Chem 3502/5502
Physical Chemistry II (Quantum Mechanics)
3 Credits
Fall Semester 2010
Laura Gagliardi
Lecture 20, November 01, 2010
Solved Homework
We determined that the two coefficients in our two-gaussian wave function were
c
1
= 0.3221 and
c
2
= 0.7621. We also determined that
ψ
1
s
r
,
θ
,
φ
;
c
1
,
α
1
,
c
2
,
α
2
( )
−
1
2
∇
2
ψ
1
s
r
,
θ
,
φ
;
c
1
,
α
1
,
c
2
,
α
2
( )
=
1.5
c
1
2
+
0.6435
c
1
c
2
+
0.3
c
2
2
Using the normalized coefficient values, we have <
T
> = 0.4873. Since <
H
> was
−
0.4819,
<
V
> must be
−
0.9692. We could also simply plug the coefficients into
ψ
1
s
r
,
θ
,
φ
;
c
1
,
α
1
,
c
2
,
α
2
( )
−
1
r
ψ
1
s
r
,
θ
,
φ
;
c
1
,
α
1
,
c
2
,
α
2
( )
=
−
1.5958
c
1
2
−
1.5908
c
1
c
2
−
0.7136
c
2
2
which is also (as it must be)
−
0.9692. If we take
−
2<
T
> we have –0.9746. So, although
the virial theorem is
almost
satisfied, it is not quite. The error is less than 1%.
Exact wave functions
must
satisfy the virial theorem, but that is
not
true of
approximate wave functions. As such, one may examine the virial ratio to assess the
quality of an approximate wave function. As one gets closer to the exact wave function,
one should get closer to a ratio of –2.
Antisymmetry
Consider a quantum mechanical system consisting of two indistinguishable
particles, e.g., two electrons. If we wanted to compute the probability that we would find
electron 1 in some volume of space characterized by
r
a
≤
r
≤
r
b
,
θ
a
≤
θ
≤
θ
b
, and
φ
a
≤
φ
≤
φ
b
, and at the same time find electron 2 in some volume of space characterized by
r
c
≤
r
≤
r
d
,
θ
c
≤
θ
≤
θ
d
, and
φ
c
≤
φ
≤
φ
d
, we would compute this probability as
P V
1
1
( )
,
V
2
2
( )
[ ] =
Ψ ψ
1
1
( )
,
ψ
2
2
( )
[ ]
φ
c
φ
d
∫
θ
c
θ
d
∫
r
c
r
d
∫
φ
a
φ
b
∫
θ
a
θ
b
∫
r
a
r
b
∫
2
r
1
2
dr
1
sin
θ
1
d
θ
1
d
φ
1
r
2
2
dr
2
sin
θ
2
d
θ
2
d
φ
2
(20-1)
where the cumbersome notation is meant to emphasize that the probability has to do with
electron 1 being in volume 1 and electron 2 being in volume 2, that
Ψ
depends (in an
unspecified way) on two individual electron wave functions
ψ
, the first of which is