1
Phys 460 Problem Set #6
Fall 2012
Problem 1
(a) The total energy is composed of the kinetic energy and the potential energy:
X 1
1
2
2
M vs + C(us )
E =
2
2
s
2
N 1
1
1 X dus
1
1X1
2
= M
+ C (u1 u0 ) +
C (us us1 )2 + (us+1 us )2 + C (uN uN 1 )2
2
dt
2
1
Phys 460 Problem Set #4
Fall 2012
Problem 1
(a) The solutions for Kronig-Penny model with periodic delta potential satisfy
P
sin Ka + cos Ka = cos Ka,
Ka
(1)
and to nd the energy of the lowest energy band, we look for the smallest K solved above.
When k
PHY460: Problem Set #1 Due: Friday 9/19 by 5pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
Please look over the homework and grading policies in the syllabus before you begin.
1. Let us define an orthonormal basis of quantum states c
PHY460: Problem Set #5 Due: Friday 11/21 by 5pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
1. In a particular semiconductor there are 1019 donors/m3 with an ionization energy (i.e., gap between donor
levels and conduction band) Ed o
1
Phys 460 Problem Set #3
Fall 2012
Problem 1
(a) For nearly free elections on a 2d square lattice, the kinetic energy of the electrons is
~2 k 2 /2m. The wave number at the corner of the first Brillion zone is 2k higher than
the wave number at the midpoi
PHY460: Problem Set #4 Due: Tuesday 11/4 by 5pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
1. Consider electrons of mass m in a periodic potential in one dimension given by
X
V (x) = aU
(x na)
(1)
n
We want to find the equation whic
1
Phys 460 Problem Set #1
Fall 2012
Problem 1
(a) Since there is no interaction for the three identical particles, we may set onsite energy as
Ea and shift energy of particles by Ea . Then the hopping part of the Hamiltonian is
0 t 0
0 1 0
X
Ht = t
|iihj|
PHY460: Problem Set #6 Due: Friday 12/12 by 5 pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
1. Monatomic linear lattice: Consider a longitudinal wave
us = u cos(t sKa)
(1)
which propagates in a monatomic linear lattice of atoms M ,
1
Phys 460 Problem Set #2
Fall 2012
Problem 1
(a) (b): If we assume there is one atom in each unit cell, a block of the lattice looks like Fig.
(1).
FIG. 1: Problem 1
(c): (a) has C2 symmetry and (b) has C4 (which contains C2 ) symmetry. The rotation cent
PHY460: Problem Set #3 Due: Friday 10/17 by 5pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
1. Free electron energies on the square lattice in the empty lattice approximation
(a) Show for a simple square lattice (two dimensions) that
PHY460: Problem Set #2 Due: Friday 10/3 by 5pm in Yellow PHY460 homework box between
Loomis and MRL on 2nd Floor
1. Draw a 5x5 lattice of unit cells for a 2d lattice with Bravais lattice vectors:
(a) ~a1 = a
x, ~a2 = 2a
y
(b) ~a1 = a
x + a
y , ~a2 = a
x a
1
Phys 460 Problem Set #7
Fall 2012
Problem 1
(a) The magnetic eld B(x) inside the plate and parallel to it. The penetration equation
2 d2 B/dx2 = B, where is the penetration depth, has the solutions of form B(x) = Aex/ +
Bex/ . Near the edge of the plate
1
Phys 460 Problem Set #5
Fall 2012
Problem 1
In low temperature, the concentration of conduction elections can be approximated as in Eq.
(53) in Kittel:
E
d
1/2 2kB T
n (n0 Nd )
d
(
with n0 2
me kB T
2~2
)3/2
and Nd is the concentration of donors. For Nd