Consider the time-dependent Schrödinger equation,
describing a wavefunction :
By definition, this equation’s
satisfies the following:
From this, we define the inverse
as follows, so that :
Note that is an operator, while is a function.
For the sake of consistency, we thus define
as a multiplication by
and integration over and :
For an arbitrary function ,
so that .
Moving on, the Schrödinger equation can be rewritten like so,
Let us assume that is simple,
such that and can be found without issues
by solving the defining equation above.
Suppose we now add a more complicated and
possibly time-dependent term ,
in which case the corresponding fundamental solution
This equation is typically too complicated to solve,
so we would like an easier way to calculate this new .
The perturbed wavefunction
satisfies the Schrödinger equation:
We know that ,
which we put on the right,
and then we apply in front:
This equation is recursive,
so we iteratively insert it into itself.
Note that the resulting equations are the same as those from
time-dependent perturbation theory:
The parenthesized expression clearly has the same recursive pattern,
so we denote it by and write the so-called Dyson equation:
Such an iterative scheme is excellent for approximating .
Once a satisfactory accuracy is obtained,
the perturbed wavefunction can be calculated from:
This relation is equivalent to the Schrödinger equation.
So now we have the operator ,
but what about the fundamental solution function ?
Let us take its definition, multiply it by an arbitrary ,
and integrate over ’s second argument pair:
Where we have hidden the arguments for brevity.
We now apply to this equation
(which contains an integral over independent of ):
Here, the shape of Dyson’s equation is clearly recognizable,
so we conclude that, as expected, the operator
is defined as multiplication by the function followed by integration:
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,