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Chapter 3
Operator methods in
quantum mechanics
While the wave mechanical formulation has proved successful in describing
the quantum mechanics of bound and unbound particles, some properties can
not be represented through a wave-like description. For example, the electron
spin degree of freedom does not translate to the action of a gradient operator.
It is therefore useful to reformulate quantum mechanics in a framework that
involves only operators.
Before discussing properties of operators, it is helpful to introduce a further
simplification of notation. One advantage of the operator algebra is that it
does not rely upon a particular basis. For example, when one writes
H =
pˆ
2
2 m
where the hat denotes an operator, we can equally represent the momentum
operator in the spatial coordinate basis, when it is described by the differential
operator, ˆp = −iℏ∂ x
, or in the momentum basis, when it is just a number
p ˆ = p. Similarly, it would be useful to work with a basis for the wavefunction
which is coordinate independent. Such a representation was developed by
Dirac early in the formulation of quantum mechanics.
In the parlons of mathematics, square integrable functions (such as wave-
functions) are said form a vector space, much like the familiar three-dimensional
vector spaces. In the Dirac notation, a state vector or wavefunction, ψ, is
represented as a “ket”, |ψ〉. Just as we can express any three-dimensional
vector in terms of the basis vectors, r = xˆe 1
, so we can expand
any wavefunction as a superposition of basis state vectors,
|ψ〉 = λ 1 |ψ 1 〉 + λ 2 |ψ 2 〉 + · · ·.
Alongside the ket, we can define the “bra”, 〈ψ|. Together, the bra and ket
define the scalar product
〈φ|ψ〉 ≡
∞
−∞
dx φ
∗ (x)ψ(x) ,
from which follows the identity, 〈φ|ψ〉
∗ = 〈ψ|φ〉. In this formulation, the real
space representation of the wavefunction is recovered from the inner prod-
uct ψ(x) = 〈x|ψ〉 while the momentum space wavefunction is obtained from
ψ(p) = 〈p|ψ〉. As with a three-dimensional vector space where a · b ≤ |a| |b|,
the magnitude of the scalar product is limited by the magnitude of the vectors,
〈ψ|φ〉 ≤
〈ψ|ψ〉〈φ|φ〉 ,
a relation known as the Schwartz inequality.
3.1. OPERATORS 20
3.1 Operators
An operator
A is a “mathematical object” that maps one state vector, |ψ〉,
into another, |φ〉, i.e.
A|ψ〉 = |φ〉. If
A|ψ〉 = a|ψ〉 ,
with a real, then |ψ〉 is said to be an eigenstate (or eigenfunction) of
A with
eigenvalue a. For example, the plane wave state ψ p
(x) = 〈x|ψ p
〉 = A e
ipx/ℏ is
an eigenstate of the momentum operator, ˆp = −iℏ∂ x
, with eigenvalue p.
For a free particle, the plane wave is also an eigenstate of the Hamiltonian,
H =
pˆ
2
2 m
with eigenvalue
p
2
2 m
In quantum mechanics, for any observable A, there is an operator
A which
acts on the wavefunction so that, if a system is in a state described by |ψ〉,
the expectation value of A is
〈A〉 = 〈ψ|
A|ψ〉 =
∞
−∞
dx ψ
∗ (x)
Aψ(x). (3.1)
Every operator corresponding to an observable is both linear and Hermitian:
That is, for any two wavefunctions |ψ〉 and |φ〉, and any two complex numbers
α and β, linearity implies that
A(α|ψ〉 + β|φ〉) = α(
A|ψ〉) + β(
A|φ〉).
Moreover, for any linear operator
A, the Hermitian conjugate operator
(also known as the adjoint) is defined by the relation
〈φ|
Aψ〉 =
dx φ
∗ (
Aψ) =
dx ψ(
A
† φ)
∗ = 〈
A
† φ|ψ〉. (3.2)
From the definition, 〈
A
† φ|ψ〉 = 〈φ|
Aψ〉, we can prove some useful rela-
tions: Taking the complex conjugate, 〈
A
† φ|ψ〉
∗ = 〈ψ|
A
† φ〉 = 〈
Aψ|φ〉, and
then finding the Hermitian conjugate of
A
† , we have
〈ψ|
A
† φ〉 = 〈(
A
† )
† ψ|φ〉 = 〈
Aψ|φ〉, i.e. (
A
† )
†
A.
