# Wavefunction

## 1 - Preface

In this essay, I will attempt to describe what a wavefunction is to the layman.

## 2 - Point Particles

Elementary particles such as electrons are wrongly believed by many to be point particles. A point particle is a particle that exists at a single point. To describe the position of a point particle in space would require only 3 real variables (x,y,z). All attempts so far to adequetly describe an electron as a point particle have failed. Physics doesn't describe elementary particles as point particles, but rather as wavefunctions.

## 3 - It is not a wave.

Something I need to get out of the way immediately is a common misconception about wavefunctions due to the spelling of the word. Although electrons are described by wavefunctions, and they share many properties with the scientific notion of a wave, they are not what are commonly referred to as waves. The common conception of a wave is very different from what an electron is like. I have seen many people take properties of waves and falsely apply them as properties of electrons. Do not do this. Instead, start a clean view with only the properties I will describe in this essay, and only add more properties after you prove them or learn about them in the textbooks.

## 4 - Funky

For pragmatic purposes, I will loosely define a new word: "Funky". I only intend to use this word to explain the concept of a wavefunction.

4.1 - Handup Funky

Take 10 people, number them from 1 to 10, and ask them to stand in a line in order. At any point in time each person may either hold a hand up or not hold a hand up. This is an example of a funky. A good description of this funky is that for each number from 1 to 10, there is a state of either "hand up" or "hand down". This is an example of a 1-dimensional funky that is valued by either "hand up" or "hand down"; a binary variable.

4.2 - Screen Funky

Now look at your computer screen. You can assign (x,y) coordinates to each position on your screen. For each position, there is a color which can be described by three real numbers (r,g,b) corresponding to the intensity of the red, green and blue phosphors at that location. This is an example of a 2-dimensional funky that is valued by three real numbers (r,g,b).

4.3 - Temperature Funky

Now look around your room. You can assign (x,y,z) coordinates to each position in your room. For each position there is a temperature (t). This is an example of a 3-dimensional funky that is valued by one real number.

4.4 - Functions

To those of you who are mathematically inclined, you would have noticed that the concept of a funky has many things in common with functions.

## 5 - A Wavefunction is a Funky.

If you have been able to understand the examples of funkys in the previous paragraph, then you should be able to understand a general description of a wavefunction and its properties. A wavefunction is most simply just a funky. Not much more. It can have any number of dimensions. It can be 2-dimensional, 3-dimensional, or even 35-dimensional. It can be valued in all sorts of ways, including: 3 real variables, 1 binary variable, or 100 real numbers plus 20 binary variables.

Now that you have an adequate understanding of a wavefunction for the purpose of this essay, I will cease to use the word "funky", and solely talk about wavefunctions.

## 6 - My First Wavefunction

Create a 3-dimensional wavefunction. For each 3-dimensional position, assign 1 positive real variable (d). The variable (d) is called the particle density. The units of this variable is in "particles per litre" or p/L. I call this function the "My First Wavefunction", or the mf wavefunction. The "mf" in "mf wavefunction" is not to be confused with "mother fucker".

The mf wavefunction is a very good description of particles, including electrons.

## 7 - States

Take your mf wavefunction and apply it to the 3 dimensional space in your room. To each position in your room, you will have to assign a real number for the particle density. The configuration that results when you assign real numbers to every position is called the state of the particle. There are many possible ways to assign the numbers, but for a particular instant of time, you can only assign the numbers in one way. Therefore, at each point in time, a particle can be in only one of many possible states.

Here are some example states:

Place an empty 1-litre bottle on your table. For every position outside the bottle, assign a particle density of 0 p/L. For every position inside your 1-litre bottle, assign a particle density of 1 particle per litre, or 1 p/L. In this example, your particle is completely contained inside the 1 litre bottle.

Place an empty 4-litre bottle on your table. For every position outside the bottle, assign a particle density of 0, and for every position inside the bottle assign a particle density of 0.25 p/L. In this example the particle is completely contained inside the 4-litre bottle, but since this bottle is bigger than the first one, the particle has to spread out more thinly, and therefore has a lesser density within the 4L bottle than it did in the 1L bottle.

