In my textbook it is given that
'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'
Can someone give me a very simple example of a wave function with explanation? (Note: This question is not a duplicate. I have searched for other questions of this type but the answers were overwhelmingly difficult to understand.)
A wave function is a complex-valued function $f$ defined on ${\mathbb R}^1$ (if your electron is confined to a line) or on ${\mathbb R}^2$ (if your electron is confined to a plane) or ${\mathbb R}^3$ (if your electron ranges over three-space), and satisfying $$\int |f|^2=1$$ (where the integral is defined over the entire line or plane or 3-space).
Every electron has an associated wave function, and any function satisfying the above can be the wave function associated to some electron.
The wave function tells you everything there is to know about the electron. For example, if $A$ is any set, and if you perform an experiment that answers the question "is the electron in the set $A$?", then the probability you'll get a "yes" answer is given by $$\int_A |f|^2$$
(So in particular, if $A$ is the entire space, you're asking "Is the electron anywhere at all?", and the probability of a yes answer is $1$.)
The next steps are to learn:
1) How do I use this wave function to predict the outcomes of questions about something other than the electron's location, such as, for example, its momentum?
and
2) How does this wave function change over time?
I don't think you're quite yet at the point of addressing those questions (though you will be soon enough).
The wave function is the solution to the Schrödinger equation, given your experimental situation. With a classical system and Newton's equation, you would obtain a trajectory, showing the path something would follow: the equations of motion. For a quantum mechanical system you get a wave function, and the rules it obeys over time. With this you can determine the odds for your particle to be someplace, which is as close as you can get to a trajectory.
You normally learn most of the math first, then Newtonian mechanics, and then quantum mechanics. Hopefully by then you will be able to put it in context.
A "wave function" is a mathematical model (or representation) of a given wave. A "function" is represented by the symbol $f$. It can be a function of distance (x), time (t), space (r), etc. and is usually represented by an equation. If the equation represents a wave, then the function is a wave function.
For example, a simple wave with constant amplitude and varying in time can be described by: $A \sin{t} $. Its wave function would be $f_{(t)} = A \sin{t}$. You can evaluate it over some interval, by integrating over the interval.
In the case of the electron, its wave function has already been described in WillO's answer.
I am taking seriously your request for a simple language answer, & would also be happy to provide a more complex answer. My answer hopefully provides, by a progressive series of simple statements and examples, a way of actually conceiving what a wave function is. You may want to scroll to the final paragraph.
A function is a mathematical representation of something. A wave function is a mathematical representation of a wave.
Consider y = sin x This might describe the momentary snapshot of a wave made by wiggling a rope.
This shape would move along the x-axis with time, so y= Sin(x-t) would be a very simple example of a sine wave moving along a rope.
I'm sure I will be dissed for answering your question literally. I do have 4 years of tertiary maths and 3 years of tertiary physics behind me, but you did seem to want a very simple example.
An electron is sometimes represented as somewhat like a short burst of waves, and the complex formula for this "shape" is called the wave function. Your quote from the textbook is not a definition of a wave function, it is telling you something about the wave function representing an electron.
You can imagine a straight line with a burst of wave activity somewhere along its length. If the line represents distance, then the position of the burst is the location of the electron. If the line represents time then it indicates when the electron was at some location. The wave function of an electron describes the time, location, and in fact everything we could know about the electron.
Because any method of looking at the electron has an effect on it (rather like using a torch to find out what rabbits do in the dark), we never really know exactly every value of every parameter associated with the electron.
The wave equation therefore can not simply tell us where the electron is, or how fast it is going. It gives us the probabilities of the electron (or other phenomenon) being at any location at any speed.
In our everyday experience, the way we imagine particles to be is quite adequate. It is easy to forget that this conception of a particle is a model, not a reality. In fact there is no such thing as a distinct surface of a particle. When we are dealing with things at a sub-atomic level, our everyday concept breaks down, and we admit that we only know probabilities of the where-and-when of a particle.
The wave function for an electron therefore includes everything we know about the electron, which includes that it is not a particle in the everyday sense, but is an array of greater and lesser probabilities of being somewhere at some time. The complexity is not a reflection of reality, but rather of how difficult it is to create a model than can be understood by a mind that generally needs to relate to what can be perceived by the senses.
First of all any physical particle is in some "state" that contains all the observable information. Classically the state of a particle is govern by position $x$ and momentum $p$, that is, every observable quantity is a function of $x$ and $p$, e.g. kinetic energy, potential energy, angular momentum etc.
In quantum mechanics definite position and momentum no longer exist but are spread out in the space. The information of both position and momentum are contained in a wave function $\psi (x,t)$ that tells how the both position and momentum are distributed at time the $t$. Therefore the state of a particle in quantum mechanics is the corresponding wave function.
In classical mechanics position and momentum tell the definite value of every observable quantity, and in quantum mechanics wave function tells the average value of every observable quantity. Additionally, in quantum mechanics those observable quatities can be discrete or continuous.
A wave function, or a set of wave functions, that corresponds to a particle can be found by solving the Schrödinger equation.
