Derivation of Nth-Order Perturbation

First of all, our goal. At the end of this page, I write:

|k} = (P!W!)^k |0}

Ek = {0| W! (P!W!)^(k-1) |0}

These are the forms we use in Feynman Perturbation : Virtual Particle Physics . These are pretty forms, and valid for any k. Let's derive them:

H = H0 + eW

H0 is the base Hamiltonian, W is a perturbation
e is the smallness parameter

|n> = |n0> + e|n1> + e^2|n2> ...
En = En0 + e En1 + e^2 En2 ...

H |n> = En |n> for any e	(eq. *1)

for equation (*1) to be true for all e, the terms in the
same exponent of e must be equal.

(H0 + eW) ( |n0> + e|n1> ... ) = (En0 + e En1 ... ) ( |n0> + e|n1> ... )

e^0 { H |n0> = En0 |n0> }
e^1 { H0 |n1> + W|n0> = En0 |n1> + En1 |n0> }
etc..

Gathering terms, it is easy to see this is:

W|n(k-1)> + H0|nk> = Sum[i=0 to k] En(i) |n(k-i)>

Since we are working with En in all cases, we will drop
the n now, so |n(k-1)> will be written |k-1> ; the letters
n and m will be used in the Ket to indicate an energy level.
Also, H will be used for H0, and (H+W) for the old H.

W|k-1> + H|k> = Sum[i,0,k] Ei |k-i>		(eq. *2)

Let's get E0 and E1 real quick.

k = 0 ->	H|0> = E0|0>
			E0 = <0|H|0>
k = 1 ->	W|0> + H|1> = E0|1> + E1|0>
			apply <0| and use hermiticity of H :
			E1 = <0|W|0>

Take (*2); We can pull the i = 0 and 1 terms out of the Sum (if k>1) and get:

(H - E0)|k> = (E1 - W)|k-1> + Sum[i,2,k] Ei |k-i>		(*3)

Now apply <0| to (*3) and use <0|(H - E0) = 0 (since H is hermitian):

0 = <0|(E1 - W)|k-1> + Ek<0|0>

Ek = <0|W - E1|k-1> = <0| W - <0|W|0> |k-1>

now <0|( W - <0|W|0> ) = <0|W! where ! is short for !n0 which is short for
"not state n0" ;

! = 1 - |0><0|
W! = W - W|0><0|

<0||j> = Wij - Wi0 d0j = Wi0 - Wi0 = 0 if j = 0
				  = Wij otherwise
|j> = Wij - W00 dij = Wij - W00 if i = j
				  = Wij otherwise

You see that these are not exactly equal :

( W - <0|W|0> ) = W!			not true!

<0|( W - <0|W|0> ) = <0|W!		true
( W - <0|W|0> )|0> = !W|0>		true

So this is all we need to get:

Ek = <0|W!|k-1>		(eq *4)

This is the final form of Ek ; unfortunately, it is a recursion,
dependent on |k-1>.  Thus, we need to find the states |k>.

Define P = propagator = Inverse[ En0 - H ]
(this is really Pn the propagator in the background of
the nth state)

Apply P to (*3) :

P{ (H - E0)|k> = (E1 - W)|k-1> + Sum[i,2,k] Ei |k-i> }

- |k> = P{ ( -W! )|k-1> + Sum[i,2,k] Ei|k-i> }

|k> = PW!|k-1> - Sum[i,2,k] Ei P |k-i>		(eq *5)

(this is almost a nice final form for a recursion on |k> , but we want to get
rid of the "sum" part).

