This document provides an introduction to elementary quantum mechanics. It begins by defining Hilbert spaces and establishing complex exponentials as an orthonormal basis for L2 spaces. It then discusses Fourier series and using a linear combination of complex exponentials to represent L2 functions. Next, it introduces the Fourier transform and Parseval's identity. It derives the one-dimensional Schrodinger equation and discusses its physical interpretation. Finally, it formally defines quantum mechanics as a Hilbert space with a Hamiltonian and presents the Heisenberg uncertainty principle.

1. ELEMENTARY QUANTUM MECHANICS
VICTOR B. CASTILLO
Abstract. In this paper we will study the mathematical framework of Quan-
tum mechanics, with the goal of understanding the one-dimensional Schr¨odinger
equation and the Heinsenberg uncertainty principle. We begin with the defi-
nition of a Hilbert space, the most important examples of which are the L2-
spaces. We will establish a set of basis vectors for L2([a, b])-space and begin to
examine Fourier series. We will explore the Fourier transform and its various
properties. We will then arrive at the Schr¨odinger equation and explore its
interpretations. Finally we will present the mathematical formalism of Quan-
tum mechanics and present an efficient proof of the Heinsenberg uncertainty
principle.
1. Introduction
Quantum mechanics is more then just a theory of physical phenomena. It is an
elegant mathematical framework, that has unbounded application to many other
areas of science beyond physics. It is a new level of abstraction which restructures
our understanding of mathematical objects. However, before we can give a formal
definition of Quantum Mechanics or begin our discussion of it’s interpretations, we
must study a few of its necessary elements.
We begin in section 2 by taking a look at Hilbert spaces. These linear vector
spaces are defined as being topologically complete and possessing an inner product
function which obeys three distinct properties. These three properties are linearity
in the second variable, Hermitian symmetry and non-degeneracy. We examine a
few examples of Hilbert space’s such as the complex vector space and the p
spaces.
In section 3 we define the Lp
spaces and the Lp
-norm of a function. We then
quickly draw our attention to the L2
spaces. This vector space is of particular
interest to us because it happens to be the only Lp
space which possess an inner
product. We define its inner product and discuss its classification as a Hilbert
space. Assuming it is a Hilbert space we proceed to find a set of orthonormal basis
vectors for L2
, specifically a set of complex exponentials.
In section 4 we discuss the Fourier series for L2
functions. We establish that
any L2
function can be represented as a linear combination of the orthonormal basis
vectors that were constructed in section 3. In this section it is important to note
that the Fourier coefficients are the inner product of the function with the basis
Date: 4/25/2010.
1

2. 2 VICTOR B. CASTILLO
vectors. Before concluding this section we spend some time examining the example
of the square-pulse. In this example we compute its Fourier series and observe the
effects of varying certain parameters (i.e. number of terms in the series, length of
the domain). We will see that the frequency spectrum of the coefficients goes from
being a discrete distribution to being a continuous one.
In section 5, motivated by the square-pulse example, we define the Fourier
transform of a function. We discuss several properties of the transform as well as
the Fourier transform on the convolution of functions. We also give a proof of the
Parseval identity, which states that the L2
norm of the original function and the
L2
norm of its Fourier transform are equal. Intuitively this suggests that any L2
function can be expressed as a sum of its component frequencies.
In section 6 we finally arrive at the one-dimensional Schr¨odinger equation. The
Schr¨odinger equation governs how quantum systems change over time. We examine
solutions to the Schr¨odinger equation mainly in the form of plane waves. Quan-
tum mechanics is motivated by the change of dynamical variables in the classical
setting to observables in the quantum setting. Using this approach we arrive at
the Schr¨odinger through making specific substitutions into the classical Hamilton-
ian. We conclude this section by giving a physical interpretation of Schr¨odinger’s
picture of wave mechanics and briefly discussing the uncertainty when evaluating
a particles position and momentum simultaneously.
In section 7 we begin the process of formalizing Quantum mechanics. We
define what it means for a linear operator to be Hermitian and anti-Hermitian,
and give several consequences of this fact. We also define what a Unitary operator
is as well as give a proof of why eA
is unitary if A is anti-Hermitian. We also
finally give a formal definition of Quantum mechanics. Specifically we define it to
be a Hilbert space with a distinguish Hamiltonian, the Schr¨odinger equation. We
define the elements of this space and how the operators act on them. Finally at the
conclusion of this paper, we discuss the Heisenberg uncertainty relation. Itstates
that the product of the standard deviation of any two observables is always greater
than or equal to the average of the norm of their commutator. We then briefly
discuss its application to wave mechanics.
2. Hilbert Space
Definition: A Hilbert space is a complex vector space which is topologically
complete and possesses an inner product function (· , ·) : H × H → C that obeys
the following properties. If v, w ∈ H and α ∈ C then
• It is linear in the second variable (v , αw) = α(v , w)
and also (v , u + w) = (v , u) + (v , w)
• Hermitian-symmetry (v , w) = (w , v)
• Nondegeneracy (v , v) ≥ 0 with equality only when v = 0
This inner product is called an Hermitian inner product.

