Chem 120A
Elementary Classical Mechanics and Probability Theory
01/18/06
Spring 2006
Lecture 1
READING:
Engel: Section 1.1 and Appendix A.1
(sections 1.21.4 offer a historical motivation for the study of quantum
mechanics; you may find those sections interesting, but you will not be responsible for that material)
Feynman (CPL): Chapter 6
Feynman (FL3): Chapter 1
Motion of a single particle in 1D
Consider a particle with mass
m
in 1D (one dimensional) space. At any given time it is at a position
x
(
t
)
and
has velocity
v
(
t
) =
dx
(
t
)
dt
. The functions
x
(
t
)
and
v
(
t
)
are obtained from Newton’s second law:
m
d
2
x
(
t
)
dt
2
=
F
(
x
(
t
))
,
(1)
or
F
=
ma
. The actual form of
F
depends on the specific circumstances.
Due to the fact that Equation 1 is a secondorder differential equation in time, if we are given a specific set
of initial conditions,
x
(
t
0
)
and
v
(
t
0
)
, the functions
x
(
t
)
and
v
(
t
)
are
uniquely
determined for all
t
. This is
called a ”‘trajectory.”’
x(t
1
)
v(t
1
)
x'(t
1
)
v'(t
1
)
time
x(t
2
)
v(t
2
)
x'(t
2
)
v'(t
2
)
The trajectories for different
initial conditions are unique;
they never cross.
Figure 1: An example of trajectories for different initial conditions
The energy for this 1D system is given by
E
(
t
)
≡
1
2
mv
2
(
t
)+
U
(
x
(
t
))
,
(2)
where
U
(
x
(
t
))
is defined in terms of
F
(
x
(
t
))
:
−
dU
(
x
(
t
))
dx
(
t
)
=
F
(
x
(
t
))
(3)
Chem 120A, Spring 2006, Lecture 1
1
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U
≡
potential energy and
1
2
mv
2
(
t
)
≡
T
≡
kinetic energy. In quantum mechanics,
F
plays little role, however,
U
plays a crucial role.
Conservation of energy
Note: in the following section we will use the ‘’dot” notation for derivatives WRT time,
· ≡
d
dt
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 Spring '07
 Whaley
 Physical chemistry, pH, Energy, Force, Kinetic Energy, Potential Energy, CHEM 120A

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