Re: Problems with functions of a Complex Variable

*To*: mathgroup at smc.vnet.net*Subject*: [mg102820] Re: Problems with functions of a Complex Variable*From*: pfalloon <pfalloon at gmail.com>*Date*: Fri, 28 Aug 2009 00:45:43 -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 Hi there, When you use ComplexExpand, Mathematica does indeed expand the expression assuming that all variables (in this case a and b) are real. However, this assumption only applies within that function call, so in the subsequent call to Conjugate a and b are once again assumed to be completely arbitrary complex quantities. There are a few ways to do the conjugation in the way you are looking for. One which I have occasionally found useful is to change the sign of explicit complex numbers using something like expr /. Complex[x_,y_] :> Complex[x,y] Alternatively, using Simplify with appropriate Assumptions will also work: Conjugate[a+I*b] // Simplify[#, Element[{a,b}, Reals]]& It can be something of a minefield trying to evaluate integrals etc with symbolic quantities that are possibly complex. It's generally simplest if you can express things in terms of real-valued symbols, apply Simplify etc to get the integrand in explicit form (i.e. free of Conjugate), and then carry out integrals. Here is a simple example related to your case (i've suppressed output): ClearAll["Global`"] (* basis function *) psi[n_,x_,t_] = Sqrt[2]*Exp[-I*n^2*t]*Sin[n*Pi*x]; (* some linear combination which will have non-trivial time-dependence *) f[x_,t_] = psi[1,x,t] + I*psi[2,x,t]; (* visualize the functions over the box 0<x<1 *) Plot[Evaluate[Table[psi[n,x,0],{n,3}]],{x,0,1}] (* carry out integrals of individual basis functions to show orthonormality *) Table[Integrate[psi[m,x,0]*Conjugate@psi[n,x,0], {x,0,1}], {m,3}, {n, 3}] // MatrixForm (* some linear combination which will have non-trivial time- dependence, with REAL coefficients a[n] *) f[x_,t_] = Sum[a[n]*psi[n,x,t], {n,3}] (* get the probability density using Conjugate and ComplexExpand (assumes a[n], x, t are all real) *) absSq[x_,t_] = f[x,t]*Conjugate@f[x,t] // ComplexExpand (* now do the integral *) Integrate[absSq[x,t], {x,0,1}] Hope this helps and good luck! Cheers, Peter.