Phasor

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

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

where i (called j by engineers) is the Imaginary Number and the Modulus and Argument (also called Phase) are

$$\vert z\vert = \sqrt{x^2+y^2}$$ (2)

$$\phi = \tan^{-1}\left({y\over x}\right).$$ (3)

Here, $\phi$ is the counterclockwise Angle from the Positive Real axis. In the degenerate case when $x=0$,

$$\phi=\cases{ -{\textstyle{1\over 2}}\pi & if $y<0$\cr {\rm... ...ined} & if $y=0$\cr {\textstyle{1\over 2}}\pi & if $y>0$.\cr}$$ (4)

It is trivially true that

$$\sum_i \Re[\psi_i] = \Re\left[{\sum_i \psi_i}\right].$$ (5)

Now consider a Scalar Function $\psi\equiv \psi_0e^{i\phi}$. Then

$$I \equiv [\Re(\psi)]^2 = [{\textstyle{1\over 2}}(\psi+\psi^*)]^2 = {\textstyle{1\over 4}}(\psi+\psi^*)^2$$

$$= {\textstyle{1\over 4}}(\psi^2+2\psi\psi^*+{\psi^*}^2).$$ (6)

Look at the time averages of each term,

$$\left\langle{\psi^2}\right\rangle{}=\left\langle{{\psi_0}^2e... ...ngle{} = {\psi_0}^2\left\langle{e^{2i\phi}}\right\rangle{} = 0$$ (7)

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

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

Therefore,

$$\left\langle{I}\right\rangle{} = {\textstyle{1\over 2}}\vert\psi\vert^2.$$ (10)

Consider now two scalar functions

$$\psi_1 \equiv \psi_{1,0}e^{i(kr_1+\phi_1)}$$ (11)

$$\psi_2 \equiv \psi_{2,0}e^{i(kr_2+\phi_2)}.$$ (12)

Then

$$I \equiv [\Re(\psi_1)+\Re(\psi_2)]^2 = {\textstyle{1\over 4}}[(\psi_1+{\psi_1}^*)+(\psi_2+{\psi_2}^*)]^2$$

$$= {\textstyle{1\over 4}}[(\psi_1+{\psi_1}^*)^2+(\psi_2+{\psi_2}^*)^2+2(\psi_1\psi_2+\psi_1{\psi_2}^*+{\psi_1}^*\psi_2+{\psi_1}^*{\psi_2}^*)]$$ (13)

$$\left\langle{I}\right\rangle{} = {\textstyle{1\over 4}}[2\psi_1{\psi_1}^*+2\psi_2{\psi_2}^*+2\psi_1{\psi_2}^*+2{\psi_1}^*\psi_2]$$

$$= {\textstyle{1\over 2}}[\psi_1({\psi_1}^*+{\psi_2}^*)+\psi_2({\psi_1}^*+{\psi_2}^*)]$$

$$= {\textstyle{1\over 2}}(\psi_1+\psi_2)({\psi_1}^*+{\psi_2}^*) = {\textstyle{1\over 2}}\vert\psi_1+\psi_2\vert^2.$$ (14)

In general,

$$\left\langle{I}\right\rangle{} = {1\over 2}\left\vert{\sum_{i=1}^n \psi_i}\right\vert^2.$$ (15)

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