Quantum Mechanics for Mathematicians

and the dynamical equations as

d

dt

Ψ(x,t) ={Ψ(x,t),h}

which can be evaluated to give

∂

∂t

Ψ(x,t) =−

i

2 m

∂^2

∂x^2

Ψ(x,t)

Note that there are other possible forms of the Hamiltonian function that
give the same dynamics, related to the one we chose by integration by parts, in
particular

Ψ(x)

d^2

dx^2

Ψ(x) =

d

dx

(

Ψ(x)

d

dx

Ψ(x)

)

−|

d

dx

Ψ(x)|^2

or

Ψ(x)

d^2

dx^2

Ψ(x) =

d

dx

(

Ψ(x)

d

dx

Ψ(x)−

(

d

dx

Ψ(x)

)

Ψ(x)

)

+

(

d^2

dx^2

Ψ(x)

)

Ψ(x)

Neglecting integrals of derivatives (assuming boundary terms go to zero at in-
finity), one could have used

h=

1

2 m

∫+∞

−∞

|

d

dx

Ψ(x)|^2 dx or h=−

1

2 m

∫+∞

−∞

(

d^2

dx^2

Ψ(x)

)

Ψ(x)dx

37.3 The propagator in non-relativistic quantum field theory

In quantum field theory the Heisenberg picture operators that provide ob-
servables will be products of the field operators, and the time-dependence of
these for the free-particle theory was determined in section 37.2. For the time-
independent state, the natural choice is the vacuum state| 0 〉, although other
possibilities such as coherent states may also be useful. States with a finite
number of particles will be given by applying field operators to the vacuum, so
such states just corresponds to a different product of field operators.
We will not enter here into details, but a standard topic in quantum field
theory textbooks is “Wick’s theorem”, which says that the calculation of expec-
tation values of products of field operators in the state| 0 〉can be reduced to
the problem of calculating the following special case:

Definition(Propagator for non-relativistic quantum field theory).The propa-
gator for a non-relativistic quantum field theory is the amplitude, fort 2 > t 1

U(x 2 ,t 2 ,x 1 ,t 1 ) =〈 0 |Ψ(̂x 2 ,t 2 )Ψ̂†(x 1 ,t 1 )| 0 〉

The physical interpretation of these functions is that they describe the am-
plitude for a process in which a one-particle state localized atx 1 is created at