Quantum Chemistry: Math Basics
I. The Inner Product
Comparing things is a human endeavor. You may have your mother’s eyes and your
father’s nose. We realize that you don’t actually have the eyes of your mother, but that you
have features similar to your parents. While such qualitative comparisons are frequently
useful, in quantum chemistry we need a more quantitative way to compare two objects.
In what follows we will learn how to compare two objects described by functions
and give a quantitative measure to their similarity. Having a firm grasp of the concepts
presented here will make what is to follow in the rest of the semester much easier. So get
a pencil or pen, a few sheets of graph paper, put on some music and let’s get going.
We know how to compare two vectors. We do this by generating their dot product.
Recall that we can define vectors in 2D in terms of their x and y components. Let ~x be a
unit vector in the x-direction and ~y be a unit vector in the y-direction. A general vector,
~u, is then represented as:
~u = a~x + b~y
where a and b are scalars (numbers). Now we can also represent ~u just in terms of its
components i.e.
~u = {a, b}
where the unit vectors are implied. ~u is said to be represented by a tuple. In general an
n-dimensional vector is represented by an n-tuple:
~u = {a1 , a2 , a3 , ..., an }
where each of the numbers ai is the component of the ith unit vector. Using this representation ~x = {1, 0} and ~y = {0, 1}
Now let ~u = {a1 , a2 } and ~v = {b1 , b2 }. We can create an operation called the dot
product between ~u and ~v as:
~u · ~v = (a1 b1 + a2 b2 )
(Q: Show that for a general n-dimensional vector, ~u = {a1 , a2 , a3 , ..., an },
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and ~v = {b1 , b2 , b3 , ..., bn }, ~u · ~v = n
1 ai bi ).
Let’s see what this operation tells us about ~u and ~v . First observe that ~u · ~x = a1 . In
words, ~u · ~x tells us how much of ~u lies along the x-direction and of course ~u · ~y = a2 tells
us how much of ~u lies along the y-direction. The dot product allows us to tell how similar
a vector is to some other vector, in this case ~u and its component vectors. In general to
find how much of ~u lies in the direction of ~v we form the dot product between ~u and a unit
vector in the direction of ~v .
(Q: For a general n-dimensional vector, ~u = {a1 , a2 , a3 , ..., an } and
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|~u| ≡ (~u · ~u) 2 show that ~u/|~u| is a unit vector lying in the same
direction as ~u.)
So the degree of similarity between ~u and a unit vector lying in the direction of ~v is
simply given by :
~u · ~v
|~v |
In quantum mechanics we generally want to compare wave-functions, not vectors, and
it is not clear how to make this comparison. However, we could compare wave-functions
or functions in general if we could express them as vectors. Let’s see if there is a way to
express a function as a vector. For example, let’s attempt to express the functions x2 and
2
sin(x) as vectors in the interval 0 ≥ x ≥ 1 and then take their dot product to determine
how much of x2 lies in the “direction” of sin(x).
We begin by defining a series of functions n fm (x),where m = 1, 2, ..., n
n f (x)
m
=
m
1 if m−1
n <x≤ n;
0 if elsewhere.
(Q: Let n = 2 and graph the two functions 2 f1 (x) and 2 f2 (x). Let
n = 4 and graph the four functions 4 f1 (x), 4 f2 (x), 4 f3 (x), and
4 f (x).
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)
Now we can use these functions, which we will call basis functions, to “approximate”
another function, let’s use x2 . (Remember as an example we are trying to write x2 as a
vector.)
n X(x)
≡
n
X
)n fm (x)
(x2 | m
n
m=1
where (x2 | m
) means evaluate x2 at the point m
n . As examples for n = 2 and n = 3,
n
2
2
2 X(x) = 1
2 f (x) + 2
2 f (x)
1
2
2
2
3 X(x)
2
2
2
1
2
3
3
3
3 f (x)
=
f1 (x) +
f2 (x) +
3
3
3
3
(Q: Graph 2 X(x),3 X(x), and 4 X(x) on top of x2 . What will happen to n X(x) as n approaches ∞?)
We can represent 2 X(x) as a vector with 2 X(x) = { 41 , 1}
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(Q: What are the basis vectors of 2 X(x)? Express n X(x) as an
n-tuple. What are the basis vectors of this n-tuple?)
As n approaches ∞, for the functions n fm (x) the region over which they are defined to
be one becomes narrower. Let’s speed things up by defining a set of basis functions which
are one only at a point. We will call these δ-functions, where
δp (x) =
n
1
0
if x = p;
if elsewhere.
Now we can exactly represent x2 as a vector over this set of δ-functions? Were there
are as many basis vectors as there are points on the line between 0 and 1.
(Q: We can just as easily think of δp as a unit vector along the p direction in our infinite dimensional vector space i.e. as ~δp . Evaluate:
~δ 1 · x2 . What does this represent?)
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Just as we can express x2 as a vector, we can also express sin(x) as a vector over our
chosen interval. Denote this set of functions by n S(x) where:
n S(x)
=
n
X
sin
m
m=1
n
n f (x)
m
(Q: Write 4 S(x) as a 4-tuple. What are the basis vectors of this
4-tuple? Are these the same basis vectors as 4 X(x)?)
As n X(x) and n S(x) are vectors, we can take their dot product, i.e
n X(x) ·n
n m
X
m 2
S(x) =
sin
n
n
m=1
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Notice that when we take a dot product the basis vectors disappear. This seems
reasonable as the similarity between two things should not depend on our point of reference.
So if we take the dot product of our two functions as exactly defined on the set of δ-functions
we really have to do a sum over an infinite number of points along an interval- we need to
do an integral. So the exact dot product between our two functions is:
x2
Z 1
· sin(x) =
x2 sin(x)dx
0
The term “dot product” implies that two vectors are being compared. When comparing
functions the more general term “inner product” is used. We use a special notation to
indicate that the inner product of g(x) and h(x) is to be taken, where if g(x) and h(x) are
real valued functions
Z ∞
hg(x)|h(x)i ≡
g(x)h(x)dx.
−∞
(Q:Now, write an expression and determine how much of x2 lies
along the direction of sin(x) in the interval 0 < x ≤ 1. (Remember to do this you have to write a unit vector lying in the direction of sin(x). What other term have we used for this operation?)
Compare your expression to the first Fourier coefficient of x2 . In
words describe what is being done when a function is expanded in
a Fourier series. Finally write a general expression to determine
how much of the function g(x) lies in the direction of h(x) in the
interval a < x < b.)
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