3. ELEMENTARY QUANTUM MECHANICS 3
2.1. Example - Space of Complex numbers. We have that the Complex vector
space, Cn
, is a Hilbert space. Given two vectors z, w ∈ Cn
their inner product,
denoted here by a dot, is
(z1, z2, z3, ...zn) · (w1, w2, w3...wn) = ¯z1w1 + ¯z2w2 + ¯z3w3 + · · · + ¯znwn.
It is easy to check that this inner product is linear in the second position so that,
z · (aw + bu) = a(z · w) + b(z · u),
Hermitian symmetry since
(w · z) = w · z = w · z = (z · w),
and obviously that z · w is zero if and only if z = 0.
2.2. Example - 2
. 2
is the space of all square-summable sequences of complex
numbers. More precisely, it is the space of infinite complex sequences z = (z1, z2, ...)
such that
∞
n=1
|zn|2
< ∞.
The inner product on 2
is defined to be
(z, w) 2 =
∞
n=1
¯znwn.
We see that it is linear in the second position
(z, αw) 2 =
∞
n=1
¯znαwn
= α
∞
n=1
¯znwn
= α(z, w) 2
and also
(z, w + u) 2 =
∞
n=1
¯zn(w + u)n
=
∞
n=1
¯znwn + ¯znun
=
∞
n=1
¯znwn +
∞
n=1
¯znun
= (z, w) 2 + (z, u) 2 .
We also see that it exhibits Hermitian-symmetry
(z, w) 2 =
∞
n=1
znwn =
∞
n=1
znwn = (w, z) 2 .
And finally, (z, z) 2 is zero if and only if z = 0. It is thus clear that 2
is a Hilbert
space.

4. 4 VICTOR B. CASTILLO
3. Lp
-Spaces
We now examine the Lp
spaces. Lp
(Ω) is the space of all functions f : Ω → C
such that,
Ω
|f|p
< ∞.
where Ω is the space over which the function is defined, and 0 < p < ∞. These
spaces possess a norm. We define the Lp
-norm of a function to be,
||f||p =
Ω
|f|p
1
p
.
A quick example of an Lp
-space is L1
. This is the space of all functions such that,
Ω
|f| < ∞.
The L1
-norm of a function is thus simply given by, ||f||1 = Ω
|f|.
3.1. L2
-Space. An Lp
-space of particular interest to us will be the space of all
square-integrable complex functions. These are functions f : Ω → C and Ω ⊂ Rn
with the property that
Ω
|f|2
< ∞.
The norm of this space is the usual Lp
norm with p = 2
||f||2 =
Ω
|f|2
1
2
.
The L2
spaces are worth examining because they are the only Lp
space with an
inner product. We can define the L2
inner product as follows,
Definition: Given two functions v, w ∈ L2
(Ω), where Ω ⊂ R, their L2
-inner
product is given by,
(v, w)L2 =
1
|Ω| Ω
vw
where |Ω| is the measure of Ω. For example, the domain Ω = [a, b] has measure
b − a.
We show that this inner product obeys all the requirements of a Hilbert space
inner product. We can check that it is linear in the second position,
(v, αw)L2 =
1
|Ω|
vαw =
α
|Ω|
vw = α(v , w)
also
(v, w+u)L2 =
1
|Ω|
v(w+u) =
1
|Ω|
vw+vu =
1
|Ω|
vw+
1
|Ω|
vu = (v , w)+(v , u).
We check for Hermitian-symmetry
(v, w) =
1
|Ω|
v w =
1
|Ω|
vw = (w , v).

