Understanding Quantum Mechanics #7: Energy Levels

[This is a transcript of the video embedded below. Parts of the text will not make sense without the graphics in the video.]

Today I want to tell you what these plots show. Has anybody seen them before? Yes? Atomic energy levels, right! It’s one of the most important applications of quantum mechanics. And I mean important both historically and scientifically. Today’s topic also a good opportunity to answer a question one of you asked on a previous video “Why do some equations even actually need calculating, as the answer will always be the same?” That’s a really good question. I just love it, because it would never have occurred to me.

Okay, so we want to calculate what electrons do in an atom. Why is this interesting? Because what the electrons do determines the chemical properties of the elements. Basically, the behavior of the electrons explains the whole periodic table: Why do atoms come in particular groups, why do some make good magnets, why are some of them good conductors? The electrons tell you.

How do you find out what the electrons do? You use quantum mechanics. Quantum mechanics, as we discussed previously, works with wave-functions, usually denoted Psi. Here is Psi. And you calculate what the wave-function does with the Schrödinger equation. Here is the Schrödinger equation.

Now, the way I have written this equation here, it’s completely useless. We know what Psi is, that’s the thing we want to calculate, and we know how to take a time-derivative, but what is H? H is called the “Hamiltonian” and it contains the details about the system you want to describe. The Hamiltonian consists of two parts. The one part tells you what the particles do when you leave them alone and they don’t know anything of each other. So that would be in empty space, with no force acting on them, with no interaction. This is usually called the “kinetic” part of the Hamiltonian, or sometimes the “free” part. Then you have a second part that tells you how the particle, or particles if there are several, interact.
 
In the simplest case, this interaction term can be written as a potential, usually denoted V. And for an electron near an atomic nucleus, the potential is just the Coulomb potential. So that’s proportional to the charge of the nucleus, and falls with one over r, where r is the distance to the center of the nucleus. There is a constant in front of this term that I have called alpha, but just what it quantifies doesn’t matter for us today. And the kinetic term, for a slow-moving particle is just the square of the spatial derivatives, up to constants.

So, now we have a linear, partial differential equation that we need to solve. I don’t want to go through this calculation, because it’s not so relevant here just how to solve it, let me just say there is no magic involved. It’s pretty straight forward. But there some interesting things to learn from it.

The first interesting thing you find when you solve the Schrödinger equation for electrons in a Coulomb potential is that the solutions fall apart in two different classes. The one type of solution is a wave that can propagate through all of space. We call these the “unbound states”. And the other type of solution is a localized wave, stuck in the potential of the nucleus. It just sits there while oscillating. We call these the “bound states”. The bound states have a negative energy. That’s because you need to put energy in to rip these electrons off the atom.

The next interesting thing you find is that the bound states can be numbered, so you can count them. To count these states, one commonly uses, not one, but three numbers. These numbers are all integers and are usually called n, l, and m.

“n” starts at 1 and then increases, and is commonly called the “principal” quantum number. “l” labels the angular momentum. It starts at zero, but it has to be smaller than n.

So for n equal to one, you have only l equal to zero. For n equal to 2, l can be 0 or 1. For n equal to three, l can be zero, one or two, and so on.

The third number “m” tells you what the electron does in a magnetic field, which is why it’s called the magnetic quantum number. It takes on values from minus l to l. And these three numbers, n l m, together uniquely identify the state of the electron.

Let me then show you how the solutions to the Schrödinger equation look like in this case, because there are more interesting things to learn from it. The wave-functions give you a complex value for each location, and the absolute value tells you the probability of finding the electron. While the wave-function oscillates in time, the probability does not depend on time.

I have here plotted the probability as a function of the radius, so I have integrated over all angular directions. This is for different principal quantum numbers n, but with l and m equal to zero.

You can see that the wave-function has various maxima and minima, but with increasing n, the biggest maximum, so that’s the place you are most likely to find the electron, moves away from the center of the atom. That’s where the idea of electron “shells” comes from. It’s not wrong, but also somewhat misleading. As you can see here, the actual distribution is more complicated.

