Absolute Square

Also known as the squared Norm. The absolute square of a Complex Number $z$ is written $\vert z\vert^2$ and is defined as

$$\vert z\vert^2 \equiv zz^*,$$ (1)

where $z^*$ denotes the Complex Conjugate of $z$. For a Real Number, (1) simplifies to

$$\vert z\vert^2=z^2.$$ (2)

If the Complex Number is written $z=x+iy$, then the absolute square can be written

$$\vert x+iy\vert^2=x^2+y^2.$$ (3)

An important identity involving the absolute square is given by

$$\vert a\pm b e^{-i \delta}\vert^2 = (a\pm b e^{-i \delta})(a\pm b e^{i \delta})$$

$$= a^2+b^2 \pm a b (e^{i \delta} + e^{-i \delta})$$

$$= a^2+b^2\pm 2 a b \cos\delta.$$ (4)

If $a=1$, then (4) becomes

$$\vert 1\pm b e^{-i\delta}\vert^2 = 1+b^2\pm 2b\cos\delta$$

$$= 1+b^2\pm 2b[1-2\sin^2({\textstyle{1\over 2}}\delta)]$$

$$= 1\pm 2b+b^2\mp 4b\sin^2({\textstyle{1\over 2}}\delta)$$

$$= (1\pm b)^2\mp 4b\sin^2({\textstyle{1\over 2}}\delta).$$ (5)

If $a=1$, and $b=1$, then

$$\vert 1- e^{-i\delta}\vert^2 = (1-1)^2+4\cdot 1\sin^2({\textstyle{1\over 2}}\delta)= 4\sin^2({\textstyle{1\over 2}}\delta).$$ (6)

Finally,

$$\vert e^{i\phi_1}+e^{i\phi_2}\vert^2 = (e^{i\phi_1}+e^{i\phi_2})(e^{-i\phi_1}+e^{-i\phi_2})$$

$$= 2+e^{i(\phi_2-\phi_1)}+e^{-i(\phi_2-\phi_1)}$$

$$= 2+2\cos(\phi_2-\phi_1) = 2[1+\cos(\phi_2-\phi_1)]$$

$$= 4\cos^2(\phi_2-\phi_1).$$ (7)