Complex Conjugate of a Function

[James Sherman Jr.]

A few comments/questions.

On Thu, Oct 9, 2008 at 12:36 AM, Thomas Markovich <thomasmarkovich at gmail.com
> wrote:

> Ok,
>
> So heres what I have found that works. The complex conjugate of e^ix
> is the same as taking the complex conjugate of the answer from that
> function.


This seems to me to the very definition of what you mean by "conjugate of a
function".  Formally (mathematically) speaking, conjugate is a function that
maps complex numbers to another complex number, so conjugate of a function
isn't valid unless you define it like you did.


> and I have found that conj(some vector) ends up being defined.

This is because in octave, the conj function operates on an element by
element basis, this isn't a mathematical conjugate definition.  conj(some
vector) = vector of the conjugates of each term.



> I have
> also discovered that I can just do, assuming that k is my vector
> generated from b(x):
>
>  conj(k) .* diff(k) - k .* diff(conj(k))
>
> And this should give me the probability current. Anyways, what I am
> struggling with now is, how do I define b(x) with x being [-2*pi: .
> 01 : 2*pi] (-2*pi < x < 2*pi) to get into a vector k. What I have now
> is k = [b(x)]; but have tried everything I can think of.
>
All the function diff is doing is taking the difference between successive
elements in a vector, or in essence dy.  However, to estimate the derivative
at a particular point in the function(empirically), you also need to
calculate dx, so you would also want to (I assume) do something more like
conj(k).*diff(k)./diff(x) - k.*diff(conj(k))./diff(x)

This won't work either, because of how the function diff works.  If we let z
= diff(w) where z and w are vectors and w is of length 3, then we get
z(1) = w(2)-w(1)
z(2) = w(3)-w(2)

But there is no way to calculate z(3) since that would require knowledge of
w(4) which doesn't exist.  So, z will have length of one less than w.  So,
back to your example, we have no way (unless you make some other
assumptions) of how to estimate the derivative at the last point, so I think
what you are aiming for is, by letting m be k without its last element:
m = k(1:end-1);
conj(m).*diff(k)./diff(x) - m.*diff(conj(k))./diff(x)

At least this is what I think you're getting at (I don't know anything about
probability current of wave functions).  If this is true, I think it would
be far better to symbolically calculate the derivative function of b, so
that you could directly calculate the derivative of b and not use the
approximation of the derivative by way of using diff.  It is just a sum of
exponentials, so the derivative should be straightforward, though maybe
tedious, to calculate.

Hope this helps.

James Sherman



>
>
>
> On Oct 8, 2008, at 3:24 PM, Thomas Markovich wrote:
>
> > Hi,
> >
> > I am using octave and I am trying to find the probability current of
> > of a set of wave functions. To do this though, I get
> >
> > (¥*) d/dx(¥) - ¥(d/dx(¥*))
> >
> > I have tried to use both functions, conj and ' to find the conjugate
> > but octave gives me the following error.
> >
> > octave-3.0.2:59> conj(b(x))
> > ans =
> >
> > Columns 1 through 6:
> >
> >   0.792789 - 0.000000i   0.521003 - 0.044985i   0.361599 -
> > 0.061587i   0.263736 - 0.026430i   0.163356 + 0.047835i   0.013050 +
> > 0.086010i
> >
> > Columns 7 through 12:
> >
> >  -0.173588 + 0.028121i  -0.377793 - 0.064103i  -0.673954 -
> > 0.079818i  -1.212822 - 0.014479i  -2.150759 + 0.048640i  -3.688296 +
> > 0.059676i
> >
> > Column 13:
> >
> >  -6.190114 + 0.027294i
> >
> > error: conj: argument undefined
> >
> > Any help would would be much appreciated.
> >
> > Thomas Markovich
> > Undergraduate Research Assistant
> > University of Houston
>
>
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