Chem 120A
Going beyond 1 particle in 1D
03/06,08/06
Spring 2006
Lecture 19
READING:
Engel: 4.3 and 4.4, also see p.60
So far we have looked at the Particle in a Potential Well and the Harmonic Oscillator in 1Dimension. These
can be used to model the electronic energy levels in conjugated alkenes and the vibrational energy levels in
diatomic molecules, respectively. However, atoms and molecules exist in 3D and in general are composed
of multiple particles. So we will now generalize some of our earlier postulates to 2D and 3D and for more
than one particle.
The commutator relation between the position and momentum operators is now
[
ˆ
x
,
ˆ
p
x
] =
i
¯
h
[
ˆ
y
,
ˆ
p
y
] =
i
¯
h
[
ˆ
z
,
ˆ
p
z
] =
i
¯
h
(1)
So in a given dimension, position and momentum do not commute. This means that if we measure the
position of the paritcle along the xcooridinate and then its momentum in the xdirection, we will not get
the same value as if we measure the momentum first in the xdirection and then the position along x. What
about position along the xcoordinate and momentum in the ydirection?
[
ˆ
x
,
ˆ
p
y
] =
0
[
ˆ
y
,
ˆ
p
x
] =
0
(2)
So position and momentum operators in different dimensions commute. If we make two measurements but
each is along a different dimension, the order of the measurements does not matter. This is a a result of the
fact that translations commute (i.e. if I move to x = 6 and then I move to y = 4, I will be in the same spot as
If I had moved first to y = 4 and then to x = 6). Our generalized commutator relationship between position
and momentum for 1 particle in a multidimensional space is then:
[
ˆ
r
i
,
ˆ
p
j
] =

i
¯
h
δ
i j
(3)
where i and j label the dimension
What about measurements on more than one particle? If we measure the position of particle 1 along the
xcoordinate, it will not affect the measurement of position along the xcoordinate for particle 2 (as long as
the particles are NOT entangled!). Measurements in the same dimension but for different particles commute.
Measurements on the same particle but in different dimensions also commute. Only measurements on the
same particle and in the same dimension do NOT commmute. Mathematically this is written as:
ˆ
r
α
,
i
,
ˆ
p
β
,
j
=

i
¯
h
δ
i j
δ
α
,
β
(4)
Chem 120A, Spring 2006, Lecture 19
1
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where i and j label the dimension and
α
and
β
label the particle.
We will now write out our position, momentum, and kinetic energy operators in 2D and 3D (using Carte
sian coordinates).
2

D
3

D
(5)
Position
ˆ
r
=
x
ˆ
i
+
y
ˆ
j
ˆ
r
=
x
ˆ
i
+
y
ˆ
j
+
z
ˆ
k
(6)
Momentum
ˆ
p
=

i
¯
h
[
∂
∂
x
ˆ
i
+
∂
∂
y
ˆ
j
] =

i
¯
h
∇
ˆ
p
=

i
¯
h
[
∂
∂
x
ˆ
i
+
∂
∂
y
ˆ
j
+
∂
∂
z
ˆ
k
] =

i
¯
h
∇
(7)
Kinetic Energy
T
=
ˆ
p
2
2
m
=

¯
h
2
2
m
∇
2
T
=
ˆ
p
2
2
m
=

¯
h
2
2
m
∇
2
(8)
You have seen that the normalized 1D Particle in a Box wavefunction has units of 1
/
√
L
(i.e.
ψ
(
x
) =
2
L
sin
(
n
π
x
L
)
). Thus

ψ

2
is a probability density or a probability per unit length (since we are in 1D). The
probability of finding the particle at a point along the line between x and x + dx is then

ψ

2
dx
. If we take
the the integral
x
2
x
1
ψ
*
(
x
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 Spring '07
 Whaley
 Physical chemistry, pH, Energy, Kinetic Energy, CHEM 120A

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