5Axioms for quantum mechanics
IB Quantum Mechanics
5.1 States and observables
–
States of a quantum system correspond to non-zero elements of a complex
vector space
V
(which has nothing to do with the potential), with
ψ
and
αψ physically equivalent for all α ∈ C \ {0}.
Furthermore, there is a complex inner product (
φ, ψ
) defined on
V
satisfying
(φ, α
1
ψ
1
+ α
2
ψ
2
) = α
1
(φ, ψ
1
) + α
2
(φ, ψ
2
)
(β
1
φ
1
+ β
2
φ
2
, ψ) = β
∗
1
(φ
1
, ψ) + β
∗
2
(φ
2
, ψ)
(φ, ψ) = (ψ, φ)
∗
kψk
2
= (ψ, ψ) ≥ 0
kψk = 0 if and only if ψ = 0.
Note that this is just the linear algebra definition of what a complex inner
product should satisfy.
An operator A is a linear map V → V satisfying
A(αψ + βφ) = αAψ + βAφ.
For any operator
A
, the Hermitian conjugate or adjoint, denoted
A
†
is
defined to be the unique operator satisfying
(φ, A
†
ψ) = (Aφ, ψ).
An operator Q is called Hermitian or self-adjoint if
Q
†
= Q.
A state χ 6= 0 is an eigenstate of Q with eigenvalue λ if
Qχ = λχ.
The set of all eigenvalues of Q is called the spectrum of Q.
A measurable quantity, or observable, in a quantum system corresponds to
a Hermitian operator.
So far, we have worked with the vector space of functions (that are sufficiently
smooth), and used the integral as the inner product. However, in general, we can
work with arbitrary complex vector spaces and arbitrary inner products. This
will become necessary when we, say, study the spin of electrons in IID Principles
of Quantum Mechanics.
Even in the case of the vector space of (sufficiently smooth) functions, we will
only work with this formalism informally. When we try to make things precise,
we will encounter a lot of subtleties such as operators not being well-defined
everywhere.