Practice Problems
1. (Schr¨ordinger equation) A quantum mechanical particle on the line with an inﬁnite potential
outside of the interval (0
,l
) is given by Schr¨
ordinger’s equation for
x
∈
(0
,l
),
t
∈
R
≥
0
, and a
positive real number
m
∈
R
>
0
by
i
∂
∂t
u
(
x,t
) =

1
2
m
∂
2
∂x
2
u
(
x,t
)
.
(a) Write down the solution as inﬁnite series for Dirichlet boundary conditions.
(b) Write down the solution as inﬁnite series for the boundary conditions
u
x
(0
,t
) = 0 and
u
(
l, t
) = 0.
2. (Eigenvalue problem) Solve the eigenvalue problem
x
2
f
00
(
x
) + 3
xf
0
(
x
) +
λ f
(
x
) = 0
.
for 1
< x < e
where
e
is the Euler number
e
= 2
.
71
.. .
with
f
(1) =
f
(
e
) = 0. Assume
that
λ >
1 and look for solutions of the form
f
(
x
) =
x
m
. Determine the eigenvalues and
eigenfunctions.
3. (Waves in resistant medium) Waves in a resistant medium, i.e., with friction present, are
described by the PDE
u
tt
=
c
2
u
xx

r u
t
where
x
∈
(0
,l
) and
r
is a constant.
(a) For
r
= 0, i.e., no friction, ﬁnd the solution explicitly in series form for the boundary
conditions
u
(0
,t
) = 0
,u
x
(
l,t
) = 0 and the initial conditions
u
(
x,
0) =
x,u
t
(
x,
0) = 0.
(b) For
r
= 0, i.e., no friction, ﬁnd the solution explicitly in series form for the inhomogenous
boundary conditions
u
(0
,t
) =
h, u
(
l,t
) =
k
(where
h, k
are constants) and the initial
conditions
u
(
x,
0) = 0
,u
t
(
x,
0) = 0.
(c) For 0
< r <
2
πc
l
write down a series solution that satisﬁes Dirichlet boundary conditions
and the initial conditions
u
(
x,
0) = 0
,
u
t
(
x,
0) =
x
(
l

x
)
.
What happens for
t
→ ∞
.
(d) For 0
< r <
2
πc
l
consider boundary conditions with a periodic source at one end, i.e.,
u
(0
,t
) = 0 and
u
(
l, t
) =
A e
iωt
for a given
ω
. Show that the PDE and the BC are satisﬁed
by
v
(
x,t
) =
A e
iωt
sin (
βx
)
sin (
βl
)
,
where
β
2
c
2
=
ω
2

irω
.
(e) In (d) show that no matter what the initial condtions for
u
(
x,t
) are,
v
(
x,t
) is the
asymptotic form of the solution for
t
→ ∞
.
4. (Complex Fourier series)
(a) Show that a complex valued function, periodic,
C
1
function
f
(
x
) is real valued if and
only if its complex Fourier coeﬃcients satisfy
c
*
n
=
c

n
for all
n
∈
Z
where
*
denotes
complex conjugation.