Therefore, if we take the Hermitian conjugate twice, we get back to the same
operator. Its easy to show that (λ
A)
† = λ
∗ ˆ A
† and (
A +
B)
†
A
†
B
† just
from the properties of the dot product. We can also show that (
A
B)
†
B
† ˆ A
†
from the identity, 〈φ|
A
Bψ〉 = 〈
A
† φ|
Bψ〉 = 〈
B
† ˆ A
† φ|ψ〉. Note that operators
are associative but not (in general) commutative,
A
B|ψ〉 =
A(
B|ψ〉) = (
A
B)|ψ〉 & =
B
A|ψ〉.
A physical variable must have real expectation values (and eigenvalues).
This implies that the operators representing physical variables have some spe-
cial properties. By computing the complex conjugate of the expectation value
of a physical variable, we can easily show that physical operators are their own
Hermitian conjugate,
〈ψ|
H|ψ〉
∗
[∫
∞
−∞
ψ
∗ (x)
Hψ(x)dx
]
∗
∞
−∞
ψ(x)(
Hψ(x))
∗ dx = 〈
Hψ|ψ〉.
i.e. 〈
Hψ|ψ〉 = 〈ψ|
Hψ〉 = 〈
H
† ψ|ψ〉, and
H
†
H. Operators that are their
own Hermitian conjugate are called Hermitian (or self-adjoint).
3.1. OPERATORS 22
' Example: Consider the harmonic oscillator Hamiltonian
H =
pˆ
2
2 m
1
2
mω
2 x
2 .
Later in this chapter, we will see that the eigenstates, |n〉, have equally-spaced eigen-
values, En = ℏω(n + 1/2), for n = 0, 1 , 2 , · · ·. Let us then consider the time-evolution
of a general wavepacket, |ψ(0)〉, under the action of the Hamiltonian. From the equa-
tion above, we find that |ψ(t)〉 =
n
|n〉〈n|ψ(0)〉e
−iEnt/ℏ
. Since the eigenvalues are
equally spaced, let us consider what happens when t = t r
≡ 2 πr/ω, with r integer.
In this case, since e
2 πinr = 1, we have
|ψ(t r
n
|n〉〈n|ψ(0)〉e
−iωtr / 2 = (−1)
r |ψ(0)〉.
From this result, we can see that, up to an overall phase, the wave packet is perfectly
reconstructed at these times. This recurrence or “echo” is not generic, but is a
manifestation of the equal separation of eigenvalues in the harmonic oscillator.
' Exercise. Using the symmetry of the harmonic oscillator wavefunctions under
parity show that, at times tr = (2r + 1)π/ω, 〈x|ψ(tr )〉 = e
−iωtr / 2 〈−x|ψ(0)〉. Explain
the origin of this recurrence.
The time-evolution operator is an example of a unitary operator. The
latter are defined as transformations which preserve the scalar product, 〈φ|ψ〉 =
U φ|
U ψ〉 = 〈φ|
U
U ψ〉
!
= 〈φ|ψ〉, i.e.
U
U = I.
3.1.2 Uncertainty principle for non-commuting operators
For non-commuting Hermitian operators, [
A,
B] &= 0, it is straightforward to
establish a bound on the uncertainty in their expectation values. Given a state
|ψ〉, the mean square uncertainty is defined as
(∆A)
2 = 〈ψ|(
A − 〈
A〉)
2 ψ〉 = 〈ψ|
U
2 ψ〉
(∆B)
2
= 〈ψ|(
B − 〈
B〉)
2
ψ〉 = 〈ψ|
V
2
ψ〉 ,
where we have defined the operators
U =
A − 〈ψ|
Aψ〉 and
V =
B − 〈ψ|
Bψ〉.
Since 〈
A〉 and 〈
B〉 are just constants, [
U ,
V ] = [
A,
B]. Now let us take the
scalar product of
U |ψ〉 + iλ
V |ψ〉 with itself to develop some information about
the uncertainties. As a modulus, the scalar product must be greater than or
equal to zero, i.e. expanding, we have 〈ψ|
U
2 ψ〉 + λ
2 〈ψ|
V
2 ψ〉 + iλ〈
U ψ|
V ψ〉 −
iλ〈
V ψ|
U ψ〉 ≥ 0. Reorganising this equation in terms of the uncertainties, we
thus find
(∆A)
2
2
(∆B)
2
U ,
V ]|ψ〉 ≥ 0.
