Class examples from Chapter 5
5.2.2. Similar to Fig. 5.11, what is the probability distribution when there are still five
fermion balls but the total energy is fifteen?
Solution
When there are five fermion balls and the total energy is fifteen, there are
Solution, HW #12
ECE 462
9.3.3. Calculate the transmission matrix similar to Eq. (9.21 b) by considering
current flow on the right side.
Solution
%
%
I 2 = I 2in I 2 out = Tr cfw_ 2 ( A1 + A2 ) f 2 Tr cfw_ 2 A1 f1 + 2 A2 f 2
= Tr [ 2 A1 ] ( f 2 f1 )
= T
Solution, HW #12
ECE 462
9.3.3. Calculate the transmission matrix similar to Eq. (9.21 b) by considering
current flow on the right side.
Solution
%
%
I 2 = I 2in I 2 out = Tr cfw_ 2 ( A1 + A2 ) f 2 Tr cfw_ 2 A1 f1 + 2 A2 f 2
= Tr [ 2 A1 ] ( f 2 f1 )
= T
Solution, HW #11
ECE 462
9.1.1 Develop an expression for the transmission in the single energy channel,
Eq. 9.6, by starting from the right side of the channel
Solution
Go back and look at the current flowing at the left partition.
q
I2 = 2 ( f2 N ) .
h
O
HW #9
Solution, ECE 462
7.2.1. A particle is in the ground state of a 10 nm infinite well. A magnetic field of 10
Tesla in the z direction is turned on. What is the difference in energy if the
particle is spin up or spin down?
Solution
The expected value
Solution, HW #8
ECE 462
7.1.1. Write the X and Y spinors, ( x ) , ( y ) as superpositions of the up and down
spinors, + and .
Solution
1
( x) =
( + + ) ( y ) = 12 ( + + i )
2
7.1.2. Find a spinor that has angles = 45o and = 30o on the Bloch sphere.
cfw_R
Solution, HW #7
ECE 462
6.1.1. Using the program se6_1.m, change to a periodic potential with spikes at +1 meV.
Are the results any different? Run this and show the results.
Solution
No, the results are the same.
o
6.1.3. If the lattice constant is 50 A ,
Solution, HW #6
ECE 462
5.2.3 Consider the system below which has six fermion particles. We include spin
degeneracy, so up to two particles can be in any given state. The example below is for E
=6. Find the probability of occupation if E = 8. (Hint: Wheth
Solution, HW # 5
ECE 462
4.3.3
Look at the matrix A:
2 1 i
A = 1 3 i
i i 2
i. Is this matrix Hermitian?
ii. Find the eigenvalues and corresponding eigenvectors. cfw_You probably want to use
MATLAB to do this!
iii. Find the matrices that will transfor
Solution, HW #4
3.4.1. Using the program se3_2.m, calculate the transmission through the following
barrier:
Figure P3.4.1 Each of the barriers is 0.15 eV high and 0.6 nm wide. The barrier centers
are 4 nm apart.
Hints: I would suggest first repeating the
Solution, HW #3
ECE 462
3.1.1. Prove the time-shifting theorem. It comes about directly from the definition of the
Fourier transform in Eq. (6.1.2).
Solution
F
( f ( t t ) ) = f ( t t ) e dt
= t + t0
it
o
0
=
f ( ) e
i ( + t0 )
d = eit0 F ( )
3.1.2. Fin
Solution, HW #2
ECE 462
1.3.1. Add the calculation of the expected value of position x to the program
se1_1.m. It should print out on the graphs, like the kinetic and potential energy
expected values. Show how this value varies as the particle propagates.
Solution, HW # 1
ECE 462
1.1.1
Look at Eq. (1.1.2). Show that h / has units of momentum.
Solution
2
h J s kg m s kg m
=
=
=
= [ p]
2
m
m s
s
1.1.2
Titanium has a work function of 4.33 eV. What is the maximum wavelength of
light that I can use to indu
Solution, HW #13
ECE 462
10.1.1. A very important canonical structure in quantum mechanics is the Harmonic
Oscillator, which has the potential
1
12
2
U HO = m0 x 2 = k0 x 2 .
2
2
2
Eref
k0 = m0
h
Find approximations to the first three eigenfunctions an