info prev up next book cdrom email home

Phasor

The representation, beloved of engineers and physicists, of a Complex Number in terms of a Complex exponential

\begin{displaymath}
x+iy=\vert z\vert e^{i\phi},
\end{displaymath} (1)

where i (called j by engineers) is the Imaginary Number and the Modulus and Argument (also called Phase) are
$\displaystyle \vert z\vert$ $\textstyle =$ $\displaystyle \sqrt{x^2+y^2}$ (2)
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \tan^{-1}\left({y\over x}\right).$ (3)

Here, $\phi$ is the counterclockwise Angle from the Positive Real axis. In the degenerate case when $x=0$,
\begin{displaymath}
\phi=\cases{
-{\textstyle{1\over 2}}\pi & if $y<0$\cr
{\rm...
...ined} & if $y=0$\cr
{\textstyle{1\over 2}}\pi & if $y>0$.\cr}
\end{displaymath} (4)


It is trivially true that

\begin{displaymath}
\sum_i \Re[\psi_i] = \Re\left[{\sum_i \psi_i}\right].
\end{displaymath} (5)

Now consider a Scalar Function $\psi\equiv \psi_0e^{i\phi}$. Then
$\displaystyle I$ $\textstyle \equiv$ $\displaystyle [\Re(\psi)]^2 = [{\textstyle{1\over 2}}(\psi+\psi^*)]^2 = {\textstyle{1\over 4}}(\psi+\psi^*)^2$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(\psi^2+2\psi\psi^*+{\psi^*}^2).$ (6)

Look at the time averages of each term,
\begin{displaymath}
\left\langle{\psi^2}\right\rangle{}=\left\langle{{\psi_0}^2e...
...ngle{} = {\psi_0}^2\left\langle{e^{2i\phi}}\right\rangle{} = 0
\end{displaymath} (7)


\begin{displaymath}
\left\langle{\psi\psi^*}\right\rangle{}=\left\langle{\psi_0e...
...si_0e^{-i\phi}}\right\rangle{} = {\psi_0}^2 = \vert\psi\vert^2
\end{displaymath} (8)


\begin{displaymath}
\left\langle{{\psi^*}^2}\right\rangle{}=\left\langle{{\psi_0...
...le{} = {\psi_0}^2\left\langle{e^{-2i\phi}}\right\rangle{} = 0.
\end{displaymath} (9)

Therefore,
\begin{displaymath}
\left\langle{I}\right\rangle{} = {\textstyle{1\over 2}}\vert\psi\vert^2.
\end{displaymath} (10)


Consider now two scalar functions

$\displaystyle \psi_1$ $\textstyle \equiv$ $\displaystyle \psi_{1,0}e^{i(kr_1+\phi_1)}$ (11)
$\displaystyle \psi_2$ $\textstyle \equiv$ $\displaystyle \psi_{2,0}e^{i(kr_2+\phi_2)}.$ (12)

Then


$\displaystyle I$ $\textstyle \equiv$ $\displaystyle [\Re(\psi_1)+\Re(\psi_2)]^2 = {\textstyle{1\over 4}}[(\psi_1+{\psi_1}^*)+(\psi_2+{\psi_2}^*)]^2$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[(\psi_1+{\psi_1}^*)^2+(\psi_2+{\psi_2}^*)^2+2(\psi_1\psi_2+\psi_1{\psi_2}^*+{\psi_1}^*\psi_2+{\psi_1}^*{\psi_2}^*)]$ (13)
$\displaystyle \left\langle{I}\right\rangle{}$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[2\psi_1{\psi_1}^*+2\psi_2{\psi_2}^*+2\psi_1{\psi_2}^*+2{\psi_1}^*\psi_2]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\psi_1({\psi_1}^*+{\psi_2}^*)+\psi_2({\psi_1}^*+{\psi_2}^*)]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\psi_1+\psi_2)({\psi_1}^*+{\psi_2}^*) = {\textstyle{1\over 2}}\vert\psi_1+\psi_2\vert^2.$ (14)

In general,
\begin{displaymath}
\left\langle{I}\right\rangle{} = {1\over 2}\left\vert{\sum_{i=1}^n \psi_i}\right\vert^2.
\end{displaymath} (15)

See also Affix, Argument (Complex Number), Complex Multiplication, Complex Number, Modulus (Complex Number), Phase



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-26