5. ELEMENTARY QUANTUM MECHANICS 5
Finally it is obvious that this inner product is zero if and only if v is zero. Aside
from the issue of topological completeness, we have verified that L2
is a Hilbert
space. We note in passing that the Riesz-Fischer theorem proves that Lp
(Ω) is
complete, although we shall not prove this here.
We should also note that there are spaces which possess a norm and are topo-
logically complete but may or may not have an inner product. These spaces are
called Banach spaces.
3.2. Orthogonal sets in L2
. We will now look at a subspace of L2
([a, b]). We
consider the inner product of two complex exponential functions on the interval
[a, b]. We define the inner product as above. We can take the inner product
between two complex exponentials (e
int
b−a , e
imt
b−a ),
(e
int
b−a , e
imt
b−a )L2 =
1
b − a
b
a
e
−int
b−a e
imt
b−a dt
=
1
b − a
b
a
e
imt−int
b−a dt
=
1
b − a
b
a
e
i(m−n)t
b−a dt
We consider the cases where n = m and n = m separately. For n = m we have
(e
int
b−a , e
imt
b−a )L2 =
1
b − a
b
a
e0
dt
=
1
b − a
b
a
1dt
=
1
b − a
t|b
a
= 1
for n = m
(e
int
b−a , e
imt
b−a )L2 =
1
b − a
b
a
e
i(m−n)t
b−a dt
=
1
b − a
b − a
i(m − n)
e
i(m−n)t
b−a |b
a
=
1
i(m − n)
(e
i(m−n)b
b−a − e
i(m−n)a
b−a )
=
1
i(m − n)
e
i(m−n)a
b−a (
e
i(m−n)b
b−a
e
i(m−n)a
b−a
− 1)
=
1
i(m − n)
e
i(m−n)a
b−a (ei(m−n)
− 1).
Since m − n is just an integer, ei(m−n)
= 1. So this is always equal to zero.

6. 6 VICTOR B. CASTILLO
We have that
(e
int
b−a , e
imt
b−a )L2 =
1 if n = m
0 if n = m
What this means is that this set of complex exponential functions form an orthog-
onal set in L2
([a, b]). This set can be thought of as orthonormal basis vectors for
L2
([a, b]).
4. Fourier Series
One should be able to construct any vector in L2
([a, b]) as a linear combination
of orthogonal basis vectors. In the case of L2
-space, we should be able to write any
L2
functions as a linear combination of our complex exponentials. So given any
L2
([a, b]) function f, we should be able to write
f(x) =
∞
−∞
f(n)e2πint
for appropriate f(n). This is indeed the case and it leads us into our next topic of
discussion, the Fourier Series.
We begin by defining the Fourier series for a given periodic function f(x). For
a function to be periodic with periodicity p, it must be true that f(x) = f(x+p) =
f(x + 2p) = ... and f(x) = f(x − p) = f(x − 2p) = ....
Definition: Given a function f(x) with period p = 2a, it’s Fourier series is,
f(x) =
∞
n=−∞
f(n)e
2πinx
p x ∈ [−a, a]
where the fourier coefficients, ˆf(n), are
f(n) =
1
p
a
−a
f(x)e
−2πinx
p = e
2πinx
p , f(x) ,
namely the inner product of the function with the basis vectors. We can relate this
to the analogous case in real vector spaces where given an orthonormal basis of
vectors (e1, e2, e3, ..en), we could construct any vector as
v = (e1, v)e1 + (e2, v)e2 + (e3, v)e3 + ...(en, v)en.
The Fourier series intends to express a function f(x) with a linear combination of
complex exponentials.
Something to note is that there is still the question of convergence. That is
to say, does the Fourier series converge to the function f(x)? This is essentially
the question of the completeness of the orthonormal set {e
2πint
p }∞
n=−∞. That is,
are there basis vectors that are not among these complex exponentials, or have
we indeed found all of the basis vectors? It turns out that the Fourier series does
indeed converge, but for the purpose of this discussion we will accept this without
proof.

