2.1 The Time-Dependent Schrödinger Equation
27
2.1.1 Observables
Let ˆ
X be a linear operator. Its adjoint ˆ
X
† is defined by requiring that the equality
all space
ψ
∗ ˆ
X
†
φdx =
all space
φ
∗ ˆ
X ψdx
(2.6)
is valid for any pair of wavefunctions φ and ψ. In Dirac notation:
ψ
ˆ
X
†
φ
=
φ
ˆ
X
ψ
∀ φ, ψ .
(2.7)
An operator ˆ
X is said to be Hermitian if ˆ
X = ˆ
X
† , or equivalently
ψ
ˆ
X
φ
=
φ
ˆ
X
ψ
∗
∀ φ, ψ .
(2.8)
The complex number λ is an eigenvalue of ˆ
X if ˆ
X φ = λφ, and φ is the corresponding
eigenfunction. The spectrum of an operator is the full set of its eigenvalues. For a
Hermitian operator we have that the eigenvalues are real numbers and the eigenvectors are mutually orthogonal (or at least they can be chosen so as to be orthogonal)
and represent a basis for the vector space on which the operator is defined.
In quantum mechanics a physical observable is associated with a Hermitian operator. For example, for position ˆ
x, linear momentum ˆ
p x and kinetic energy ˆ
T we
have:
ˆ
x = x
ˆ
p = −i
∂
∂ x
ˆ
T = −
2
2m
∇
2
.
(2.9)
The spectrum of a Hermitian operator ˆ
X can be discrete, Eq. (2.10), or continuous,
Eq. (2.11):
ˆ
X φ i = λ i φ i i = 1, 2, . . .
(2.10)
ˆ
X φ λ = λφ λ λ ∈ R .
(2.11)
The eigenfunctions of an observable ˆ
O with a discrete spectrum are normalizable in
the sense of Eq. (2.3), while those of an observable with a continuous spectrum require
a different normalization because the integral (2.2) does not exist (see Appendix C).
In fact, a wavefunction is normalizable only if it tends to zero when any space coordinate takes values far from a limited interval (“bound state”): but this requirement
is satisfied only by some “special” eigenfunctions (those satisfying the boundary
conditions), making the spectrum discrete.
We now consider a system in a physical state described by the normalized wavefunction Ψ (t), and an observable ˆ
O with a discrete spectrum: ˆ
Oφ i = λ i φ i . We
assume that ˆ
O is not explicitly dependent on time. We choose the eigenfunctions φ i
to be orthonormal, i.e.,
27
2.1.1 Observables
Let ˆ
X be a linear operator. Its adjoint ˆ
X
† is defined by requiring that the equality
all space
ψ
∗ ˆ
X
†
φdx =
all space
φ
∗ ˆ
X ψdx
(2.6)
is valid for any pair of wavefunctions φ and ψ. In Dirac notation:
ψ
ˆ
X
†
φ
=
φ
ˆ
X
ψ
∀ φ, ψ .
(2.7)
An operator ˆ
X is said to be Hermitian if ˆ
X = ˆ
X
† , or equivalently
ψ
ˆ
X
φ
=
φ
ˆ
X
ψ
∗
∀ φ, ψ .
(2.8)
The complex number λ is an eigenvalue of ˆ
X if ˆ
X φ = λφ, and φ is the corresponding
eigenfunction. The spectrum of an operator is the full set of its eigenvalues. For a
Hermitian operator we have that the eigenvalues are real numbers and the eigenvectors are mutually orthogonal (or at least they can be chosen so as to be orthogonal)
and represent a basis for the vector space on which the operator is defined.
In quantum mechanics a physical observable is associated with a Hermitian operator. For example, for position ˆ
x, linear momentum ˆ
p x and kinetic energy ˆ
T we
have:
ˆ
x = x
ˆ
p = −i
∂
∂ x
ˆ
T = −
2
2m
∇
2
.
(2.9)
The spectrum of a Hermitian operator ˆ
X can be discrete, Eq. (2.10), or continuous,
Eq. (2.11):
ˆ
X φ i = λ i φ i i = 1, 2, . . .
(2.10)
ˆ
X φ λ = λφ λ λ ∈ R .
(2.11)
The eigenfunctions of an observable ˆ
O with a discrete spectrum are normalizable in
the sense of Eq. (2.3), while those of an observable with a continuous spectrum require
a different normalization because the integral (2.2) does not exist (see Appendix C).
In fact, a wavefunction is normalizable only if it tends to zero when any space coordinate takes values far from a limited interval (“bound state”): but this requirement
is satisfied only by some “special” eigenfunctions (those satisfying the boundary
conditions), making the spectrum discrete.
We now consider a system in a physical state described by the normalized wavefunction Ψ (t), and an observable ˆ
O with a discrete spectrum: ˆ
Oφ i = λ i φ i . We
assume that ˆ
O is not explicitly dependent on time. We choose the eigenfunctions φ i
to be orthonormal, i.e.,