Chem 120A
Particle in a Box, Continued II and the Finite SquareWell
02/15/06
Spring 2006
Lecture 13
READING:
Engel (QCS): Chapter 4 (you should finish both Ch. 4 and 5 before the end of next week)
Properties of PIB solutions, continued
5. The kinetic energy is related to the curvature or wiggliness of the wavefunction. We can see this by
examining the particle in a box wavefunctions. The wavefunctions,
ψ
(
x
)
, shown in Figure5 become
Figure 2.
The wavefunctions and the square of the
wavefunctions for a particle in a box of length L.
increasingly wiggly as the energy for the wavefunction increases. For example,
E
1
<
E
2
<
E
3
and
we know that the potential energy is zero inside of the box. This implies that the kinetic energy,
KE
,
is increasing as
n
increases since the total energy is the sum of the kinetic and potential terms. The
reason why the curvature is directly related to kinetic energy is that
KE
is proportional to
d
2
ψ
(
x
)
/
dx
2
.
6. Symmetry is very important in quantum mechanics. It helps us to simplify complicated problems.
Parity
is associated with the symmetry with respect to the origin in a coordinate system. For example,
the finite square well potential in Figure 6 has
V
(
x
) =
V
(
−
x
)
. This symmetry is called ’inversion
symmetry.’ The same symmetry applies for the infinite square well.
Let’s consider the Schr¨odinger equation for the infinite square well potential,
−
¯
h
2
2
m
d
2
ψ
(
x
)
dx
2
+
V
(
x
)
ψ
(
x
) =
E
ψ
(
x
)
.
(1)
Chem 120A, Spring 2006, Lecture 13
1
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V
0
x
V(x)
Figure 1: Potential for a quantum particle in a 1D well of
length a.
V(x) = V
0
,
a/2 < x < a/2
V(x) = 0,
x
r
a/2
a/2
a/2
I
II
III
V=0
In order to investigate the symmetry of this equation with respect to the origin, we replace
x
with
−
x
,
E
ψ
(
−
x
)
=
−
¯
h
2
2
m
d
2
ψ
(
−
x
)
d
(
−
x
)
2
+
V
(
−
x
)
ψ
(
x
)
E
ψ
(
−
x
)
=
−
¯
h
2
2
m
d
2
ψ
(
x
)
dx
2
+
V
(
x
)
ψ
(
−
x
)
∴
ψ
(
−
x
)
is also a solution to the Schr¨odinger equation with energy
,
E
(2)
In Lecture 12, we saw that the solutions to the Schr¨odinger equation for a particle in an infinite box
in 1D are all nondegenerate, and that the eigenstates must also be real and normalized. The fact that
ψ
(
x
)
and
ψ
(
−
x
)
have the same energy implies that
ψ
(
x
) =
A
ψ
(
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 Spring '07
 Whaley
 Physical chemistry, pH, Energy, odd parity, Schr¨ dinger equation

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