Quantum Mechanics for Mathematicians

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