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February 26, 2010
Chemistry 120A Hour Examination
Useful formulas
Schr¨
odinger’s timedependent equation
in one dimension:
i
~
∂
Ψ(
x,t
)
/∂t
=
H
Ψ(
x,t
)
,
where
~
is Planck’s constant over 2
π
,
H
is the Hamiltonian operator, and
Ψ(
x,t
) is the wavefunction as a function of position,
x
, and time,
t
.
The Hamiltonian
for a particle of mass
m
in
d
= 1 is
H
=
1
2
m
ˆ
p
2
+
V
(
x
) =

~
2
2
m
d
2
dx
2
+
V
(
x
)
,
where ˆ
p
= momentum operator, and
V
(
x
) = potential energy.
Stationary solutions
to Schr¨
odinger’s equation are of the form
Ψ
stationary
(
x,t
) =
ψ
(
x
)
e

iEt/
~
,
where
ψ
(
x
) is the solution to the eigenvalue equation
H
ψ
(
x
) =
E ψ
(
x
)
.
The momentum operator
, ˆ
p
= (
~
/i
)
d/dx
, obeys
x
ˆ
p

ˆ
px
≡
[
x,
ˆ
p
] =
i
~
Time derivative of expectation value
for a general observable
G
is
d
dt
Z
h
Ψ
*
(
x,t
)
ˆ
G
Ψ(
x,t
)
i
dx
=
d
h
ˆ
G
i
/dt
= (1
/i
~
)
h
[
ˆ
G,
H
]
i
.
1
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 Spring '10
 DAVIDCHANDLER

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