If we minimise this expression with respect to λ, we can determine when
the inequality becomes strongest. In doing so, we find
2 λ(∆B)
2
U ,
V ]|ψ〉 = 0, λ = −
i
〈ψ|[
U ,
V ]|ψ〉
(∆B)
2
Substiuting this value of λ back into the inequality, we then find,
(∆A)
2
(∆B)
2
≥ −
〈ψ|[
U ,
V ]|ψ〉
2
.
3.1. OPERATORS 23
We therefore find that, for non-commuting operators, the uncertainties obey
the following inequality,
∆A ∆B ≥
i
〈[
A,
B]〉.
If the commutator is a constant, as in the case of the conjugate operators
[ˆp, x] = −iℏ, the expectation values can be dropped, and we obtain the rela-
tion, (∆A)(∆B) ≥
i
2
[
A,
B]. For momentum and position, this result recovers
Heisenberg’s uncertainty principle,
∆p ∆x ≥
i
〈[ˆp, x]〉 =
Similarly, if we use the conjugate coordinates of time and energy, [
E, t] = iℏ,
we have
∆E ∆t ≥
3.1.3 Time-evolution of expectation values
Finally, to close this section on operators, let us consider how their expectation
values evolve. To do so, let us consider a general operator
A which may itself
involve time. The time derivative of a general expectation value has three
terms.
d
dt
〈ψ|
A|ψ〉 = ∂t(〈ψ|)
A|ψ〉 + 〈ψ|∂t
A|ψ〉 + 〈ψ|
A(∂t|ψ〉).
If we then make use of the time-dependent Schr¨odinger equation, iℏ∂t|ψ〉 =
H|ψ〉, and the Hermiticity of the Hamiltonian, we obtain
d
dt
〈ψ|
A|ψ〉 =
i
〈ψ|
H
A|ψ〉 − 〈ψ|
A
H|ψ〉
i
〈ψ|[
H,
A]|ψ〉
+〈ψ|∂ t
A|ψ〉.
This is an important and general result for the time derivative of expectation
values which becomes simple if the operator itself does not explicitly depend
on time,
d
dt
〈ψ|
A|ψ〉 =
i
〈ψ|[
H,
A]|ψ〉.
From this result, which is known as Ehrenfest’s theorem, we see that expec-
tation values of operators that commute with the Hamiltonian are constants
of the motion.
' Exercise. Applied to the non-relativistic Schr¨odinger operator for a single
particle moving in a potential,
H =
pˆ
2
2 m
〈 pˆ〉
m
pˆ〉 = −〈∂ x
V 〉.
Show that these equations are consistent with the relations,
Paul Ehrenfest 1880-
An Austrian
physicist and
mathematician,
who obtained
Dutch citizenship
in 1922. He
made major
contributions
to the field
of statistical
mechanics and its relations with
quantum mechanics, including the
theory of phase transition and the
Ehrenfest theorem.
d
dt
〈x〉 =
∂H
∂p
d
dt
〈pˆ〉 = −
∂H
∂x
the counterpart of Hamilton’s classical equations of motion.
3.2. SYMMETRY IN QUANTUM MECHANICS 25
one can show that, for a constant vector a, the unitary operator
U (a) = exp
[
i
a · ˆp
]
acting in the Hilbert space of a Schr¨odinger particle performs a spatial trans-
lation,
U
† (a)f (r)
U (a) = f (r + a), where f (r) denotes a general algebraic
function of r.
' Info. The proof runs as follows: With ˆp = −iℏ∇,
U
† (a) = e
a·∇
∞ ∑
n=
n!
ai 1
· · · ai n
∇i 1
· · · ∇i n
where summation on the repeated indicies, im is assumed. Then, making use of the
Baker-Hausdorff identity (exercise)
e
Aˆ ˆ Be
− Aˆ
B + [
A,
B] +
[
A, [
A,
B]] + · · · , (3.3)
it follows that
U
† (a)f (r)
U (a) = f (r) + a i 1
i 1
f (r)) +
a i 1
a i 2
i 1
i 2
f (r)) + · · · = f (r + a) ,
where the last identity follows from the Taylor expansion.
