We have seen that the wave function can be complex, so we now review some properties of complex numbers.
A complex number $z$ is a number of the form
\(
\begin{equation}
z=x+i y, \quad \text { where } i \equiv \sqrt{-1} \tag{1.25}
\end{equation}
\)
and where $x$ and $y$ are real numbers (numbers that do not involve the square root of a negative quantity). If $y=0$ in (1.25), then $z$ is a real number. If $y \neq 0$, then $z$ is an imaginary number. If $x=0$ and $y \neq 0$, then $z$ is a pure imaginary number. For example, 6.83 is a real number, $5.4-3 i$ is an imaginary number, and $0.60 i$ is a pure imaginary number. Real and pure imaginary numbers are special cases of complex numbers. In (1.25), $x$ and $y$ are called the real and imaginary parts of $z$, respectively: $x=\operatorname{Re}(z) ; y=\operatorname{Im}(z)$.
The complex number $z$ can be represented as a point in the complex plane (Fig. 1.3), where the real part of $z$ is plotted on the horizontal axis and the imaginary part on the vertical axis. This diagram immediately suggests defining two quantities that characterize the complex number $z$ : the distance $r$ of the point $z$ from the origin is called the absolute value or modulus of $z$ and is denoted by $|z|$; the angle $\theta$ that the radius vector to the point $z$ makes with the positive horizontal axis is called the phase or argument of $z$. We have
\(
\begin{gather}
|z|=r=\left(x^{2}+y^{2}\right)^{1 / 2}, \quad \tan \theta=y / x \tag{1.26}\
x=r \cos \theta, \quad y=r \sin \theta
\end{gather}
\)
So we may write $z=x+i y$ as
\(
\begin{equation}
z=r \cos \theta+i r \sin \theta=r e^{i \theta} \tag{1.27}
\end{equation}
\)
since (Prob. 4.3)
\(
\begin{equation}
e^{i \theta}=\cos \theta+i \sin \theta \tag{1.28}
\end{equation}
\)
The angle $\theta$ in these equations is in radians.
If $z=x+i y$, the complex conjugate $z^{*}$ of the complex number $z$ is defined as
\(
\begin{equation}
z^{} \equiv x-i y=r e^{-i \theta} \tag{1.29}
\end{equation}
\)
FIGURE 1.3 (a) Plot of a complex number $z=x+i y$. (b) Plot of the number $-2+i$.
If $z$ is a real number, its imaginary part is zero. Thus $z$ is real if and only if $z=z^{}$. Taking the complex conjugate twice, we get $z$ back again, $\left(z^{}\right)^{*}=z$. Forming the product of $z$ and its complex conjugate and using $i^{2}=-1$, we have
\(
\begin{gather}
z z^{}=(x+i y)(x-i y)=x^{2}+i y x-i y x-i^{2} y^{2} \
z z^{}=x^{2}+y^{2}=r^{2}=|z|^{2} \tag{1.30}
\end{gather}
\)
For the product and quotient of two complex numbers $z{1}=r{1} e^{i \theta{1}}$ and $z{2}=r{2} e^{i \theta{2}}$, we have
\(
\begin{equation}
z{1} z{2}=r{1} r{2} e^{i\left(\theta{1}+\theta{2}\right)}, \quad \frac{z{1}}{z{2}}=\frac{r{1}}{r{2}} e^{i\left(\theta{1}-\theta{2}\right)} \tag{1.31}
\end{equation}
\)
It is easy to prove, either from the definition of complex conjugate or from (1.31), that
\(
\begin{equation}
\left(z{1} z{2}\right)^{}=z{1}^{*} z{2}^{} \tag{1.32}
\end{equation}
\)
Likewise,
\(
\begin{equation}
\left(z{1} / z{2}\right)^{}=z{1}^{*} / z{2}^{}, \quad\left(z{1}+z{2}\right)^{}=z{1}^{*}+z{2}^{}, \quad\left(z{1}-z{2}\right)^{}=z{1}^{*}-z{2}^{} \tag{1.33}
\end{equation}
\)
For the absolute values of products and quotients, it follows from (1.31) that
\(
\begin{equation}
\left|z{1} z{2}\right|=\left|z{1}\right|\left|z{2}\right|, \quad\left|\frac{z{1}}{z{2}}\right|=\frac{\left|z{1}\right|}{\left|z{2}\right|} \tag{1.34}
\end{equation}
\)
Therefore, if $\psi$ is a complex wave function, we have
\(
\begin{equation}
\left|\psi^{2}\right|=|\psi|^{2}=\psi^{} \psi \tag{1.35}
\end{equation}
\)
We now obtain a formula for the $n$th roots of the number 1 . We may take the phase of the number 1 to be 0 or $2 \pi$ or $4 \pi$, and so on. Hence $1=e^{i 2 \pi k}$, where $k$ is any integer, zero, negative, or positive. Now consider the number $\omega$, where $\omega \equiv e^{i 2 \pi k / n}, n$ being a positive integer. Using (1.31) $n$ times, we see that $\omega^{n}=e^{i 2 \pi k}=1$. Thus $\omega$ is an $n$th root of unity. There are $n$ different complex $n$th roots of unity, and taking $n$ successive values of the integer $k$ gives us all of them:
\(
\begin{equation}
\omega=e^{i 2 \pi k / n}, \quad k=0,1,2, \ldots, n-1 \tag{1.36}
\end{equation}
\)
Any other value of $k$ besides those in (1.36) gives a number whose phase differs by an integral multiple of $2 \pi$ from one of the numbers in (1.36) and hence is not a different root. For $n=2$ in (1.36), we get the two square roots of 1 ; for $n=3$, the three cube roots of 1 ; and so on.