operators as
P̂=
∫∞
−∞
Ψ̂†(x)(−id
dx
Ψ(̂x))dx
=
∫∞
−∞
∫∞
−∞
∫∞
−∞
1
√
2 π
e−ip
′x
a†(p′)(−i)(ip)
1
√
2 π
eipxa(p)dpdp′dx
=
∫∞
−∞
∫∞
−∞
δ(p−p′)pa†(p′)a(p)dpdp′
=
∫∞
−∞
pa†(p)a(p)dp
For more discussion of this operator and its relation to spatial translations,
see section 38.3.1.
- The HamiltonianĤ. As an operator quadratic in the field operators, this
can be chosen to be
Ĥ=
∫∞
−∞
Ψ̂†(x)
(
−
1
2 m
d^2
dx^2
)
Ψ(̂x)dx=
∫∞
−∞
p^2
2 m
a†(p)a(p)dp
The dynamics of a quantum field theory is usually described in the Heisen-
berg picture, with the evolution of the field operators given by Fourier trans-
formed versions of the discussion in terms ofa(p),a†(p) of section 36.5. The
quantum fields satisfy the general dynamical equation
d
dt
Ψ̂†(x,t) =−i[Ψ̂†(x,t),Ĥ]
which in this case is
∂
∂t
Ψ̂†(x,t) =− i
2 m
∂^2
∂x^2
Ψ̂†(x,t)
Note that the field operatorΨ̂†(x,t) satisfies the (conjugate) Schr ̈odinger
equation, which now appears as a differential equation for distributional oper-
ators rather than for wavefunctions. Such a differential equation can be solved
just as for wavefunctions, by Fourier transforming and turning differentiation
into multiplication, and we find
Ψ̂†(x,t) =√^1
2 π
∫∞
−∞
e−ipxei
p^2
2 mta†(p)dp
Just as in the case of 36.5, this formal calculation involving the quantum
field operators has an analog in terms of the Ψ(x) and a quadratic function on
the phase space. One can write
h=
∫+∞
−∞
Ψ(x)
− 1
2 m
∂^2
∂x^2
Ψ(x)dx