When you describe the state of a physical system, you collect all the pa-
rameters required to fully specify it: for instance, the state of a ball may be
specified by its position in space, its velocity, and maybe its rate of spin; the
state of a volume of gas by its temperature and its pressure. If you are trying
to describe a quantum system like a hydrogen atom, you may think specifying
the position and velocity of the atom and its constituents, the nucleus and the
electron would give the quantum state. Whether this is true, or even possible
in principle, depends on how you are trying to see the atom: which properties
you are trying to measure and what experiments you are using to measure its
properties,
So we first identify a system, an isolated set of physical properties that we
have experimental access to and are trying to describe. The quantum state of
the system, denoted by the notation |statei,
1
is represented by measured values
1
The notation due to Dirac that we use in quantum mechanics may need some more
clarification. A state is labelled abstractly as |ψi, or as |0i, or as |x
i
i. The labels are just
mnemonics to tag the state. They may be numbers but are not the components of the
vector in any basis. For instance, |0i does not mean the zero vector, for which we will
use the notation
~
0. The 0 used as a label is an indication of a first basis vector in the
computational basis.
33
34 Introduction to Quantum Physics and Information Processing
of the physical properties used to describe it. It is important to know which
properties are independent of each other, measuring which do not interfere
with the other properties. The outcome of the measurement could be one of
many possibilities. Each possibility labels a different state. The set of all these
forms the state space of the system.
In the last chapter, the particular property of spin of an electron was
targeted for study, by designing the Stern–Gerlach experiment. This led to a
state space of two states.
The properties of a quantum state turn out to conform to those of a vector
in the mathematical sense: a member of a complex vector space (see Box 3.1),
with a notion of norm or inner product defined on it. Such a vector space
is called a Hilbert space (see Box 3.2). The vector describing a state must
also have unit norm, since we will be attaching a notion of probabilities to
the state. The complex vector may also have imaginary components, but an
overall phase factor is unimportant since we have no means of measuring it.
Thus a state is unit vector in complex space, modulo an overall phase factor.
Postulate 1. The state of an isolated quantum mechanical system is a unit
vector in Hilbert space.
Box 3.1: Linear Vector Space
A vector space V is a set of objects v, called vectors, that abstractly
satisfy the properties of closure under an operation of addition, and under
multiplication by a scalar which belongs to a field F, that for example could
be real or complex numbers. In what follows, a vector is designated by a
boldface, such as v, while a scalar is not.
The axioms defining a vector space are
1. Addition: one can define an operation “+” such that for any vectors
v
i
∈ V
n
,
(A1) V
n
is closed under +: v
1
+ v
2
∈ V
n
,
(A2) + is commutative: v
1
+ v
2
= v
2
+ v
1
,
(A3) + is associative: (v
1
+ v
2
) + v
3
= v
1
+ (v
2
+ v
3
) .
(A4) ∃ a zero vector or additive identity 0 ∈ V
n
such that v + 0 = v.
(A5) For each v ∈ V, ∃ an additive inverse −v ∈ V
n
such that v+(−v) =
0.
2. Scalar Multiplication: for any scalar α ∈ F and vector v
i
∈ V
n
,
(M1) V is closed under scalar multiplication: αv ∈ V
n
,
(M2) For the multiplicative identity 1, we have 1v = v,
(M3) Multiplication by the scalar 0 gives the zero vector: 0v = 0,
The Essentials of Quantum Mechanics 35
(M4) Associativity: α(βv) = (αβ)v,
(M5) Distributivity over vector addition: α(v
1
+ v
2
) = αv
1
+ αv
2
,
(M6) Distributivity over scalar addition: (α + β)v = αv + βv
The element −v = (−1)v is the additive inverse of v.
Vectors can be represented by components if we choose a set of “coor-
dinates” or basis vectors for the representation. A basis for a vector space
consists of a set of vectors {e
i
} whose defining properties are:
1. they are linearly independent: no basis vector can be expressed as a
linear combination of the other basis vectors; no set of numbers {a
i
}
can be found such that
X
i
a
i
e
i
= 0.
2. they span the vector space V: any vector v ∈ V can be expressed as a
linear combination of the basis vectors:
v = c
1
e
1
+ c
2
e
2
+ ··· + c
n
e
n
.
The index i counts the basis vectors: i = 1…n. The total number n of basis
vectors is the dimension of the vector space. This dimension can be finite or
infinite. The index i can be discrete or continuous. We will only be dealing
here with finite-dimensional complex vector spaces.
A vector space in general has more than one basis. A vector represented
by its components is also represented as a column matrix:
v =
c
1
c
2
.
.
.
c
n
.
To save space, we will also represent this as the transpose of a row vector
v =
h
c
1
c
2
. . . c
n
i
T
.
This representation is extremely useful when we consider transformations of
a vector space into another by linear maps, which can be represented by
matrices.
Vector spaces are familiar to us from 3-dimensional spacial vectors, but
the above definitions generalize such properties to a larger class of objects.
We find that even continuous functions of complex numbers that are infinitely
differentiable and vanish fast at infinity form a vector space
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