7. ELEMENTARY QUANTUM MECHANICS 7
4.1. Example. A particularly useful example that we will return to often through-
out this study is that of the square-pulse wave. It is the function
f(x) =
1 if x ∈ [−1
2 , 1
2 ]
0 if otherwise
where f ∈ L2
([−1, 1]).
To obtain it’s Fourier series we begin by computing the fourier coefficients.
f(n) =
1
2
1
2
− 1
2
e−iπnx
=
1
−2iπn
e−iπnx
1
2
− 1
2
=
1
−2iπn
(e
iπn
2 − e− iπn
2 )
=
1
−2iπn
(cos(
πn
2
) + i sin(
πn
2
) − cos(
πn
2
) + i sin(
πn
2
))
=
sin(πn
2 )
πn
Its Fourier series is thus
f(x) =
k
n=−k
sin(πn
2 )
πn
eiπnx
x ∈ [−a, a].
It is instructive to see how increasing the value of n changes the approximation
of f(x). Figures 1, 2, and 3 show the approximation for k = 1, k = 3 and k = 10,
respectively. We observe that the Fourier series for k = 1 is the best approximation
1.0 0.5 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Figure 1. for k=1

8. 8 VICTOR B. CASTILLO
1.0 0.5 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Figure 2. for k=3
1.0 0.5 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Figure 3. for k=10
of the square-pulse using just a single sinusoid. We also see that as the number of
terms, k, in the Fourier series increase we obtain a better and better approximation
of the original function f(x).

9. ELEMENTARY QUANTUM MECHANICS 9
There is also something more to be said about the Fourier coefficients f(n).
we can analyze the plot of f(n) versus 2πn
p . This plot is called the frequency
spectrum of the Fourier series of f(x). In Figure 4 we plot the frequency spectrum
for k = 10. The previous example and illustrations were of a pulse defined within
Figure 4. for k=10
a small period. We will now examine what happens to the frequency spectrum as
we increase the length of the period. In other words, we would like to see what
happens as the end points of the interval are stretched out towards infinity.
Using the method outlined in the above example, it is easy to see that the
Fourier coefficients of the square-pulse for x ∈ [−a, a] is:
f(n) =
1
2πan
sin(
πn
2a
)
=
1
4a2
sinc(
πn
2a
)
As we shall see, in Figures 5 and 6 the frequency spectrum is a sinc function. It
is very important to notice that as the fundamental period of the pulse becomes
larger the frequency spectrum appears to go from being a discrete distribution to
being a continuous one.
5. The Fourier Transform
We now define the Fourier transform of a complex function.

10. 10 VICTOR B. CASTILLO
Figure 5. for a=10
Figure 6. for a=100
Definition: Let f be an L2
function on [−a, a] for every positive a. Also
suppose
∞
−∞
|f|dt < ∞. The Fourier transform of f is defined to be
F{f(x)}(ω) =
∞
−∞
f(x)e−iωx
dx

11. ELEMENTARY QUANTUM MECHANICS 11
5.1. Example - the square-pulse. We can take the Fourier transform of the
square-pulse wave from the previous section as an example. We recall that this
function was defined:
f(x) =
1 if x ∈ [−1
2 , 1
2 ]
0 if otherwise
for x ∈ [−1, 1].
We can take F{f(x)} to get
F{f(x)}(ω) =
1
−1
e−iωx
dx
=
−1
iω
eiωx
|1
−1
=
−1
iω
(e−iω
− eiω
)
=
−1
iω
(cos(ω) − i sin(ω) − cos(ω) − sin(ω)
=
1
ω
sin(ω)
= sinc(ω).
As expected, the Fourier transform of the square-pulse is a sinc function.
5.2. Properties of the Fourier Transform. The Fourier Transform has many
properties. A few of the properties that will be useful to us throughout this study
and their proofs are listed below.
5.2.1. Linearity. If a and b are numbers and f, g are complex valued functions then:
F{af(x) + bg(x)} = aF{f(x)} + bF{g(x)}
proof:
F{af(x) + bg(x)} =
∞
−∞
(af(x) + bf(x))e−iωx
dx
=
∞
−∞
af(x)e−iωx
dx +
∞
−∞
bg(x)e−iωx
dx
= a
∞
−∞
f(x)e−iωx
dx + b
∞
−∞
g(x)e−iωx
dx
= aF{f(x)} + bF{g(x)}
5.2.2. Time-shifting. If F{f(x)} = F(ω) and x0 is some number then:
F{f(x − x0)} = e−iωx0
F(ω)

