Is this Gaussian Integral calculation correct?

[Jay R. Yablon]

In the file linked below, I have written out a particular calculation of
a Gaussian integral, and would like to know if (6) in this file is is
correct. (If link does not work, right click to download and then
open.)

http://jayryablon.files.wordpress.com/2008/05/is-this-calculation-of-the-gaussian-integral-correct.pdf

This is not rocket science nor is it particularly earthshaking. But, I
am having some conceptual trouble thinking about this integral (6) in
which the plane wave coefficient B=0, and would appreciate if someone
can help me straighten out my thinking about this.

Best regards,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

[mathman]

When you did the change of variables x -> x-B/A, you did not change the argument of V(x). I believe you should.

[neuropulp@yahoo.com.au]

Hey again J-boy!

> and would like to know if (6) in this file is is correct.
> http://jayryablon.files.wordpress.com/2008/05/is-this-calculation-of-...

Your eq(5) looks wrong. You can't just add the exponents
like that unless they commute. (d/dB doesn't commute with B^2).

> This is not rocket science nor is it particularly
> earthshaking. But, I am having some conceptual trouble
> thinking about this integral (6) in which the plane wave
> coefficient B=0, and would appreciate if someone can help me
> straighten out my thinking about this.

You're guaranteed to get more help with this sort of boring
basic stuff over on www.physicsforums.com (try the quantum
physics forum). Also, it's latex-capable, so you don't need
to rely on auxiliary files like you're doing here on spr.

- LOL from Neuropulp!

P.S: I tried to reply to your previous posting, but it
attempt failed. Don't know why.

P.P.S: Try to think of a more original salutation than
"Pulp-boy". You can do better than that! :-) :-)
(Wrong gender, btw.)

[Ian Parker]

On 15 May, 22:51, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> In the file linked below, I have written out a particular calculation of
> a Gaussian integral, and would like to know if (6) in this file is is
> correct. (If link does not work, right click to download and then
> open.)
>
> http://jayryablon.files.wordpress.com/2008/05/is-this-calculation-of-...
>
> This is not rocket science nor is it particularly earthshaking. But, I
> am having some conceptual trouble thinking about this integral (6) in
> which the plane wave coefficient B=0, and would appreciate if someone
> can help me straighten out my thinking about this.
>
It would seem OK up to a point but it is not the way I would tackle
it.

Now e^x = 1+x+x^2/2! + x^3/3! + etc

Hence e^f(x) = 1+f(x)+f(x)^2/2! + f(x)^3/3! + etc

We know that Int (x^n)dx = x^(n+1)/(n+1) Hence if we have a power
series integration is trivial.

Now f(x) = -x^2 + V(x) we have a power series for f(x) and getting a
term by term evaluation is a matter of a term by term multiplication.
This gives your your equations for what you have labelled as C(n).

You can also constuct a series using Taylor's theorem, that is to say
you differentiate e^(f(x)) repeatedly.

You get f'(x)e^(f(x)) differentiating this again (f''(x) +
(f'(x)^2)e^(f(x))

Can you see one thing? The process of Taylor differentiation is
isomorphic with binomial multiplication (we would be surprised if it
wasn't).

You want clarification. I think the isomorphism of binomial
multiplication and Taylor's theorem does clarify things consiserably.

The physical meaning. There isn't a way to do things that isn't a
little bit messy.

- Ian Parker