Place three 1-litre bottles on your table. Assign 0 p/L to all positions outside of those bottles. Assign 0.5 p/L completely inside the first bottle, and 0.25 p/L completely inside the other two bottles. In this case we have a particle density that is different depending on which bottle you are looking at. Also, unless the bottles are touching each other, there will be empty space in between the bottles in which the particle density is zero.

## 8 - Total Particle Quantity

You may have noticed that I was careful with the densities I assigned to the bottles in my previous examples. When you multiply the particle density in a bottle (p/L) times the volume of the bottle (L) you get the particle quantity within that bottle. That is to say, if you multiply the amount of litres times the amount of particles per litre, you get the total amount of particles. If a bottle is 1 litre, and contains a density of 0.25 p/L, the total quantity of particles in that bottle is 0.25. That means that we have 25% of a particle inside that bottle. If you look at the example with 3 bottles above, you will notice that if you add up the quantity of particles in each bottle, that the total sum is one. I was careful to make sure that this sum was always one in each of my examples above. When you add up the sum of particle quantities in every bottle that your particle is partly in, you get the total particle quantity.

Say you have a 4-litre bottle on your table, and you assign a density of 0.5 p/L within that bottle. The total amount of particles would be 4 times 0.5, which is 2. This isn't right, because our wavefunction is only supposed to describe one particle. We therefore add the following restriction to the allowable states of our wavefunction:

The total particle quantity of the wavefunction must always be 1.

For those who want to know, wavefunctions whose total particle quantity is 1 are called normalized wavefunctions. Physicists always work with normalized wavefunctions. Sometimes when doing calculations, they will generate a wavefunction that isn't normalized, in which case they quickly normalize it.

## 9 - Electrons in Atoms

What would happen if we take an electron and put it completely inside a bottle, and then squeeze the bottle to make it smaller? The particle density within the bottle will rise. If an electron is evenly spread inside of a 0.1 L bottle, the particle density within that bottle would be 10 p/L. Do not be fooled into thinking that there are ten particles in this bottle. If you multiply the particle density times the volume, 10*0.1, you get a total particle quantity of 1.

Electrons that are bound to atoms are normally squeezed very very tightly near the nucleus of the atom. It's squeezed so tightly, that it is possible to approximate it somewhat adequately as a point particle. It must be remembered though that an electron is not a point particle, but a wavefunction, and therefore is actually spread out in a volume, even if it's a very small volume.

## 10 - Sounds like gas

You may have noticed that the mf wavefunction sounds very much like it has a lot of the same properties as a gas. It can be spread out over different volumes with variable densities. But the total amount of that wavefunction (or particle) is always 1, and never changes. This is the same property that gases have: their total mass is always the same, but they can be spread out over different volumes with variable densities.

## 11 - My Second Wavefunction

Our mf wavefunction is so limited. It's only valued by a single real number. Even our computer screens are valued by at least 3 real numbers. There's only so much you can do with a single real number. So let's create a new wavefunction that is more versatile.

Our second wavefunction will also be 3-dimensional like the first. But it will be valued by 2 real numbers (d,p) instead of 1. The first real number d is just the particle density as we had before. The second real number p is the phase. We also add the restriction that the phase must always be greater than or equal to 0, and must always be less than 360. Think of the phase as an angle: the direction an arrow is pointing at. The phase may be 0, or 90, or 270.5, or 359.999, but it may not be 360, nor -20, nor 400. We also have the added restriction that wherever the particle density is zero, the phase must also be zero. So pairs of values such as (0.25, 90), (2, 0) are valid, but (0, 90) is invalid.

This wavefunction is very popular and already has a name. It's called the spinless one-particle non-relativistic wavefunction. People who take a first course on quantum mechanics will spend most of their time studying and working with this wavefunction.

## 12 - The Phase

The phase is not something I want to explain in detail in this essay. It would take a whole new essay for me to explain the phase with enough detail for your understanding of it to be as good as your understanding of the particle density. But I do want to say a few things about it.

The particle density of the wavefunction tells us very much about the electron. If you know that the electron is in a particular state, say for example a particular density within a bottle of a specified volume, then you may predict that a small time into the future the density will not be very different. Unless you wait a very long time, if the particle spreads out, it will not spread out too far from its original position. You use the particle density information to figure this out.