12. 12 VICTOR B. CASTILLO
proof: We let x = x − x0 so we have
F{f(x − x0)} =
∞
−∞
f(x )e−iω(x +x0)
dx
= e−iωx0
∞
−∞
f(x )e−iωx
dx
= e−iωx0
F(ω)
5.2.3. Symmetry. We have that if F{f(x)} = F(ω) then, F{F(x)} = 2πf(−ω)
proof: By the inverse of Fourier transform,
f(x) = F−1
{F(ω)} =
1
2π
∞
−∞
F(ω)eiωx
Letting x = −x, we get
f(−x ) =
1
2π
∞
−∞
F(ω)e−iωx
dω
finally we interchange x and ω to get
2πf(−ω) =
∞
−∞
F(x )e−iωx
dx = F{F(x)}
5.2.4. Time Reversal.
F{f(−x)} = F(−ω)
proof: We have that
F{f(−x)} =
∞
−∞
f(−x)e−iωx
dx
Letting x = −x , we have
F{f(−x)} = −
−∞
∞
f(−x )e−iωx
dx =
∞
−∞
f(−x)e−iωx
dx = F(−ω)
5.3. Convolution. There is one more property that will be useful to us in later
developments.
Definition: If f, g ∈ L2
([−∞, ∞]) then the convolution f ∗ g of f and g is
defined by
(f ∗ g)(x) =
∞
−∞
f(τ)g(x − τ)dτ

13. ELEMENTARY QUANTUM MECHANICS 13
Theorem 5.1. Let f, g ∈ L2
([−∞, ∞]) and suppose that
∞
−∞
|f(x)|dx and
∞
−∞
|g(x)|dx
both converge. Let F{f(x)} = F(ω) and F{g(x)} = G(ω). Then we have
Time-Convolution
F{[f ∗ g](x)} = F(ω)G(ω)
and Frequency Convolution
F{f(x)g(x)} =
1
2π
[F ∗ G](ω)
This theorem states that the Fourier transform of the convolution of two func-
tions is just the product of their Fourier transforms. Also the Fourier transform
of the product of two functions is just the convolution of their Fourier transforms.
The proof of this proceeds as follows:
proof: Time-convolution
F{[f ∗ g](x)} =
∞
−∞
∞
−∞
f(τ)g(x − τ)dτ e−iωx
dx
=
∞
−∞
∞
−∞
f(τ)g(x − τ)e−iωx
dτdx
=
∞
−∞
∞
−∞
f(τ)g(x − τ)e−iωx
dxdτ
=
∞
−∞
f(τ)
∞
−∞
g(x − τ)e−iωt
dx dτ
By applying time-shifting to
∞
−∞
g(x − τ)e−iωx
dx, we have that
∞
−∞
g(x − τ)e−iωx
dt = F{g(x − τ)} = e−iωx
G(ω)
Therefore,
F{[f ∗ g](x)} =
∞
−∞
f(τ) e−iωx
G(ω) dτ
= G(ω)
∞
−∞
f(τ)e−iωx
dτ
= G(ω)F(ω)
Theorem 5.2. The Parseval identity for an L2
function f(x) and its Fourier
transform F(ω) is
∞
−∞
[f(x)]2
dx =
∞
−∞
|F(ω)|2
dω.

