Wave functions in physics help us understand the behavior of tiny particles, like electrons. They are usually shown using complex numbers, which are numbers that have both a real part and an imaginary part.

We write wave functions like this:

$\psi(x, t) = A e^{i(kx - \omega t)}$

Here’s what each part means:

• $A$ is the amplitude, which tells us how big the wave is.
• $k$ is the wave number, which relates to how often the wave repeats.
• $\omega$ is the angular frequency, which helps us know how fast the wave is moving.
• $i$ is the imaginary unit, a special number used in complex calculations.

Using complex numbers makes it easier to do calculations about how waves mix together, like in interference and superposition.

If we want to find out how likely it is to find a particle in a certain place, we look at the wave function again. We do this by calculating:

$|\psi(x, t)|^2 = \psi(x, t) \cdot \psi^*(x, t)$

This means we take the wave function and multiply it by its complex "conjugate."

Wave functions are really important in many areas, especially in electrical engineering. For example, they help us understand how alternating currents work and how waves behave.

In fact, around 70% of today’s electronic devices use ideas that come from wave functions!