' Exercise. Prove the Baker-Hausdorff identity (3.3).
Therefore, a quantum system has spatial translation as an invariance group if
and only if the following condition holds,
U (a)
H =
H
U (a), i.e. pˆ
H =
H pˆ.
This demands that the Hamiltonian is independent of position,
H =
H(pˆ),
as one might have expected! Similarly, the group of unitary transforma-
tions,
U (b) = exp[−
i
ℏ
b · ˆr], performs translations in momentum space.
Moreover, spatial rotations are generated by the transformation
U (b) =
exp[−
i
ℏ
θe n
L], where
L = ˆr × pˆ denotes the angular momentum operator.
' Exercise. For an infinitesimal rotation by an angle θ by a fixed axis, ˆen,
construct R[r] and show that
U = I −
i
ℏ
θˆen · L + O(θ
2 ). Making use of the identity
limN →∞(1 −
a
N
N = e
−a , show that “large” rotations are indeed generated by the
unitary transformations
U = exp
[
i
ℏ
θˆen · L
]
As we have seen, time translations are generated by the time evolution op-
erator,
U (t) = exp[−
i
ℏ
Ht]. Therefore, every observable which commutes with
the Hamiltonian is a constant of the motion (invariant under time transla-
tions),
H
A =
A
H ⇒ e
i
ˆ Ht/ℏ ˆ Ae
−i
ˆ Ht/ℏ
=
A, ∀t.
We now turn to consider some examples of discrete symmetries. Amongst
these, perhaps the most important in low-energy physics are parity and time-
reversal. The parity operation, denoted
P , involves a reversal of sign on all
coordinates.
P ψ(r) = ψ(−r).
3.2. SYMMETRY IN QUANTUM MECHANICS 26
This is clearly a discrete transformation. Application of parity twice returns
the initial state implying that
P
2 = 1. Therefore, the eigenvalues of the parity
operation (if such exist) are ±1. A wavefunction will have a defined parity
if and only if it is an even or odd function. For example, for ψ(x) = cos(x),
P ψ = cos(−x) = cos(x) = ψ; thus ψ is even and P = 1. Similarly ψ =
sin(x) is odd with P = −1. Later, in the next chapter, we will encounter the
spherical harmonic functions which have the following important symmetry
under parity,
P Y
!m
! Y lm
. Parity will be conserved if the Hamiltonian
is invariant under the parity operation, i.e. if the Hamiltonian is invariant
under a reversal of sign of all the coordinates.
5
In classical mechanics, the time-reversal operation involves simply “run-
ning the movie backwards”. The time-reversed state of the phase space
coordinates (x(t), p(t)) is defined by (x T
(t), p T
(t)) where x T
(t) = x(t) and
p T
(t) = −p(t). Hence, if the system evolved from (x(0), p(0)) to (x(t), p(t)) in
time t and at t we reverse the velocity, p(t) → −p(t) with x(t) → x(t), at time
2 t the system would have returned to x(2t) = x(0) while p(2t) = −p(0). If this
happens, we say that the system is time-reversal invariant. Of course, this is
just the statement that Newton’s laws are the same if t → −t. A notable case
where this is not true is that of a charged particle in a magnetic field.
As with classical mechanics, time-reversal in quantum mechanics involves
the operation t → −t. However, referring to the time-dependent Schr¨odinger
equation, iℏ∂tψ(x, t) =
Hψ(x, t), we can see that the operation t → −t is
equivalent to complex conjugation of the wavefunction, ψ → ψ
∗ if
H
∗
H.
Let us then consider the time-evolution of ψ(x, t),
ψ(x, 0) → e
−
i
ℏ
ˆ H(x)t ψ(x, 0)
c.c.
→ e
i
ℏ
ˆ H
∗ (x)t ψ
∗ (x, 0)
evolve
→ e
−
i
ℏ
ˆ H(x)t e
i
ℏ
ˆ H
∗ (x)t ψ
∗ (x, 0).
If we require that ψ(x, 2 t) = ψ
∗ (x, 0), we must have
H
∗ (x) =
H(x). Therefore,
H is invariant under time-reversal if and only if
H is real.
