1. Department of Chemistry
Indian Institute of Technology Madras, Chennai
CY101: Part A II Semester
Assignment 1
Quantum Chemistry – Elementary quantum mechanical models
1) Consider an electron with a kinetic energy 2 eV. Calculate its momentum,
velocity and the de Broglie wavelength. Is it reasonable to assume that the
mass of the electron is independent of its speed?
2) Assume that an electron moves in a box of length 200 pm. Estimate the
kinetic energy of electron using a particle in a box model. State your
approximations. What is the de Broglie wavelength of the electron?
3) What are the eigenfunctions of a free particle kinetic energy operator?
Obtain the momentum eigenvalues and eigenfunctions for the free
particle. Are these wave functions “acceptable” in quantum mechanics? If
not, how do you make them “acceptable”?
4) Verify that the eigenfunctions Ψn
(x) =
2
L
sin
nπx
L
are eigenfunctions of the
Hamiltonian for the particle in a box but not its momentum.
5) The wave function for a particle in a one dimensional box is given as
ψ (x) =
2
L
c1 sin
πx
L
+ c2 sin
2πx
L
⎧
⎨
⎩
⎫
⎬
⎭
. Calculate the expectation value for the
energy and momentum of this particle. If the energy of the particle is found
to be five times the smallest eigenvalue for the particle in a box, obtain a
value for c1 and c2 Are they unique?
6) You have used a procedure for separation of variables to derive the
eigenfunctions of a particle in a 2-D or a 3-D box. Obtain by a similar
procedure the solution of the time dependent Schrodinger equation

2. i
∂ψ
∂t
= ˆHψ where ˆH is independent of time and ( )n xΦ are eigenfunctions
of ˆH , i.e. ˆHΦn
(x) = En
Φn
(x).
7) What are the energy eigenfunctions and eigenvalues for a particle in a 1-D
box whose boundaries are
2
L
x = ± (Box length is L and the mass of the
particle is m)?
8) For the one dimensional box, obtain an expression for the maxima of the
probability densities
2
( )nP xψ= for all n .
9) Consider the wave functions
1
( ) ikx
k x e
a
ψ = in the interval ,
2 2
a a⎛ ⎞
−⎜ ⎟
⎝ ⎠
. Are
these wave functions normalized and orthogonal in the limit a → ∞ ? Show
your answer explicitly.
10) Calculate the expectation values x , 2
x , p , 2
p for a free particle in
a 1-D box, with
2
sinn
n x
L L
π
ψ = . Calculate the product
Δx Δp = x2
− x
2
p2
− p
2
where xΔ and pΔ are
uncertainties in the particle position and momentum. Show that this
verifies the Heisenberg’s uncertainty principle.
11) For the free particle in a two dimensional box, calculate the expectation
values for the following quantities:
2 2
,xy x and y −i x
∂
∂y
− y
∂
∂x
What is your interpretation of the operator −i x
∂
∂y
− y
∂
∂x
?
12) An arbitrary normalized wave function for a particle in a one dimensional
box is given by ψ (x) = Cx2
(L − x)2
.
a) Obtain an expression for C.

3. b) Express ( )xψ as
1
2
( ) sinn
n
n x
x C
L L
π
ψ
∞
=
= ∑ . Obtain an expression for
the coefficient Cn.
c) What is the value of the sum
2
1
n
n
C
∞
=
∑ ?
d) What is the average value for the energy of the particle in this
state?
13) Consider the rectangular wave function
ψ (x) =
1
L
, 0 < x < L
ψ (x) = 0, otherwise
Express
1
2
( ) sinn
n
n x
x C
L L
π
ψ
∞
=
= ∑ and obtain an expression for Cn and
2
1
n
n
C
∞
=
∑ . Is this an acceptable wave function?
14) A linear one dimensional chain of conjugated polyene contains 12 C-C
links (-C=C-C=C-)3 with an average bond distance of 1.33 Angstrom.
Ignore the end effect and assume that the pi electrons are free to move
along the bonds. Determine the energy difference between the highest
occupied energy level and the lowest unoccupied energy level. Assume
that at most two electrons can occupy each energy level.
15) Calculate the commutator ˆ ˆ ˆ ˆ ˆˆ, x y zx y z p p p⎡ ⎤+ + + +⎣ ⎦.
16) If ˆA , ˆB and ˆC are three operators, show that the commutator
ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , ,A BC B A C A B C⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
and ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , ,AB C A B C A C B⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
. Use these
to calculate the commutator 2 2
ˆ ˆ, xx p⎡ ⎤⎣ ⎦.
17)For the free particle in a cubical box of side L, calculate the energy E for
the first eight states and identify the degeneracy of each of them.
18) Calculate the ground state energy of the electron in the hydrogen atom
using the wave functionψ (r) = ce−r/a0
. The Hamiltonian is

4. H = −
2
2m
∂2
∂x2
+
∂2
∂y2
+
∂2
∂z2
⎛
⎝⎜
⎞
⎠⎟ −
e2
4πε0r
.
19) Assuming a simple 3d cubic box model, calculate the longest wave length
that an electron can have in a box of size 118 pm. Calculate the average
kinetic energy and determine what fraction this constitutes of the total
energy calculated using the exact wave function in the previous problem?
20) Examine carefully all the fifteen angular wave functions (l=1, 2 and 3) and
the animations given in your web site. Identify all angular nodes.