133
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
so that the state |n> defined by (5.81) is an eigenstate of the number operator
n ˆ = ˆ
a
† a ˆ, with integer eigenvalue n:
n ˆ|n> = n|n>.
(5.83)
It is straightforward to generalize all the foregoing to a system whose
Lagrangian is a sum of N independent oscillators, as in (5.48):
N
∑
2
2
L ˆ =
(
1 mq ˆ ˙ −
1 mω
2 q ˆ ).
(5.84)
2
r
2
r r
r=1
The required generalization of the basic commutation relations (5.57) is
[ˆ q r , p ˆ s ] = iδ rs
(5.85)
[ˆ q r , q ˆ s ] = [ˆ p r , p ˆ s ] = 0
since the different oscillators labelled by the index r or s are all independent.
The Hamiltonian is (cf (5.56))
N
eigenvalues of each number operator n ˆ = ˆ
a a ˆ r are n by the previous results,
ˆ
H =
∑
2
1
2
r
r r
2
[(1/2m)ˆ p + mω
2 q ˆ ]
(5.86)
r=1
N
∑
=
(ˆ a
† a ˆ r +
1 )ω r
r
2
(5.87)
r=1
with ˆ
a r and ˆ
a
†
r defined via the analogues of (5.69) and (5.70). Since the
†
r
r
r ,
the eigenvalues of H ˆ indeed have the form (5.29),
N
∑
1
E =
(n r + )ω r .
(5.88)
2
r=1
The corresponding eigenstates are products |n 1 >|n 2 > · · · |n N > of N individual oscillator eigenstates, where |n r > contains n r quanta of excitation, of frequency ω r ; the product state is usually abbreviated to |n 1 , n 2 , . . . , n N >. In the
ground state of the system, each individual oscillator is unexcited: this state
is |0, 0, . . . , 0>, which is abbreviated to |0>, where it is understood that
a ˆ r |0> = 0
for all r.
(5.89)
The operators ˆ
a
† create oscillator quanta; the operators ˆ
a r destroy oscillator
r
quanta.
5.2.4 Lagrange–Hamilton classical field mechanics
We now consider how to use the Lagrange–Hamilton approach for a field,
starting again with the classical case and limiting ourselves to one dimension
to start with.
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
so that the state |n> defined by (5.81) is an eigenstate of the number operator
n ˆ = ˆ
a
† a ˆ, with integer eigenvalue n:
n ˆ|n> = n|n>.
(5.83)
It is straightforward to generalize all the foregoing to a system whose
Lagrangian is a sum of N independent oscillators, as in (5.48):
N
∑
2
2
L ˆ =
(
1 mq ˆ ˙ −
1 mω
2 q ˆ ).
(5.84)
2
r
2
r r
r=1
The required generalization of the basic commutation relations (5.57) is
[ˆ q r , p ˆ s ] = iδ rs
(5.85)
[ˆ q r , q ˆ s ] = [ˆ p r , p ˆ s ] = 0
since the different oscillators labelled by the index r or s are all independent.
The Hamiltonian is (cf (5.56))
N
eigenvalues of each number operator n ˆ = ˆ
a a ˆ r are n by the previous results,
ˆ
H =
∑
2
1
2
r
r r
2
[(1/2m)ˆ p + mω
2 q ˆ ]
(5.86)
r=1
N
∑
=
(ˆ a
† a ˆ r +
1 )ω r
r
2
(5.87)
r=1
with ˆ
a r and ˆ
a
†
r defined via the analogues of (5.69) and (5.70). Since the
†
r
r
r ,
the eigenvalues of H ˆ indeed have the form (5.29),
N
∑
1
E =
(n r + )ω r .
(5.88)
2
r=1
The corresponding eigenstates are products |n 1 >|n 2 > · · · |n N > of N individual oscillator eigenstates, where |n r > contains n r quanta of excitation, of frequency ω r ; the product state is usually abbreviated to |n 1 , n 2 , . . . , n N >. In the
ground state of the system, each individual oscillator is unexcited: this state
is |0, 0, . . . , 0>, which is abbreviated to |0>, where it is understood that
a ˆ r |0> = 0
for all r.
(5.89)
The operators ˆ
a
† create oscillator quanta; the operators ˆ
a r destroy oscillator
r
quanta.
5.2.4 Lagrange–Hamilton classical field mechanics
We now consider how to use the Lagrange–Hamilton approach for a field,
starting again with the classical case and limiting ourselves to one dimension
to start with.