EE211 Spring 2025
Homework - Problem Set #2
Due: March 31th, 08:59 am
Please scan your homework and upload it on the KLMS website.
Problem 1. A Principle of quantum mechanics (8 pts)
The wavelength of green light is 580 nm. If an electron has the same wavelength,
(a) Determine the electron velocity and momentum.
(b) Repeat part (a) for blue light with a wavelength of 470 nm.
Problem 2. Applications of Schrodinger’s Wave equation (10pts)
For a 1-dimensional infinite potential well for an electron, where the potential energy is given
by V(x) = 0 for −L/2 < x < L/2, and V(x) = ∞ elsewhere. By solving Schrodinger’s wave
equation,
(a) Find the wave function of the electron for the potential well (show the major
derivation steps and the normalized constants for the wave function),
(b) Determine the energy states of the electron.
Problem 3. Applications of Schrodinger’s Wave equation (12pts)
(a) An electron in a potential well of thickness 2nm, with infinitely high potential barriers
on either side. It is in the lowest possible energy state in this well. What would be the
probability of finding the electron between 0.3 and 0.5 nm from one side of the well?
(b) Consider an electron in a three-dimensional cubic box of side length 𝐿𝑧 . The walls of
the box are presumed to correspond to infinitely high potentials. Find an expression
for the allowed energies of the electron in this box. Express the result in terms of the
lowest allowed energy, 𝐸1∞ , of a particle in a one-dimensional box.
(c) State the energies and describe the form of the wavefunctions for the four lowest
energy states.
(d) Are any of these states degenerate? If so, say which, and also give the degeneracy
associated with any of the degenerate eigenenergies you have found.
Problem 4. Applications of Schrodinger’s Wave equation (5pts)
Electrons with energy 𝐸 are incident in the direction perpendicular to the barrier, on an
infinitely thick potential barrier of height 𝑉0 , where 𝐸 > 𝑉0 . Show that the fraction of
electrons reflected from this barrier is
1−𝑎 2
𝑅=[
]
1+𝑎
𝐸−𝑉
where 𝑎 = √ 𝐸 0
Problem 5. Allowed and forbidden energy bands (5 points)
The bandgap energy in a semiconductor is usually a slight function of temperature. In some
cases, the bandgap energy versus temperature can be modeled by (Varshini equation)
𝛼𝑇 2
𝑇+𝛽
𝐸𝑔 = 𝐸𝑔 (0) −
where 𝐸𝑔 (0) is the value of the bandgap energy at 𝑇 = 0 K. For GaAs, the parameter values
are 𝐸𝑔 (0) = 1.519 eV, 𝛼 = 5.4 × 10−4 eV/K, and 𝛽 = 204 K. Plot 𝐸𝑔 versus 𝑇 over the
range 0 ≤ 𝑇 ≤ 400 K. Calculate the values at 𝑇 = 250 K, 290 K, and 330 K.
Problem 6. Allowed and forbidden energy bands (12 points)
(a) Explain the effective mass of electrons in semiconductors briefly. (less than 1/4 pages)
(b) Explain two kinds of charge carriers in the semiconductors. Why do we introduce holes in
semiconductors to count the electron behavior in the valance band? (less than 1/4 pages)
(c) Do the electrons in the fully filled valance band contribute to the current? Why? (less than
1/4 pages)
Problem 7. Electrical conduction in Solids (10 points)
Figure 1 shows the parabolic E versus k relationship in the valence band for a hole in two
particular semiconductor materials. Determine the effective mass (in units of the free electron
mass) of the two holes.
Figure 1
Problem 8. Density of states function (10 points)
Considering a free electron confined to a two-dimensional infinite potential well, derive the
density of allowed quantum states.
Problem 9. Density of states function (8 points)
Starting with the three-dimensional infinite potential well function given by Equation 1
below, and using the separation of variables technique, derive Equation 2
𝑉(𝑥, 𝑦, 𝑧) = 0 for 0 < x < a
0<y<a
0<z<a
<Equation 1>
2𝑚𝐸
𝜋2
= 𝑘𝑥 2 + 𝑘𝑦 2 + 𝑘𝑧 2 = (𝑛𝑥 2 + 𝑛𝑦 2 + 𝑛𝑧 2 )( 2 )
ℏ
𝑎
<Equation 2>
Problem 10. Density of states function (10 points)
Repeat deriving the density of states for the one-dimensional case.
Problem 11. Statistical Mechanics (10 points)
(a) Determine for what energy above EF (in terms of kT ) the Fermi–Dirac probability
function is within 1 percent of the Boltzmann approximation.
(b) Give the value of the probability function at this energy.