14. 14 VICTOR B. CASTILLO
We notice here that these are the norm squared of L2
functions. Indeed what
this identity is saying is that the norms of the original function and its Fourier
transform are equal. The proof of the Parseval identity proceeds as follows
Proof: Using Frequency convolution with x = 0
F{f(x)f(x)} =
∞
−∞
[f(x)]2
dx
=
1
2π
[F ∗ F](τ)
=
1
2π
∞
−∞
F(τ)F(−τ)
=
1
2π
∞
−∞
F(τ)F(τ)dτ
=
1
2π
∞
−∞
|F(τ)|2
dτ
With the use of the Fourier Transform we are able to see any L2
function, on either
a bounded or unbounded domain, expressed as a sum of its component frequency
contributions. By the Parseval Identity we see that the sum of the energies of
the component frequencies equals that of the original function. We thus have
enough to begin a preliminary discussion of the Schr¨odinger equation and Quantum
Mechanics.
6. The Schr¨odinger equation and Quantum Mechanics
6.1. The Wave function and the Schr¨odinger equation. In Quantum me-
chanics particles are no longer considered to be point-like objects but wave-like
distributed objects. The most basic wave-like object that can be studied is the
plane wave
Ψ(x, t) = e
i
(kx−Et)
.
This is in one dimension, but the arguments that follow are analogous in the multi-
dimensional case. The step into the Quantum world is characterized by the change
of dynamical variables in the classical setting to operators in the Quantum setting.
We can observe the reasoning behind this change by taking a closer look at the
wave function Ψ. We see that by taking derivatives of Ψ, we get
∂
∂t
Ψ =
1
i
Ee
i
(kx−Et)
=
1
i
EΨ
and
∂
∂x
Ψ =
−1
i
ke
i
(kx−Et)
=
−1
i
kΨ.
We see that taking the time derivative of the wave function is the same as mul-
tiplying it by 1
i E and taking the spacial derivative is the same as multiplying it
by −1
i k. In Quantum Mechanics these dynamical variables, i ∂
∂t and −i ∂
∂x , be-
come operators and are identified as the energy and momentum operators, E and
k respectively.

15. ELEMENTARY QUANTUM MECHANICS 15
In the classical setting, particles obey the following equation relating the total
energy to the sum of the kinetic and potential energies,
E =
|k|2
2m
+ V (x).
Making the appropriate substitutions from above we get,
i
∂
∂t
=
− 2
2m
∂2
∂x2
+ V
Thus any wave function must satisfy this equation,
i
∂
∂t
Ψ =
− 2
2m
∂2
∂x2
Ψ + V Ψ.
This is the time-dependent, one dimensional Schr¨odinger equation.
6.2. Interpretation of the Schro¨odinger equation. Quantum mechanics sug-
gests that the wave function is the most complete description that can be given of
a physical system. The Schr¨odinger equation describes how the quantum state of a
physical system evolves over time. We found above that the one-dimensional time
dependent Schr¨odinger equation without potential is
i
∂
∂t
Ψ =
− 2
2m
∂2
∂x2
Ψ
We are interested in the possible solutions to this equation. And as we suggested
above the wave function,
Ψ(x, t) = e
i
(kx−Et)
is one such solution. We can quickly check to verify that this is indeed a solution.
(We can take 2m = 1 for our purposes) Evaluating the left side we get,
i
∂
∂t
e
i
(kx−Et)
= −Ee
i
(kx−Et)
and the right side gives,
2 ∂2
∂x2
e
i
(kx−Et)
= 2 i
k
∂
∂x
e
i
(kx−Et)
=
i 2
2
k2
e
i
(kx−Et)
= −k2
e
i
(kx−Et)
after equating both sides and cancelling out terms we see that Ψ(x, t) = e
i
(kx−Et)
is a solution if E = k2
.
In the Schr¨odinger interpretation of quantum mechanics we see that particles
can be represented by L2
([−∞, ∞]) functions, commonly known as wave-functions.
We discussed above that L2
([−∞, ∞]) functions can be represented as a super-
position of complex exponentials. Therefore we can interpret quantum states as
elements of a Hilbert space.

