1
Physical Chemistry
Lecture 16
Using the particle in a box to
understand special problems
Particle in a finite 1D box
With a finite potential in the
“external” regions, the wave
function can be nonzero in
those regions
Must solve Schroedinger’s
equation piecewise in three
regions
Match the solutions at the
boundaries between regions
Match derivatives at the
boundaries
Uses symmetry to simplify
problem
Puts origin at the center of
the box, rather than one
edge
Solutions can be classified
as even or odd under
inversion through zero
2
/
2
/
0
2
/
2
/
)
(
0
a
x
a
for
a
x
and
a
x
for
V
x
V
Particle in a finite 1D box
Schroedinger’s equation in the
three regions looks similar
With a finite potential in the
“external” regions, the solutions
–evenforE<V
0
include the
possibility of the particle being
in those regions
Very different from predictions
of classical mechanics
Equations in the three regions
look similar
Solutions are sums of
exponential functions
Different kinds of solutions
for
E
<
V
0
and
>
0
Quantum conditions are
determined by boundary
conditions
0
2
2
2
2
2
2
2
"
"
2
2
/
2
/
V
E
dx
d
m
regions
external
For
E
dx
d
m
a
x
a
For
Particle in a finite 1D box
For E > V
0
Wave functions are oscillatory
functions of x
Sums of exponential functions with
imaginary arguments
Amplitudes are related by boundary
conditions
For E
, it approaches the
behavior of the
free particle in one
dimension
Oscil atory behavior
Deviation from the free particle
depends on energy relative to the
potential
2
0
2
'
'
)
(
2
'
2
'
'
)
(
"
"
)
(
2
/
2
/
V
E
m
k
mE
k
where
e
k
k
B
e
k
k
A
x
regions
external
For
Be
Ae
x
a
x
a
For
x
ik
x
ik
ikx
ikx
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 Fall '10
 Staff
 Physical chemistry, Electron, pH, finite 1D box

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