gas. If the gas is
in a potential V(x), then
the energies of the states are not just the kinetic energy. But if
the typical wavelength is
uch shorter than the length scale for changes of V(x), we can define
a local Fermi wavevector
k_F(x). If the situation is static, then we must have [hbar^2 k_F(x)^2/2m
+ V(x) ]= constant,
since the electrons at the top of the Fermi sea at different spatial
locations will flow until
this energy is equalized. This idea is used in the Thomas-Fermi
model of many electron
atoms, and is systematically improved in the density functional
approach to molecular
structure that won the Nobel prize last year. It is also used in the
thoery of metals, and in many
other settings.