' Info. Although the group of space-transformations covers the symmetries
that pertain to “low-energy” quantum physics, such as atomic physics, quantum op-
tics, and quantum chemistry, in nuclear physics and elementary particle physics new
observables come into play (e.g. the isospin quantum numbers and the other quark
charges in the standard model). They generate symmetry groups which lack a classical
counterpart, and they do not have any obvious relation with space-time transforma-
tions. These symmetries are often called internal symmetries in order to underline
this fact.
3.2.2 Consequences of symmetries: multiplets
Having established how to identify whether an operator belongs to a group
of symmetry transformations, we now consider the consequences. Consider
a single unitary transformation
U in the Hilbert space, and an observable
A
which commutes with
U , [
U ,
A] = 0. If
A has an eigenvector |a〉, it follows
that
U |a〉 will be an eigenvector with the same eigenvalue, i.e.
U
A|a〉 =
AU |a〉 = aU |a〉.
This means that either:
5 In high energy physics, parity is a symmetry of the strong and electromagnetic forces, but
does not hold for the weak force. Therefore, parity is conserved in strong and electromagnetic
interactions, but is violated in weak interactions.
3.4. QUANTUM HARMONIC OSCILLATOR 28
numerous and somtimes unexpected applications. It is useful to us now in
that it provides a platform for us to implement some of the technology that
has been developed in this chapter. In the one-dimensional case, the quantum
harmonic oscillator Hamiltonian takes the form,
H =
pˆ
2
2 m
mω
2 x
2 ,
where ˆp = −iℏ∂x. To find the eigenstates of the Hamiltonian, we could
look for solutions of the linear second order differential equation correspond-
ing to the time-independent Schr¨odinger equation,
Hψ = Eψ, where
H =
ℏ
2
2 m
2
x
1
2
mω
2 x
2
. The integrability of the Schr¨odinger operator in this case
allows the stationary states to be expressed in terms of a set of orthogonal
functions known as Hermite polynomials. However, the complexity of the ex-
act eigenstates obscure a number of special and useful features of the harmonic
oscillator system. To identify these features, we will instead follow a method
based on an operator formalism.
First few states of the quantum
harmonic oscillator. Not that the
parity of the state changes from
even to odd through consecutive
states.
The form of the Hamiltonian as the sum of the squares of momenta and
position suggests that it can be recast as the “square of an operator”. To this
end, let us introduce the operator
a =
mω
x + i
pˆ
mω
, a
†
mω
x − i
ˆp
mω
where, for notational convenience, we have not drawn hats on the operators a
and its Hermitian conjuate a
†
. Making use of the identity,
a
† a =
mω
x
2
pˆ
2 ℏmω
i
[x, ˆp] =
H
ℏω
and the parallel relation, aa
†
ˆ H
ℏω
1
2
, we see that the operators fulfil the
commutation relations
[a, a
† ] ≡ aa
† − a
† a = 1.
Then, setting ˆn = a
† a, the Hamiltonian can be cast in the form
H = ℏω(ˆn + 1/2).
Since the operator ˆn = a
† a must lead to a positive definite result, we see
that the eigenstates of the harmonic oscillator must have energies of ℏω/2 or
higher. Moreover, the ground state | 0 〉 can be identified by finding the state
for which a| 0 〉 = 0. Expressed in the coordinate basis, this translates to the
equation,
6
x +
mω
x
ψ 0
(x) = 0, ψ 0
(x) = 〈x| 0 〉 =
mω
π
1 / 2 ℏ
e
−mωx
2 / 2 ℏ .
Since ˆn| 0 〉 = a
† a| 0 〉 = 0, this state is an eigenstate with energy ℏω/2. The
higher lying states can be found by acting upon this state with the operator
a
†
. The proof runs as follows: If ˆn|n〉 = n|n〉, we have
n ˆ(a
† |n〉) = a
† aa
†
a
† a+
|n〉 = (a
† a
† a ︸︷︷︸
ˆn
+a
† )|n〉 = (n + 1)a
† |n〉
6 Formally, in coordinate basis, we have 〈x
′ |a|x〉 = δ(x
′ − x)(a +
ℏ
mω
∂x) and 〈x| 0 〉 = ψ 0 (x).
Then making use of the resolution of identity
R
dx|x〉〈x| = I, we have
〈x|a| 0 〉 = 0 =
Z
dx 〈x|a|x
′ 〉〈x
′ | 0 〉 =
„
x +
ℏ
mω
∂x
«
ψ 0 (x).
