Since Huygens and Newton, physicists have known that light is described by (electromagnetic) waves and particles (photons). Such peculiar nature of light is called wave-particle duality. What about material particles such as electrons? de Broglie proposed a bold hypothesis that what is true for light is also true for material particles i.e. they will also exhibit wave nature. (This was indeed confirmed by experiments.) So how do we mathematically model such a wave? What physicists thought of using to study material particles was a complex plane wave called de Broglie wave. It looks like $$\psi(x,t)=Ae^{i(kx-\omega t)}$$ for 1-dimensional case. For 3-dimensional case, it would be \begin{equation}\label{eq:planewave}\psi({\bf r},t)=Ae^{i({\bf k}\cdot{\bf r}-\omega t)}\end{equation} Before we continue, one may wonder how physicists came up with this kind of wave. I can only speculate but such a complex plane wave was already well-known to physicists as it is a solution of Maxwell’s equation in electromagnetism. Hence, complex plane wave may describe electromagnetic wave, and naturally it became the first candidate for modeling material particles. In fact, it worked out well as we shall see and consequently complex numbers played a crucial role in building quantum mechanics.

Let us first study some properties of plane waves. The plane wave \eqref{eq:planewave} describes a free particle, more accurately a free particle in a state. In order for a plane wave to behave like a particle, we want it to be localized i.e. the wave is defined in a tiny region. (There is a more mathematically subtle reason why we require this.) We can achieve this by redefining $\psi({\bf r},t)$ as $$\psi({\bf r},t)=\left\{\begin{array}{ccc}Ae^{i({\bf k}\cdot{\bf r}-\omega t)} & \mbox{for} & {\bf r}\ \mbox{within a volume}\ V=L^3\\0 & \mbox{for} & {\bf r}\ \mbox{outside a volume}\ V=L^3\end{array}\right.$$ Physicists call this box renormalization. Physically the state of a particle must not depend on a particular location of the tiny box, so we require the periodicity condition $$\psi(x,y,z,t)=\psi(x+L,y,z,t)=\psi(x,y+L,z,t)=\psi(x,y,z+L,t)$$ Here, $L$ is called wave length. The periodicity condition implies that ${\bf k}$ is quantized as $${\bf k}=\frac{2\pi}{L}{\bf n}$$ where ${\bf k}=(k_x,k_y,k_z)$, ${\bf n}=(n_x,n_y,n_z)$, and $n_i=0,1,2,\cdots$, $i=x,y,z$. The vector ${\bf k}$ is called wave vector and for 1-dimensional case, $k$ is called wave number. If the wave is periodic in time, say $\psi(x,t)=\psi(x,t+T)$, then we obtain $e^{-i\omega T}=1$. The nonzero minimum value of $T$ is $T=\frac{2\pi}{\omega}$. $\omega=\frac{2\pi}{T}$ is called angular frequency. $kx-\omega t$ is called phase and if $kx-\omega t$ is constant, the wave moves at the speed $v_p=\frac{dx}{dt}=\frac{\omega}{k}$. This $v_p$ is called phase velocity. For 3-dimensional case, \begin{align*}\psi({\bf r},t)&=Ae^{i({\bf k}\cdot{\bf r}-\omega t)}\\&=Ae^{i{\bf k}\cdot\left({\bf r}-\frac{\omega t}{|{\bf k}|^2}{\bf k}\right)}\\&=Ae^{i{\bf k}\cdot\left({\bf r}-\frac{\omega t}{|{\bf k}|}\hat{\bf k}\right)}\end{align*}So, the phase velocity would be $${\bf v}_p=\frac{d{\bf r}}{dt}=\frac{\omega}{|{\bf k}|}\hat{\bf k}$$

The image of wave function $\psi(x,t)=Ae^{i(kx-\omega t)}$ is a circle. We are in fact quite familiar with this kind of waves. On a beautiful day, you go to a lake. You would be then tempted to throw a rock into the cam water. When you do, you would see circular water waves spreading out from the point of impact.

References:

[1] Walter Greiner, Quantum Mechanics, An Introduction, 4th Edition, Springer, 2001

[2] Quantum Mechanics, H.-S. Song (in Korean)