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Re: Problems with functions of a Complex Variable

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102821] Re: Problems with functions of a Complex Variable
  • From: pfalloon <pfalloon at gmail.com>
  • Date: Fri, 28 Aug 2009 00:45:54 -0400 (EDT)
  • References: <h75nn9$pgc$1@smc.vnet.net>

On Aug 27, 8:38 pm, Sergio Miguel Terrazas Porras <sterr... at uacj.mx>
wrote:
> Dear Group:
>
> I am trying to illustrate with Mathematica 7.0.1 some of the concepts of Quantum Mechanics.
>
> In particular, for the particle in an Infinite Square Well, I define the time dependent Wave Function (Sum of Sines times Complex Exponentials ) Ok.
>
> Using only the first 3 terms of the series, I run into the following:
>
> If I use ComplexExpand[ ] I get things of the form a + I b, which, I would assume means to Mathematica that a and b are Real.
>
> I then ask for the Conjugate of the preceding expresi=F3n, and I get things like Conjugate[a] - I*Conjugate[b].  (??)
>
> If I plot the Probability Density Function, things are Ok. (Its Real)
>
> If I try to calculate the (time dependent) expected value of x, Assuming[Element[{x,t},Reals],Integrate[x*psi*Congugate[psi],{x,0,a}]
>
> Matem=E1tica hangs forever even though its only nine integrals.
>
> Any help will be very much appreciated.
>
> Sergio Terrazas

Apologies, in the example from my first response I just discovered a
slightly smarter way to do the simplification which doesn't require a
[n] to be real:

In[79]:= (* get the probability density using FunctionExpand -- allows
a[n] to be COMPLEX *)
absSq[x_,t_] = f[x,t]*Conjugate@f[x,t] // FunctionExpand[#, Element
[{x,t},Reals]] &

In[80]:=  (* now the integral returns sum of a[n]*Conjugate[a[n]] *)
Integrate[absSq[x,t],{x,0,1}]

Out[80]= a[1] Conjugate[a[1]] + a[2] Conjugate[a[2]] +
 a[3] Conjugate[a[3]]

The function FunctionExpand is very useful in this kind of situation...


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