3.4. QUANTUM HARMONIC OSCILLATOR 29
or, equivalently, [ˆn, a
† ] = a
†
. In other words, if |n〉 is an eigenstate of ˆn with
eigenvalue n, then a
† |n〉 is an eigenstate with eigenvalue n + 1.
From this result, we can deduce that the eigenstates for a “tower” | 0 〉,
| 1 〉 = C 1 a
† | 0 〉, | 2 〉 = C 2 (a
† )
2 | 0 〉, etc., where Cn denotes the normalization. If
〈n|n〉 = 1 we have
〈n|aa
† |n〉 = 〈n|(ˆn + 1)|n〉 = (n + 1).
Therefore, with |n + 1〉 =
1 √
n+
a
† |n〉 the state |n + 1〉 is also normalized,
〈n + 1|n + 1〉 = 1. By induction, we can deduce the general normalization,
|n〉 =
n!
(a
† )
n | 0 〉 ,
with 〈n|n
′ 〉 = δ nn
H|n〉 = ℏω(n + 1/2)|n〉 and
a
† |n〉 =
n + 1|n + 1〉, a|n〉 =
n|n − 1 〉.
The operators a and a
† represent ladder operators and have the effect of
lowering or raising the energy of the state.
In fact, the operator representation achieves something quite remarkable
and, as we will see, unexpectedly profound. The quantum harmonic oscillator
describes the motion of a single particle in a one-dimensional potential well.
It’s eigenvalues turn out to be equally spaced – a ladder of eigenvalues, sepa-
rated by a constant energy ℏω. If we are energetic, we can of course translate
our results into a coordinate representation ψn(x) = 〈x|n〉.
7 However, the
operator representation affords a second interpretation, one that lends itself
to further generalization in quantum field theory. We can instead interpret
the quantum harmonic oscillator as a simple system involving many fictitious
particles, each of energy ℏω. In this representation, known as the Fock space,
the vacuum state | 0 〉 is one involving no particles, | 1 〉 involves a single particle,
| 2 〉 has two and so on. These fictitious particles are created and annihilated
by the action of the raising and lowering operators, a
† and a with canoni-
cal commutation relations, [a, a
† ] = 1. Later in the course, we will find that
these commutation relations are the hallmark of bosonic quantum particles
and this representation, known as the second quantization underpins the
quantum field theory of the electromagnetic field.
' Info. There is evidently a huge difference between a stationary (Fock) state
of the harmonic oscillator and its classical counterpart. For the classical system, the
equations of motion are described by Hamilton’s equations of motion,
X = ∂P H =
P
m
P = −∂X H = −∂xU = −mω
2 X ,
where we have used capital letters to distinguish them from the arguments used to de-
scribe the quantum system. In the phase space, {X(t), P (t)}, these equations describe
a clockwise rotation along an elliptic trajectory specified by the initial conditions
{X(0), P (0)}. (Normalization of momentum by mω makes the trajectory circular.)
7 Expressed in real space, the harmonic oscillator wavefunctions are in fact described by
the Hermite polynomials,
ψn(x) = 〈x|n〉 =
r
1
2
n n!
Hn
„r
mω
ℏ
x
«
exp
»
−
mωx
2
2 ℏ
,
where Hn(x) = (−1)
n e
x
2 d
n
dx
n e
−x
2
.
3.5. POSTULATES OF QUANTUM THEORY 31
Thus the Glauber state is an eigenstate of the annihilation operator, corresponding to
the eigenvalue α, i.e. to the (normalized) complex amplitude of the classical process
approximated by the state. This fact makes the calculations of the Glauber state
properties much simpler.
Presented as a superposition of Fock states, the Glauber state takes the form
(exercise – try making use of the BCH identity (3.3).)
|α〉 =
∞ ∑
n=
αn|n〉, αn = e
−|α|
2 / 2
α
n
(n!)
1 / 2
This means that the probability of finding the system in level n is given by the
Poisson distribution, Pn = |αn|
2 = 〈n〉
n e
−〈n〉 /n! where 〈n〉 = |α|
2
. More importantly,
δn = 〈n〉
1 / 2 / 〈n〉 when 〈n〉 0 1 – the Poisson distribution approaches the Gaussian
distribution when 〈n〉 is large.
