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  • Writer's pictureNoa Feldman

3 - The Wave Function

*Is it your first visit here? Welcome! My name is Noa and I am a Physics Ph.D. student at Tel Aviv University. I write here about quantum mechanics for non-physicists. No background in mathematics or physics is required to read the blog, but I highly recommend reading the first two posts before reading this one. The first post is here and the second is here. Enjoy!*

In the previous posts, we learned about the basic principles of quantum mechanics: superposition, which means that quantum particles do not have a definite position, but instead some position probability distribution, which can be different than zero in more than one place. We saw that observing, or measuring, the position of the particle, makes it collapse into one of the positions that are allowed by its probability distributions, and that it is impossible to predict exactly which. And we saw that these principles apply also to speed, energy, and any other physical property, in a way that if the probability is collapsed into a small uncertainty in position, the speed probability is smeared into a large uncertainty.

The above leads quantum physicists to discuss not the position of the particle, but instead its probability distribution of position. In classical physics, physicists have revealed rules that determine how a particle’s position and speed can change as a reaction to forces (interactions). Likewise, in quantum mechanics, we study the rules that determine the change of the probability distributions are a result of forces. In this post, we will discuss some of these rules, which can explain some interesting phenomena of quantum physics, and be useful in the future. We will also discuss an experiment that was able to demonstrate these principles in the lab.

You may have heard previously that in quantum mechanics, an electron is both a particle and a wave. I guess this meant nothing to you back then, but you know better now. Now that we understand that the properties of the electron are described by a probability distribution, this sentence can become meaningful: A particle is still a particle, but its probability distributions (of the position, speed, etc.) behave like a wave’s power.

Before we understand the phrase above, we first need to understand what a wave is. We all know a lot of examples of waves - waves in water, light waves, or sound waves.

The definition of waves in physics is a phenomenon, described by a function of space and time, which obeys some equation called the wave equation. Any function can be plotted in a graph. A graph of a wave looks something like this:

Just like we see in water.

The thing moving in a wave can be all sorts of things - the height of football fans or water, a vibration in the air (sound waves), or the power of the electromagnetic field (light waves). Let’s think of light waves for now.

Classically, that is, in our day-to-day world, light behaves like a wave. We don’t see it in our eyes, because the wavelength of visible light is very short (wavelength is what we see in red in the drawing below). A ray of light is an electromagnetic wave moving in space. It can change directions when it hits some surface and eventually reach our eyes. The power of light we see depends not on the height of the wave function, but on the height of the function squared. So if the wave function is negative, we still get a positive power.

Now we go back to the statement above - the probability distribution of a quantum particle’s position behaves like the power of a wave. This means that the state of a quantum particle can be described by some function that behaves like a wave. If we take this wave and square it, the result will be larger than or equal to zero in all possible positions, and it will be the probability distribution of the particle’s position - so if we know the wave function, we know the probability (which is all there is to know about the particle). When physicists want to study the behavior of quantum particles, the wave function is the basic property we study.

Why do we study the wave function instead of its square, which is the probability itself? Two reasons:

  1. We know a lot about waves and how to study them. Waves were studied by classical physicists for hundreds of years before quantum mechanics was suggested.

  2. When we square a number, we lose information - its sign, positive or negative. If you have learned about complex numbers before, then you know that we can lose the number’s complex phase. This sign (or phase) is very important, as we will now see, and it can affect the particle significantly.

An important property of waves is called interference. Whenever two waves act in the same space, such as in a pool of water or on a string in a guitar, their combined shape will be their sum.

So this wave:

Created with

combined with this wave:

Created with

Will create this wave:

Created with

As waves can be positive or negative in different positions, we can observe that in some positions the interference results in a high point in the wave, and in other positions, the combined wave has a lower point or even a zero point, like in x=1 and x=-1 in the example.

A good example of wave interference, that you may have encountered in school, is the double-slit experiment, first conducted by Thomas Young in 1801. We light a laser on a board with two slits with a screen behind it:

Behind the slits, the light spreads like a wave in water, and when they come out through the slits, each slit becomes a source of the wave. The stripes pattern we see on the screen is the interference of the two waves sourced at both slits as can be seen in the figure on the right. If we know the wavelength, we can calculate a prediction of what we see on the screen, and then check whether it works in an experiment. If our prediction matches the experiment’s result, it is a good sign that we understand how waves work.

Now we can combine everything we saw today: In the previous post, I promised a small proof that quantum mechanics is true (as far as science knows). Such proof was obtained in a double-slit experiment, except that electrons were shot at the slits instead of laser light. It was conducted in 1927 by physicists called Clinton Davisson and Lester Germer. They used something that acted as a board with slits for electrons, and a screen that counted the number of electrons at each position. Electrons, as far as anyone knew before quantum mechanics, were particles and not waves, so nothing about them resembles interference. There is no such thing as a ‘minus particle’ that cancels out a ‘plus particle’, and particles do not spread across the screen, they hit it at a certain position. So based on classical mechanics, we expect that each electron shot at the board would have one of two options - either it doesn’t go through any slit and never reaches the screen, or it goes through one of the slits and hits the screen more or less in front of that slit. In this case, the screen counts will be something like this:

That was the classical assumption, in any case. The quantum assumption was that the electron can pass simultaneously through both of the slits, that is, be in a superposition of going through both slits. This probability looks just like the wave of light going through the slit in the purple drawing. Only when the electron meets the screen, it is measured by the screen and gets a definite position on the screen. Therefore, we expect to get a pattern on the screen that looks like the one we got for light.

It worked! The pattern created on the screen looked like a wave pattern, with many stripes, like in the purple figure, and not like a particle pattern, like the one in green. This was an important success of the predictions of quantum mechanics. But this is not even the coolest part.

The cool part was extensions of the experiment, done in the Heiblum lab at Weizmann institute in 1998 (it has been done before by several labs, but not in the exact version I describe now). What they did was to add a little detector that senses whether the electron went through the upper slit. That is, they measured which slit the electron went through and got it to collapse into a single slit. This collapse, in fact, gives the electron a particle-like behavior since it now does not have two wave sources that can interfere. And indeed, in this way, the pattern on the screen was like the figure in green! Who ‘chose’ the slit each electron would go through? We don’t know. But we know that the detector’s measurement provoked this choice.

The double-slit experiment is considered to be one of the fundamental experiments of quantum mechanics, and I think it demonstrates nicely everything we have seen so far: We see superposition in the behavior of electrons passing ‘simultaneously’ through both slits, and the wave-like behavior of their probability distribution. We see that observing particles changes their probability distribution and makes them collapse into a definite position. We also see that after measurement, around half of the electrons pass through the upper slit and the other half through the bottom slit, without any way to predict which electron ‘chooses’ which slit.

So, to summarize: quantum particles do not have a definite position, speed, or energy but instead some probability distribution of each of their physical properties (superposition). Physicists study these probability distributions, and it turns out that they behave like the power of a wave. This means that quantum particles are not exactly classical particles, but also not exactly waves, but some hybrid of the two, and this was shown in the double-slit experiment. Classically, we see waves (like light) and particles (like electrons), but in fact, both of these are this hybrid of waves and particles. In the next post, I’ll talk a little about what led physicists to develop the theory of quantum mechanics.

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