131
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
The amount a quantum particle can ‘stray’ from the classical path depends
on the magnitude of the corresponding action relative to ħ, the quantum of
action: the scale of coherence is set by ħ.
In summary, then, the quantum mechanical amplitude to go from q(t 1 ) to
q(t 2 ) is proportional to
( ∫
)
∑
t2
i
exp
L(q(t), q˙(t)) dt .
(5.67)
ħ t1
all paths q(t)
There is an evident generalization to quantum field theory. We shall not,
however, make use of the ‘path integral’ approach to quantum field theory in
this volume. Its use was, in fact, decisive in obtaining the Feynman rules for
non-Abelian gauge theories; and it is the only approach suitable for numerical
studies of quantum field theories (how can operators be simulated numerically?). Nevertheless, for a first introduction to quantum field theory, there
is still much to be said for the traditional approach based on ‘quantizing the
modes’, and this is the path we shall follow in the rest of this volume. Not the
least of its advantages is that it contains the intuitively powerful ‘calculus’ of
creation and annihilation operators, as we now describe. We shall return to
the path integral formalism in chapter 16 of volume 2.
5.2.3 Interlude: the quantum oscillator
As we saw in section 5.1, we need to know the energy spectrum and associated
states of a quantum harmonic oscillator. This is a standard problem, but there
is one particular way of solving it – the ‘operator’ approach due to Dirac (1981,
chapter 6) – that is so crucial to all subsequent development that we include
a discussion here in the body of the text.
For the oscillator Hamiltonian
2
2
ˆ
H =
1 p ˆ +
1 mω
2 q ˆ
(5.68)
2m
2
if ˆ
p and ˆ
q were not operators, we could attempt to factorize the Hamiltonian
in the form ‘(q + ip)(q − ip)’ (apart from the factors of 2m and ω). In the
quantum case, in which ˆ
p and ˆ
q do not commute, it still turns out to be very
helpful to introduce such combinations. If we define the operator
(
)
1 √
i
a ˆ = √
mωq ˆ + √
p ˆ
(5.69)
2
mω
and its Hermitian conjugate
(
)
†
1 √
i
a ˆ = √
mωq ˆ − √
p ˆ
(5.70)
2
mω
the Hamiltonian may be written as (see problem 5.4)
ˆ
1 † ˆ
† ˆ
1
H = (ˆ a a + ˆ
aa ˆ
† )ω = (ˆ a a + )ω.
(5.71)
2
2
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
The amount a quantum particle can ‘stray’ from the classical path depends
on the magnitude of the corresponding action relative to ħ, the quantum of
action: the scale of coherence is set by ħ.
In summary, then, the quantum mechanical amplitude to go from q(t 1 ) to
q(t 2 ) is proportional to
( ∫
)
∑
t2
i
exp
L(q(t), q˙(t)) dt .
(5.67)
ħ t1
all paths q(t)
There is an evident generalization to quantum field theory. We shall not,
however, make use of the ‘path integral’ approach to quantum field theory in
this volume. Its use was, in fact, decisive in obtaining the Feynman rules for
non-Abelian gauge theories; and it is the only approach suitable for numerical
studies of quantum field theories (how can operators be simulated numerically?). Nevertheless, for a first introduction to quantum field theory, there
is still much to be said for the traditional approach based on ‘quantizing the
modes’, and this is the path we shall follow in the rest of this volume. Not the
least of its advantages is that it contains the intuitively powerful ‘calculus’ of
creation and annihilation operators, as we now describe. We shall return to
the path integral formalism in chapter 16 of volume 2.
5.2.3 Interlude: the quantum oscillator
As we saw in section 5.1, we need to know the energy spectrum and associated
states of a quantum harmonic oscillator. This is a standard problem, but there
is one particular way of solving it – the ‘operator’ approach due to Dirac (1981,
chapter 6) – that is so crucial to all subsequent development that we include
a discussion here in the body of the text.
For the oscillator Hamiltonian
2
2
ˆ
H =
1 p ˆ +
1 mω
2 q ˆ
(5.68)
2m
2
if ˆ
p and ˆ
q were not operators, we could attempt to factorize the Hamiltonian
in the form ‘(q + ip)(q − ip)’ (apart from the factors of 2m and ω). In the
quantum case, in which ˆ
p and ˆ
q do not commute, it still turns out to be very
helpful to introduce such combinations. If we define the operator
(
)
1 √
i
a ˆ = √
mωq ˆ + √
p ˆ
(5.69)
2
mω
and its Hermitian conjugate
(
)
†
1 √
i
a ˆ = √
mωq ˆ − √
p ˆ
(5.70)
2
mω
the Hamiltonian may be written as (see problem 5.4)
ˆ
1 † ˆ
† ˆ
1
H = (ˆ a a + ˆ
aa ˆ
† )ω = (ˆ a a + )ω.
(5.71)
2
2