16. 16 VICTOR B. CASTILLO
6.3. Fourier Transform and Uncertainty. We have established above that in
the Schr¨odinger picture of quantum mechanics particles can be represented as wave-
functions. The probability of finding a particle in a certain region, Ω, is given by
Ω
¯ΨΨdx
If the region Ω consist of all of R, we recognize that this is merely the L2
inner
product of Ψ with itself. This must equal 1 in order for this interpretation to hold.
We can thus imagine a wave-function which is a Dirac delta function. The Dirac
delta function is,
δ(x) =
∞ if x = 0
0 if otherwise
with the stipulation that
∞
−∞
δ(x)dx = 1.
As a consequence of the above
∞
−∞
δ(x)f(x)dx = f(0).
If the wave function were given as the Dirac delta function then this can be inter-
preted as knowing the location of a particle with complete certainy. This is because
the probability is completely localized at zero. We can take the Fourier transform
of this function.
F{δ(x)}(ω) =
∞
−∞
δ(x)e−iwx
dx
Using the fact that
∞
−∞
δ(x)f(x)dx = f(0), we can see that the transform is a
constant function. The Fourier transform of the position function can be interpreted
as the momentum distribution. In this example we see that if the position is known
with complete accuracy then the momentum is completely undetermined. We will
visit the concept of uncertainty again in a later section.
7. General Formalism of Quantum mechanics
7.1. Hilbert space revisited. We will now expand on the notion of Hilbert space
that was defined at the beginning of this study. As a reminder the definition of a
Hilbert space is Definition: A Hilbert space, H, is a complex vector space which
is topologically complete and possesses an inner product (· , ·) : H × H → C that
obeys the following properties. If v, w ∈ H and α ∈ C then
• Linear in the second variable (v , αw) = α(v , w)
and also (v , u + w) = (v , u) + (v , w)
• Hermitian-symmetry (v , w) = (w , v)
• Nondegeneracy (v , v) ≥ 0 with equality only when v = 0
This inner product is called a Hermitian inner product.

17. ELEMENTARY QUANTUM MECHANICS 17
The dual space of H, denoted H*, is also a vector space. It is the space of all
bounded linear operators on H. A linear operate A : H → A is called Hermitian if
the following holds:
Definition: Given any two vectors v, w ∈ H, the linear operator A : H → H
is called Hermitian if,
(Av, w) = (v, Aw).
It is called Anti-Hermitian if
(Av, w) = −(v, Aw).
We can prove that if v, w ∈ H and A, B are Hermitian operators, then the bracket
of A and B, [A, B] is anti-Hermitian,
([A, B]v, w) = (ABv, w) − (BAv, w)
= (Bv, Aw) − (Av, Bw)
= (v, BAw) − (v, ABw)
= (v, −[A, B]w) = −(v, [A, B]).
In fact if A and B are anti-Hermitian then the bracket of A and B is also
anti-Hermitian,
([A, B]v, w) = (ABv, w) − (BAv, w)
= −(Bv, Aw) + (Av, Bw)
= (v, BAw) − (v, ABw)
= (v, −[A, B]w) = −(v, [A, B]).
We can also show that if A is Hermitian, iA is anti-Hermitian,
(iAv, w) = −i(Av, w) = −i(v, Aw) = −(v, iAw)
Definition: Given v, w ∈ H, a linear operator U : H → H is called Unitary if,
(Uv, Uw) = (v, w).
In other words, Unitary operators are inner product preserving. These Unitary
operators essentially play the same role in complex vector spaces as orthogonal
transformations play in real vector space. They are a rigid transformation of the
vector space. We can check that,
Theorem 7.1. If A is an anti-Hermitian operator, then
eA
= I + A +
1
2!
A2
+ ...
is Unitary.
Proof: It is obvious that e0
= I.