P(En0 - H) = [ En0 - H ]^-1 (En0 - H) = "1" = Sum[m] |m> = !W|0> , and drop the Sum :

|1> = P!W|0>

for k = 2 :

E2 = <0|W!|1> = <0|W!P!W|0>

|2> = PW!|1> - E2P|0>

	= PW!P!W|0> - <0|W!P!W|0>P|0>

The rearrangement of this type of equation is what marks the rest of
the development.

define z(x) = x - <0|x|0> , (we used z(W) before, and found
that z(W)|0> = !W|0> ; this is a general result.

choose x = W!P!W , so:

|2> = Px|0> - <0|x|0> P|0> = P z(x) |0> = P!x|0> = P!W!P!W|0>

We can play this trick over and over.  It is more difficult to find
a general form for |k> and Ek (we can make an educated guess at
this point, and we have the recursion relation, but even plugging
in the guess to the recursion we gain nothing, because the
recursion contains the form |k> = Sum[i=0 to k-1] |i> which creates
more z(x) style forms).

Let's show something by pulling out the top (k) term in the sum:
start with:

|k> = PW!|k-1> - Sum[i,2,k] Ei P |k-i>		(eq *5)

|k> = PW!|k-1> - Ek P|0> - Sum[i,2,k-1] Ei P |k-i>

	= PW!|k-1> - P<0|W!|k-1>|0> - Sum[i,2,k-1] Ei P |k-i>

	= P[1-|0><0|]W!|k-1> - Sum[i,2,k-1] Ei P |k-i>

|k> = P!W!|k-1> - Sum[i,2,k-1] Ei P |k-i>

So you see that pulling off the top term just puts a '!' between the W and the P
(where we hypothesize there will always be one (otherwise we get a 1/0 term in
the sum of 1/(En-Em) in P)).

Now, pull off another term:

|k> = P{!W!P! - P!W|0><0|}W!|k-2> - Sum[i,2,k-2] Ei P |k-i>

	= P!W!P!W!|k-2> + PP!W(!-1)W!|k-2>  - Sum[i,2,k-2] Ei P |k-i>

Each time you pull a term off of the Sum[] the |k-1| goes to |k-2>
then to |k-3> ... down to |0> ; also, the term in front picks up
one more power of P and W.  The equation also tries its best to make fancy
forms of ! and |0><0| and z(x) .

BTW |3> = P[!W!P! - P!W|0><0|]W!P!W|0>
		= P!W!P!W!P!W|0> - PP!W|0><0|W!P!W|0>
		
The hypothesis is that these forms will reduce to a pretty
one :

|k> = (P!W!)^k |0>

Ek = <0| W! (P!W!)^(k-1) |0>

These are the forms we use in  Feynman
Perturbation : Virtual Particle Physics .



You can actually get these forms using the Path Integral formulation, in
a somewhat simpler manner, because the ! becomes a principal part contour
integral - i.e. the ! in our formulation is really an artifact of the
theory, which the theory provides for itself so as to avoid infinities;
if the ! were not present, quantum mechanics would be invalid (or at
least, perturbation theory would be).  In the path integral method, this
same effect occurs but instead of doing

Sum[m not = n] 1 / (En - Em)

with the "not =" provided by the (!m) , we do:

Integral[] 1 / (deltaE + i*d)

where "deltaE" = energy of excitation over ground, and
d is allowed to go to zero.  This formulation is somewhat easier
to handle,



Let me note one more thing.  Using the recursion:

|k> = (P!W!)^k |0>

and the original formula for the expansion:

|Y> = |0> + |1> + |2> ...

we simply insert:

|Y> = |0> + P!W!|0> + P!W!P!W!|0> + ...

|Y> = ( |0> + (P!W!) ( |0> + (P!W!) ( |0> + (P!W!)( |0> + ...
	  = |0> + (P!W!)|Y>

this is now an EXACT equation for any state.  This equation is
actually surprisingly useful (with the help of some cleverness) and
we can frequently avoid perturbation approximations, and instead
get exact answers for the state |Y>

more mathematically:

f = Sum[i>=0] a^i g

	= g + Sum[i>0] a^i g 

	= g + a Sum[i>=0] a^i g

f	= g + a f

where f,g, and a are anything (functions,operators,matrices,hippopotimi).


Charles Bloom / cb at my domain
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