3.If
?
=
𝐴?
, find g for each of the following
𝐴
and f. Also indicate which solutions are an
eigenvalue/eigenfunction.
a.
𝐴
=
?
/
?𝑥
, f = cos(-x)
g=-sin(x)
b.
𝐴
=
?
2
/
?𝑥
2
, f = cos(-x)
g=-cos(x)
eigenvalue/eigenfunction.
c.
𝐴
=()
2
, f = sin(x)
g=sin
2
(x)
d.
𝐴
=
?𝑥𝑝
, f = ln(x)
g=x

4.Write out the exact wavefunction for the 2p orbitals of a hydrogen atom. Draw/sketch the
electron density that would be derived from these wavefunctions.
𝜑
2𝑝𝑥
=
𝑅
2𝑝
∗ ?
2𝑝𝑥
𝑅
2𝑝
=
1
2
√
1
6
∗ ? ∗ ?
3
2
∗ ?
−
𝜌
2
?
2𝑝𝑥
=
√3
𝑟
∗ (
1
4
∗ ?)
2

5.True or false?
a. A probability density can never be negative. True
b. The wavefunction,
𝜓
, can never be negative. False
c. The wavefunction must be a real function. False
d.
∫
𝜓?𝑥
=1for a one-particle, one-dimensional system. False
e. Electrons orbit around the nucleus of an atom. True
f. Molecular orbitals are often described using atomic orbitals. True
g. The most stable way for bonding between atoms to occur is through shared electrons. False