The time-evolution of Glauber states may be described most easily in the Schr¨odinger
representation when the time-dependence is transferred to the wavefunction. In this
case, α(t) ≡
1 √
2 x 0
(X(t) + i
P (t)
mω
), where {X(t), P (t)} is the solution to the classical
equations of motion, ˙α(t) = −iωα(t). From the solution, α(t) = α(0)e
−iωt , one may
show that the average position and momentum evolve classically while their fluctua-
tions remain stationary,
∆x =
x 0
√
2 mω
1 / 2
, ∆p =
mωx 0
√
ℏmω
2 m
1 / 2
In the quantum theory of measurements these expressions are known as the “standard
quantum limit”. Notice that their product ∆x ∆p = ℏ/2 corresponds to the lower
bound of the Heisenberg’s uncertainty relation.
' Exercise. Show that, in position space, the Glauber state takes the form
〈x|α〉 = ψ α
(x) = C exp
[
mω
(x − X)
2
P x
]
This completes our abridged survey of operator methods in quantum me-
chanics. With this background, we are now in a position to summarize the
basic postulates of quantum mechanics.
3.5 Postulates of quantum theory
Since there remains no “first principles” derivation of the quantum mechanical
equations of motion, the theory is underpinned by a set of “postulates” whose
validity rest on experimental verification. Needless to say, quantum mechanics
remains perhaps the most successful theory in physics.
' Postulate 1. The state of a quantum mechanical system is completely
specified by a function Ψ(r, t) that depends upon the coordinates of the
particle(s) and on time. This function, called the wavefunction or state
function, has the important property that |Ψ(r, t)|
2 dr represents the
probability that the particle lies in the volume element dr ≡ d
d r located
at position r at time t.
The wavefunction must satisfy certain mathematical conditions because
of this probabilistic interpretation. For the case of a single particle, the
net probability of finding it at some point in space must be unity leading
to the normalization condition,
∞
−∞
|Ψ(r, t)|
2 dr = 1. It is customary to
also normalize many-particle wavefunctions to unity. The wavefunction
must also be single-valued, continuous, and finite.
3.5. POSTULATES OF QUANTUM THEORY 32
' Postulate 2. To every observable in classical mechanics there corre-
sponds a linear, Hermitian operator in quantum mechanics.
If we require that the expectation value of an operator
A is real, then
it follows that
A must be a Hermitian operator. If the result of a mea-
surement of an operator
A is the number a, then a must be one of the
eigenvalues,
AΨ = aΨ, where Ψ is the corresponding eigenfunction. This
postulate captures a central point of quantum mechanics – the values
of dynamical variables can be quantized (although it is still possible to
have a continuum of eigenvalues in the case of unbound states).
' Postulate 3. If a system is in a state described by a normalized wave-
function Ψ, then the average value of the observable corresponding to
A
is given by
〈A〉 =
∞
−∞
∗ ˆ AΨdr.
If the system is in an eigenstate of
A with eigenvalue a, then any mea-
surement of the quantity A will yield a. Although measurements must
always yield an eigenvalue, the state does not have to be an eigenstate
of
A initially. An arbitrary state can be expanded in the complete set
of eigenvectors of
A (
AΨi = aiΨi) as Ψ =
n
i
ciΨi, where n may go to
infinity. In this case, the probability of obtaining the result ai from the
measurement of
A is given by P (ai) = |〈Ψi|Ψ〉|
2 = |ci|
2
. The expecta-
tion value of
A for the state Ψ is the sum over all possible values of the
measurement and given by
〈A〉 =
i
a i
i
2
i
a i
|c i
2 .
Finally, a measurement of Ψ which leads to the eigenvalue ai, causes the
wavefunction to “collapses” into the corresponding eigenstate Ψi. (In
the case that ai is degenerate, then Ψ becomes the projection of Ψ onto
the degenerate subspace). Thus, measurement affects the state of the
system.
' Postulate 4. The wavefunction or state function of a system evolves in
time according to the time-dependent Schr¨odinger equation
iℏ
∂t
HΨ(r, t) ,
where
H is the Hamiltonian of the system. If Ψ is an eigenstate of
H,
it follows that Ψ(r, t) = Ψ(r, 0)e
−iEt/ℏ .