18. 18 VICTOR B. CASTILLO
We also see that d
dt eAt
= AeAt
. So we can take the time derivative of the inner
product (eAt
v, eAt
w) to get,
d
dt
(eAt
v, eAt
w) = (
d
dt
eAt
v, eAt
w) + (eAt
v,
d
dt
eAt
w)
= (AeAt
v, eAt
w) + (eAt
v, AeAt
w) = 0
This is equal to 0 since A is anti-Hermitian. These facts indicate that as time
evolves (eAt
v, eAt
w) experiences no change, thus (eAt
v, eAt
w) = (v, w) for all t.
We have arrived at the point where we can give a formal definition of Quantum
mechanics.
7.2. Definition of Quantum Mechanics. Quantum mechanics is a complex
Hilbert space H (state space) which possesses a set of Hermitian operators. El-
ements of H are called kets and are denoted |v , where v is just an index. Elements
of the dual space H∗ are called bras and are denoted v|. We should note the
following definitions
v| (|v , ·)
v|w (|v , |w ).
This Hilbert space includes a particularly important Hermitian operator, H,
called the Hamiltonian. Classically this Hamiltonian is equal to the energy, which
we recall is
|k|2
2m
+ V (x).
Thus making the appropriate identifications in the Schr¨odinger equation, we can
rewrite it as
i
∂
∂t
|v = H|v
or simply
E|v = H|v
7.3. Probability and the Heisenberg uncertainty relation. If A : H → H is
an Hermitian operator (an observable) and |v is an element of H (also known as a
quantum state), then the expected value of A on the quantum state |v is defined
to be
A = v|A|v .
We recall from our statistical interpretation that we assume v|v = 1. We can also
define the variance of the observable A, denoted ∆A, as follows:
∆A (A − A )2 .
We can compute the expected value of the commutator of two Hermitian operators
A and B as follows.
[A, B] = v|AB|v − v|BA|v

19. ELEMENTARY QUANTUM MECHANICS 19
Using properties of Hermitian operators we see that,
v|BA|v = v|(BA|v )
= (|v , BA|v )
= (B|v , A|v )
= (AB|v , |v )
= (|v , AB|v )
= v|AB|v
So we have that
[A, B] = v|AB|v − v|BA|v
= v|AB|v − ( v|AB|v )
= 2iIm v|AB|v
Using the fact that |Im(z)| ≤ |z| we may conclude that
| [A, B] | = |2iIm v|AB|v | ≤ 2| v|AB|v |.
The Schwartz inequality The Schwartz inequality says that given |v and
|w , we have that v|w 2
≤ v|v w|w . This is equal if and only if |v is a (complex)
multiple of |w .
This inequality can be used on A|v and B|v . Doing so we can see that,
| [A, B] | ≤ 2| v|AB|v |
≤ 2|(A|v )||(B|v )|
≤ 2 v|A2|v v|B2|v
= 2 A2 B2
We may replace A and B above with A − a and B − b, where a and b are
constants to get,
| [A − a, B − b] | ≤ 2 (A − a)2 (B − b)2 .
We note however that
[(A − a), (B − b)] = [A − a, B] − [A − a, b]
= [A, B] − [a, B] − [A, b] − [a, b]
The last three terms all zero so we are left with [(A−a), (B−b)] = [A, B]. Therefore
we have that,
| [A, B] | = | [A − a, B − b] | ≤ 2 (A − a)2 (B − b)2
We may indeed take a = A and b = B so the the above becomes
| [A, B] | ≤ 2 (A − A )2 (B − B )2 = 2∆A∆B
This is the Heisenberg uncertainty relation. It says that the product of the standard
deviation of any two observables, which are associated with a quantum system, is
greater than the average of the norm of the commutators. Applying this to wave
mechanics, we consider the set of momentum and position operators regenerated

20. 20 VICTOR B. CASTILLO
by (pi)m
i=1 and (qj
)n
j=1, where pi : H → H and qj
: H → H.We can compose them
to find,
[pi, qj
]f = −i [
∂
∂qi
, qj
]f = −i (
∂
∂qi
(qj
f) − qj ∂
∂qi
f) = −i f.
thus we find [pi, qj
] = −i δj
i . By using the Heisenberg uncertainty relation we can
concluded that,
2
≤ ∆pi∆qj
Intuitively this means that you can not know the momentum and the position
simultaneously to an arbitrary degree of accuracy. The uncertainty will always be
greater than /2
References
[1] A. Messiah. Quantum Mechanics I, II. Dover Publications (1999)
[2] T. Frankel. The Geometry of Physics. Cambridge University Press (2003)
[3] W. Rudin. Real and Complex Analysis. McGraw-Hill Science/Engineering/Math Third Ed.
(1986)
[4] D. Powers. Boundary Value Problems. Academic Press. Fifth Ed. (2005)