### Quantum Mechanics in a nutshell

This takes a little time to get to the point of the main question:

How come we can’t say that a particle is *right there* and that it is moving *that fast* in *that direction* ?

So bear with me. I’m going through these steps:

- 1. The electron is a particle.
- 2. The electron acts like a particle
- 3. Sometimes the electron doesn’t act like a particle
- 4. Sometimes the electron acts like a wave
- 5. What is the location of the electron in QM?
- 6. If the QM location is a probability distribution, what is the QM velocity?
- 7. The location and velocity are both imprecise in QM.

**1. The electron is a particle**

In a very basic sense, QM is a math theory that describes what happens when electrons interact with matter – that is, with atoms. For instance, when you send a stream of electrons from an electron gun (which is actually at the back of all your CRT computer monitors), the electron goes through a vacuum and hits some target. In the CRT [Cathode Ray Tube], the target is the display screen. Precise experiments show that each electron hits in a single spot. Additional experiments show that the electrons are all of the same mass, and they all carry the same charge. So, they are particles. Nothing new there – that’s what we expected.

**2. The electron acts like a particle**

Then we come to the two-hole experiment. Here we shoot electrons from the electron gun through a partition that has two holes in it, and then measure the electrons when they hit a CRT screen on the far side. Here’s a really lame diagram that shows the experimental setup:

Diagram 2.1

Well, let’s work up to this experiment in steps. First, we’ll only keep the top hole open. When we do that, we find that the electrons hit the CRT screen mostly at the level of the top hole. However, some of them are a little higher, some of them are a little lower. If you measure the # of electrons at each point on the CRT screen, you get something like this (the dashed lines on the CRT screen show how many electrons you measure at each spot):

On the CRT screen, you’ll see a fuzzy dot, brighter in the middle and fading towards the edges. This is easy to understand. Think of the electrons coming out of the gun and going through the hole as though they were a stream of bullets shooting out of a machine gun. The further the stream of bullets gets from the gun, the more it spreads out. The stream of bullets spreads because some bullets move slightly faster and slightly off-course from the center. The electron beam spreads for the same reason. Some shoot a little high, some shoot a little low, but most shoot right towards the center.

Now, repeat the experiment with only the bottom hole open, and you get this:

Again, we have the same pattern – a fuzzy dot on the screen – but this time it is centered down at the level of the bottom hole.

So, we have a lot of experiments that show that the electron is a particle, and we have two experiments that show what these particles do when they go through a hole in a partition. So … what happens if we open both holes?

IF these electrons act like the particles we are used to, the CRT screen pattern should be the simple addition of the two one-hole patterns, like this:

If we shoot two machine guns at a wall, the wall gets more bullet holes where both guns hit. If the electrons act like that, the CRT screen would show a larger fuzzy dot, with a bright spot smeared between the center of the top and bottom hole.

**3. Sometimes the electron doesn’t act like a particle:**

But when we run the experiment, we don’t get the simple addition pattern! Instead, we get this crazy pattern:

This diagram is trying to show that there is a bright spot in the middle of the CRT screen, with two very dim spots right next to it, and then alternating bright and dim spots as you move further from the center of the CRT screen. As you go further from the center, the more distant bright spots are dimmer than the center spot, until the whole pattern dies out far from the center of the CRT.

What’s up with that? At some points on the screen, the patterns add up, and at other points they cancel each other out. Richard Feynman was awarded a Nobel Prize in physics, and he said that this one experiment defines the essence of the quantum mystery. I’ll go with his evaluation, since he’s the one with the Nobel Prize, not me.

**4. Sometimes the electron acts like a wave**

The experimental results above are perfectly natural in another part of physics – wave physics. For a moment, let the diagram above represent the surface of a swimming pool, with someone making waves on the far left. The surface waves go through the two holes in the partition, and then meet at the far end of the pool where we have the CRT screen. At some points on the screen, a wave crest from the top hole meets a wave crest from the bottom hole. The resulting wave is higher than either incoming wave: the wave heights add up, and that’s one of the bright spots. At other points on the screen, a wave crest from one hole meets a trough from the other. The wave heights cancel each other out, and that’s one of the dim spots. At a point where a trough from one meets a trough from the other, the wave depths add up: the trough at that point is deeper that either incoming trough. If the screen measures crests and troughs the same way (or if the screen somehow measures the SQUARE of the wave height, with depth being a negative height), the wave heights make a pattern on the pool wall just like the pattern we actually measure for electron density on the CRT screen.

Well, that right there is the quantum mystery: The electron, when it is measured at the screen, is a particle. The motion of the electron from the electron gun through the holes to the screen is the same as a wave. In QM, this wave is the “probability amplitude”, and the actual probability of the electron hitting the screen at any point is the probability amplitude squared.

