We saw in the previous chapter how to describe the spin state of an elec-
tron.
2
The “system” in this case is just that property of an electron that
responds to a gradient in an applied magnetic field. The state of this system
is a member of a 2-dimensional vector space. This is because this spin can take
one of only two possible values, ±~/2. An electron in either of these states is
described by the basis vectors
|0i = |+
~
2
i, |1i = |−
~
2
i.
A general state |ψi is a linear combination of these basis vectors with complex
coefficients:
|ψi = α
0
|0i + α
1
|1i.
As we will show, these basis states are mutually orthogonal and are nor-
malized, so that they form an orthonormal basis. This is similar to repre-
senting a physical 2-dimensional vector in terms of its components along two
orthogonal directions. This vector is represented as the column matrix of its
components: [
α
0
α
1
]
T
.
We can easily generalize this to higher dimensions. Such a picture is rel-
evant when the set of basis states for the system is larger. For example, the
system may be the magnetic moment of a spin-
3
2
atomic nucleus. This object
would have four possible states distinguished in a non-uniform magnetic field:
{|ji} = {|
3
2
i, |
1
2
i, |−
1
2
i, |−
3
2
i}.
Another example is the electronic energy of the hydrogen atom. This system
actually has a countable infinity of possible energy states labelled by the so-
called “principal quantum number” n:
{|ni}, n = 0, 1, 2 . . . .
This Hilbert space is actually infinite dimensional, though we might say the
dimensionality is “countable.” If we were concentrating on the position states
of a particle confined to a line then the possible states are a continuous infinity
labelled by the values of the position x:
{|xi}, − ∞ ≤ x ≤ +∞.
This Hilbert space is also infinite dimensional, and the dimensionality is con-
tinuous and uncountable.
2
The spin space is a subspace of the total state space of an electron, which contains
descriptors of all possible compatible measurable properties of the electron. This Hilbert
space can be expressed as a direct product of the independent subspaces.
The Essentials of Quantum Mechanics 37
Box 3.2: Hilbert Space
The linear vector space of Box 3.1 turns into something rich enough to
represent states of a physical system if a little more structure is added to it.
We now have a Hilbert space H
n
which is defined as a complex vector space
with an inner product (., .) ∈ which satisfies
(I1) (v, v) ≥ 0, (v, v) = 0 iff v = 0.
(I2) (u, v) = (v, u)
∗
(I3) (u, αv) = α(u, v)
(I4) (v
1
+ v
2
, v
3
) = (v
1
, v
3
) + (v
2
, v
3
).
With this structure in place, a vector space becomes a pre-Hilbert space,
and is a Hilbert space if the dimension is finite. For infinite-dimensional Hilbert
spaces, one needs the additional criterion of the space being complete under
the inner product, which we will not discuss here
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