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MathGroup Archive 2005

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Re: Double integral of a piecewise-constant function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61447] Re: [mg61395] Double integral of a piecewise-constant function
  • From: Chris Chiasson <chris.chiasson at gmail.com>
  • Date: Wed, 19 Oct 2005 02:16:49 -0400 (EDT)
  • References: <200510180644.CAA11181@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Sometimes Mathematica refuses to do things because it assumes all
symbols could be complex. I didn't look at your integral, but I was
wondering if you made any assumptions on the domain or range of any of
the symbols when you did the integration by hand. Also, others might
find it helpful to see your pencil and paper result.

On 10/18/05, Chris Rodgers <rodgers at physchem.nospamox.aremovec.uk> wrote:
> Dear All,
>
> I am trying to use Mathematica to calculate some quantum mechanical
> expressions involving integrals of commutators. In the simplest case,
> the Hamiltonian of the system is piecewise constant.
>
> My problem is that Mathematica can integrate this expression:
>
> (-(1/2))*Integrate[Piecewise[{{0, t[2] < 0 || t[2] >= 3 || (t[2] < 1 &&
> t[1] < 1) || (t[2] < 2 && t[1] >= 3) || (t[2] >= 1 && t[1] < 0) ||
>          (Inequality[1, LessEqual, t[2], Less, 2] && Inequality[1,
> LessEqual, t[1], Less, 2]) || (t[2] >= 2 && t[1] >= 2)},
>        {Comm[Subscript[H, 1], Subscript[H, 2]], t[2] < 1 && t[1] < 2},
> {Comm[Subscript[H, 1], Subscript[H, 3]], t[2] < 1},
>        {Comm[Subscript[H, 2], Subscript[H, 1]], t[2] < 2 && t[1] < 1},
> {Comm[Subscript[H, 2], Subscript[H, 3]], t[2] < 2},
>        {Comm[Subscript[H, 3], Subscript[H, 1]], t[1] < 1}},
> Comm[Subscript[H, 3], Subscript[H, 2]]] /. t[2] -> 0, {t[1], 0, 3}]
>
> but it can't do the two dimension integral of the same expression:
>
> -Integrate[Integrate[Piecewise[{{0, t[2] < 0 || t[2] >= 3 || (t[2] < 1
> && t[1] < 1) || (t[2] < 2 && t[1] >= 3) || (t[2] >= 1 && t[1] < 0) ||
>          (Inequality[1, LessEqual, t[2], Less, 2] && Inequality[1,
> LessEqual, t[1], Less, 2]) || (t[2] >= 2 && t[1] >= 2)},
>        {Comm[Subscript[H, 1], Subscript[H, 2]], t[2] < 1 && t[1] < 2},
> {Comm[Subscript[H, 1], Subscript[H, 3]], t[2] < 1},
>        {Comm[Subscript[H, 2], Subscript[H, 1]], t[2] < 2 && t[1] < 1},
> {Comm[Subscript[H, 2], Subscript[H, 3]], t[2] < 2},
>        {Comm[Subscript[H, 3], Subscript[H, 1]], t[1] < 1}},
> Comm[Subscript[H, 3], Subscript[H, 2]]], {t[2], 0, t[1]}], {t[1], 0, 3}]/2
>
> To me these look pretty similar, and both are easy to do with pencil and
> paper. How should I set things up so that Mathematica can perform the
> second integral?
>
> Many thanks,
>
> Chris.
>
>


--
Chris Chiasson
http://chrischiasson.com/contact/chris_chiasson


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