OK, now we have an equation that explains the experimental result. It’s called the Schrödinger wave equation. It’s way too advanced for this little note. However, standard physics theory states that the electron can be found, probably, where the wave equation says it will be, and that’s all the theory states. But the theory works really well. For example, it predicts (probably) where the electron is located in the hydrogen atom, and how much energy it has when it leaves the hydrogen atom, and what energy light it emits and absorbs as it moves around in the hydrogen atom. The Dirac equation is an extension of the Schrödinger equation that accounts for special relativity (the electron can’t go faster than the speed of light). The Dirac equation explains all of the above plus something called the “spin” of the electron. So the math equations do explain the experimental results.

Finally I’m getting to your question. So what’s all this about “can’t find the position and velocity at the same time?”

**5. What is the location of the electron in QM?**

The location of the electron, in QM, is the square of the probability amplitude. That’s all the theory tells us. Of course, when the electron hits the screen, we know exactly (pretty close) where it is then.

Until we actually measure the electron, however, QM theory says the electron is located probably at the most likely spot, but maybe a little left or right or ahead or behind of the most likely spot. The QM location is a probability curve that looks like the bell-shaped curve. Here are a couple of examples (consider the electron is moving toward us, and the bell-shaped curve measures the probability that the electron is really a little left or right of the most likely position):

Diagam 5.1 – A wide bell-shaped curve.

Diagram 5.2 – A narrow bell-shaped curve:

The wide bell-shaped curve represents a particle where we expect a big dim fuzzy spot on the CRT screen. The electron has a pretty good chance of being far off-center. The narrow bell-shaped curve represents a particle where we expect a small bright fuzzy spot. The electron is more likely to be close to the center.

The QM equations always explain experimental results on the basis of probability. You can run any experiment with many electrons (or you can run it with only one electron at a time, but run it many times). In all cases, the count and distribution of electrons on a CRT screen will match the probability profile predicted by the QM equation.

**6. If the QM location is a probability distribution, what is the QM velocity?**

Instead of velocity, QM actually talks about momentum. (Momentum is simply mass times velocity.) The location and the momentum, in QM, are “conjugate variables”. In one sense, this means that location and momentum are enough to specify the motion of the particle. In another sense, this means that the location and momentum are tightly bound up in the equation of motion for the particle.

There’s nothing really new here. In traditional mechanics, for example, if you have the original location of a particle, and you know its initial velocity and its mass, you can calculate its trajectory. All the high-falutin’ words like “conjugate variables” just mean the same thing as that.

(Oh yes, to calculate the trajectory, you also have to know the force on the particle. You need to know the force for QM, too.)

Now, when the math tells us that the location is a bell-shaped curve, then it also says that the momentum is a bell-shaped curve. It turns out that these two curves are related by something called a Fourier Transform. Given the location curve, you calculate a Fourier Transform and you get the momentum curve.

Finally: The narrower one curve is, the wider the other curve is. That’s just how Fourier Transforms work.

**7. The location and velocity are both imprecise in QM.**

If we accept that QM explains everything about an electron, then the location at any time is not precise – it is a bell-shaped curve around the most likely position. Similarly, the momentum is also not precise – it is a (related) bell-shaped curve around the most likely momentum. And, since the curves are Fourier Transforms of each other, the narrower one is, the wider the other one is. This relationship between location and momentum is called the Heisenberg Uncertainty Principle. It states that you cannot know both the QM position and QM momentum exactly. In fact, it says more than that. It says you cannot know either variable exactly, and that the more precisely you know one, the less precisely you know the other.

Additional experiments have been made, and they all confirm the equations of Quantum Mechanics. How can this be, when we can actually measure the position of the electron on the CRT screen, and we know that the electron is particle with a very specific mass and electric charge? That’s the quantum mystery, straight and simple.

Physicists fall into (roughly) 3 groups: Group #1 have tried to develop actual physical models of how the electron moves, with an actual trajectory with location and momentum at all times specified. They’ve tied themselves in knots, and come up with some theories, but they are considered to be on the fringe of physics. Group #2 take a neutral stance and say that if the equations work, use them — and don’t ask the equations for an explanation of what’s actually happening. Group #3 take the stance that the electron doesn’t have a well-defined position or momentum, and you can’t even ask the question — the only thing you can ever know is the two probability curves.

Well, that’s it. How is it possible? No one really knows. But now, at least you know something about the experiments and the math that lie behind the statement:

“Quantum theory tells us that we have to abandon the familiar image of a particle (such as an electron or a photon) as having, at any time, a definite position or a definite velocity. […] This is not simply a matter of needing more precise measurements. The uncertainty is a fundamental aspect of physical reality.”

The writer of that quote is part of physicist group #3.

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