Physicists use the phase in addition to the particle density to make even more detailed and more precise predictions of the particle's behaviour. That is the most important thing you need to know about the phase.

At each position in space, in addition to the density, imagine a little arrow pointing in a particular direction. The particle's behaviour is highly governed by not only the density at each position, but also the phase at each position, just like how a human being's behaviour is highly governed by its sexual orientation.

## 13 - Success!

If you take this (d,p)-valued wavefunction and make it dynamic, That is, have it change over time, you need an equation that describes how it changes with time. When you use an equation called the Schrodinger equation, you can predict many physical phenomena in physics. In addition to all the phenomena that was described before quantum mechanics was invented, the schrodinger equation is also capable of describing:

• All the energy states and levels in the hydrogen atom.
• Interference in the double slit experiment.

## 14 - More Precision

We started with an mf wavefunction valued by a single real variable. It was able to describe the electron very adequately, including how it can be spread out over differently sized volumes with different densities. We then added an extra variable called the phase, which made our description of an electron much more precise. Depending on how much precision you need in your description of particles, you will need to add more variables as well as more dimensions.

One good example is the Dirac wavefunction, which is a 3-dimensional wavefunction with 8 real variables! Only one of those variables is the particle density. The other variables relate to different properties electrons have such as their momentum and spin.

Another good example is when you have two or more particles in an entangled state. In this case a 3-dimensional wavefunction cannot adequately describe the particles and we need a wavefunction with many more dimensions. More dimensions are also needed when you take into account the possibility of particle creation and annihilation.

Regardless of how many variables and dimensions physicists use in their wavefunctions, the particle density is central and the most important concept in all of their wavefunctions. Because of this, understanding the mf wavefunction is not only a key concept that needs to be mastered before moving onto more advanced topics, but is also adequate for understanding the most basic properties of particles.

## 15 - That's it!

Well, that's it. Now that you understand the mf wavefunction, I hope you have a much better understanding of what particles are. Particles are not "points" in space. They are also not "waves propagating through a medium". Particles are wavefunctions.

Whenever you think of particles from now on, just imagine an mf wavefunction. Although you don't really need the other variables for a general imagination of particles, you just need to remember that there are other variables.

## 16 - Advanced Info

For the more advanced readers, I want to make the following points. The particle density and phase are usually not used as is by physicists, but instead replaced by a single complex number psi, which is equivalent. This complex number psi is called the probability amplitude (a misleading name). The phase in my description is just the argument of psi, that is p = arg(psi). The particle density in my description is just the modulus squared of psi, that is d = |psi|^2. The restriction that the total particle density must always be 1 is made precise mathematically by requiring that the integral of the modulus squared of the wavefunction over all space be 1.

Most books write "wave function" as two words rather than one word as I do. The evolution of wavefunctions over time are described by wave equations. If you've made it this far and you know some calculus, you should be able to look up the Schrodinger equation which is a wave equation for spinless particles and start studying its properties. Once you are satisfied with your understanding of the Schrodinger equation, you should study some of the other one-particle wave equations. Here is a brief list of wave equations and the kinds of particles they successfully describe:

• Schrodinger equation. Non-relativistic, spin-0.
• Pauli equation. Non-relativistic, spin-1/2.
• Klein-Gordon equation. Relativistic, spin-0.
• Dirac equation. Relativistic, spin-1/2. (eg.: electrons)
• Maxwell-Proca equation. Relativistic, spin-1. (eg.: photons)
• Rarita-Schwinger equation. Relativistic, spin-3/2.
• Bargmann-Wigner equation. Relativistic, any spin.

The non-relativistic wave equations are just approximations of the relativistic versions for the same spin. The Bargmann-Wigner equation is a relativistic wave equation of which all the previous relativistic equations are special cases of. If you learn the Bargmann-Wigner equation, you will know all you need to know about one-particle wave equations.

Have fun!

Wavefunction, by The Humanoid

• 2005-01-29 - Draft Version 1
• 2006-01-04 - Draft Version 2