A super interesting property of these probability distributions is that they are perfectly well-behaved at r equals zero. That’s interesting because, if you remember, we used a Coulomb potential that goes as 1 over r. This potential actually diverges at r equal zero. Nevertheless, the wave-functions avoids this divergence. Some people have argued that actually something similar can avoid that a singularity forms in black holes. Please check the information below the video for a reference.

But these curves show only the radial direction, what about the angular direction? To show you how this looks like, I will plot the probability of finding the electron with a color code for slices through the sphere.

And I will start with showing you the slices for the cases of which you just saw the curves in the radial direction, that is, different n, but with the other numbers at zero.

The more red-white the color, the more likely you are to find the electron. I have kept the radius fix, so this is why the orbitals with small n only make a small blip when we scan through the middle. Here you see it again. Note how the location of the highest probability moves to a larger radius with increasing n.

Then let us look at a case where l is nonzero. This is for example for n=3, l=1 and m equals plus minus 1. As you can see, the distribution splits up in several areas of high probability and now has an orientation. Here is the same for n=4, l=2, m equals plus minus 2. It may appear as if this is no longer spherically symmetric. But actually if you combine all the quantum numbers, you get back spherical symmetry, as it has to be.

Another way to look at the electron probability distributions is to plot them in three dimensions. Personally I prefer the two-dimensional cuts because the color shading contains more information about the probability distribution. But since some people prefer the 3-dimensional plots, let me show you some examples. The surface you see here is the surface inside of which you will find the electron with a probability of 90%. Again you see that thinking of the electrons as sitting on “shells” doesn’t capture very well what is going on.

Now that you have an idea how we calculate atomic energy levels and what they look like, let me then get to the question: Why do we calculate the same things over and over again?

So, this particular calculation of the atomic energy levels was frontier research a century ago. Today students do it as an exercise. The calculations physicists now do in research in atomic physics are considerably more advanced than this example, because we have made a lot of simplifications here.

First, we have neglected that the electron has a spin, though this is fairly easy to integrate. More seriously, we have assumed that the nucleus is a point. It is not. The nucleus has a finite size and it is neither perfectly spherically symmetric, nor does it have a homogeneous charge distribution, which makes the potential much more complicated. Worse, nuclei themselves have energy levels and can wobble. Then the electrons on the outer levels actually interact with the electrons in the inner levels, which we have ignored. There are further corrections from quantum field theory, which we have also ignored. Yet another thing we have ignored is that electrons in the outer shells of large atoms get corrections from special relativity. Indeed, fun fact: without special relativity, gold would not look gold.

And then, for most applications it’s not energy levels of atoms that we want to know, but energy levels of molecules. This is a huge complication. The complication is not that we don’t know the equation. It’s still the Schrödinger equation. It’s also not that we don’t know how to solve it. The problem is, with the methods we currently use, doing these calculations for even moderately sized molecules, takes too long, even on supercomputers.

And that’s an important problem. Because the energy levels of molecules tell you whether a substance is solid or brittle, what its color is, how good it conducts electricity, how it reacts with other molecules, and so on. This is all information you want to have. Indeed, there’s a whole research area devoted to this question, which is called “quantum chemistry”. It also one of the calculations physicists hope to speed up with quantum computers.

So, why do we continue solving the same equation? Because we are improving how good the calculation is, we are developing new methods to solve it more accurately and faster, and we are applying it to new problems. Calculating the energy levels of electrons is not yesterday’s physics, it’s still cutting edge physics today.

If you really want to understand how quantum mechanics works, I recommend you check out Brilliant, who have been sponsoring this video. Brilliant is a website that offers a large variety of interactive courses in mathematics and science, including quantum mechanics, and it’s a great starting point to dig deeper into the topic. For more background on what I just talked about, have a look for example at their courses on quantum objects, differential equations, and linear algebra.  

To support this channel and learn more about Brilliant go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get twenty percent off the annual premium subscription.

Thanks for watching, see you next week.

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