1.1 The function and its constraint are
f (x, y )
3x2 4xy + y 2 ,
C (x, y )
3x + y = 0.
According to the method of Lagrange multipliers, we have
(f (x, y ) C (x, y ) = 6x 4y 3 = 0,
(f (x, y ) C (x, y ) = 4x + 2y = 0,
(f (x, y ) C
Physics 831 Quiz #2 - Friday, October 2
1. Consider a MASSLESS( p = p) three-dimensional gas of spinless bosons which is kept at
temperature T . Solve for the density of Bose Condensation, c (T ). You can set h and c = 1
to save ink, and express sums
Physics 831 Quiz #1 - Friday, Sep. 5
1. Consider 2 non-interacting electrons populating two single-particle levels of energy and .
(a) What is the average energy for T = 0?
(b) What is the entropy for T = 0?
(c) What is the average energy as T ?
Physics 831 Quiz #3 - Wednesday, Oct. 14
1. A molecule of mass m has internal excitations consistent with that of a TWO-DIMENSIONAL
harmonic oscillator with tightly packed levels, h < T . Initially, a gas of such molecules
is at temperature Ti before expa
Physics 831 Quiz #4 - Monday, Nov. 2
1. Consider a gas of non-relativistic one-dimensional zero-temperature spin-1/2 fermions of mass
m, lling up all states with momenta, pf < p < pf . The system is also conned to a region,
L < x < L. This gives a phase s
Physics 831 Quiz #7 - Friday, Oct. 17
1. Assume the free energy for a complex eld in ONE dimensions is given by:
A |2 + |x |2 .
Dene the correlation as
(x) (0)(x) .
Fourier transforms in one dimensions are dened by:
dx eikx (x),
Physics 831 Quiz #5 - Monday, Nov. 9
1. A two-dimensional lattice is made of coupled oscillators that are constrained to move only
within the two-dimensional plane of the lattice. The oscillators have mass m, the speed of
sound is cs and the Debye frequen