Saturday, November 07, 2020

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 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.

You can join the chat about this video today (Saturday, Nov 7) at 6pm CET or tomorrow at the same time.


  1. typo. the link to does not work. it now points to

  2. Thanks,interesting to learn the complexity of the calculations for even fairly simple atomic & molecular models.

  3. The solution to the hydrogen atom with the Schrodinger equation requires a separation of variables. The wave function ψ(r, θ, φ) is separated into a product

    ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ)

    The Hamiltonian in the Schrödinger equation is H(r) + H(θ) + H(φ), and because the Hamiltonian has eigenvalues with the wave function these terms act separately on the parts in the product.

    This is where things do become a bit complicated. The radial Hamiltonian includes the Coulomb potential between the electron and proton and a radial kinetic energy. The angular parts include the L^2/2mr^2 for angular momentum or equivalently kinetic energy for motion perpendicular to the radius. The terms in the wave product then separately obey what are called Sturm-Liouville equations that have complete sets of polynomial solutions. The first wave function R(r) is solved as Laguerre polynomial solutions. The Θ(θ) has Legendre functions or polynomial solutions and the final part is just Φ(φ) = e^{imφ), for m the projection of the angular momentum number along the z direction. The angular parts in a way break spherical symmetry by choosing an axis the angular momentum is measured and the Legendre functions are cylindrical polynomials.

    The radial part is the principal quantum number, and it corresponds to the Rydberg series of the old Bohr atom. If begins with n = 1 and predicts energy eigenvalues says r_n = (nħ)^2/ke^2m. This all gets plugged into the energy to give E_n = -(ke^2)^2m/2(nħ)^2. I should mention that k = 1/4πε_0. The angular part gives energy eigenvalues ℓ(ℓ + 1)ħ for ℓ having values 0 1, …, n-1. The radial part has energy eigenvalues mħ, for m the projection of the angular momentum onto z. Physically the has the odd interpretation of saying the electron for n = 1 has zero angular momentum and so in a classical idea it means the electron is just sitting still relative to the proton. People in the Bohm world sort of go into some fits over this, for their classical-like electron is not behaving as expected in classical physics. But, since there is no angular momentum there can be no dipole transition that emits a photon. This is the minimal s-shell referred to in chemistry.
    This is just the hydrogen atom. What about other atoms? This is where things become much more difficult. One must use perturbation methods. This issue goes all the way back to Newton who found it difficult to model the Earth-moon-sun system together in some analytic way. This 3-body problem is not solvable in general, and Poincaré proved this and won the Sweden Prize on solving the stability of the solar system. He showed such stability cannot be proven. This haunts quantum physics as well. The traditional method has been with the Hartree-Fock method, which in years gone by I have worked with. More modern methods with QED are also used, where Feynman diagrams are the perturbation terms. While atomic physics of this sort is regarded by many as rather otiose or “grandmotherly” it is a highly active area of work.

  4. continued due to length limits:

    In the web discussion yesterday, I indicated how the hydrogen atom had symmetries isomorphic to conformal gravitation. I wrote in Unicode a discussion of this that I have on file. I could post this, or parts of it, but it is a lot of Lie group theory and geometry/topology. I suspect few would get much, and Sabine may kill it anyway as “theory mongering.” The above paragraph though indicates how atoms in general are solved with perturbation methods as these are many body problems. My group theoretic work with analogues with conformal gravitation also works this. The AdS-CFT correspondence is a many-body theory. Also, a lot of many body systems do have exact solutions. There is a great book I found by Perelomov Integrable Systems of Classical Mechanics and Lie Algebras that is a great discussion on this. The connections to AdS/conformal gravitation QFT etc is rich.

    This issue with the gold atom is interesting. This involves the 5d-6s transition in the atom with the absorption of blue light. Silver is above gold on the period table and in the same series. The analogue of the 5d-6s in gold is the 4d-5s transition in silver. However, the 4d-5s distance in silver is much greater than the 5d-6s distance in gold. With gold the electron is influenced more strongly by the nuclear Coulomb charge and in a classical idea it has a semi-relativistic motion. This means this transition absorbs higher wavelengths of light. For blue light with a wavelength ~ 350nm the energy E = ħω = 2πħc/λ. This is around 10^{-19}j or an electron volt. This corresponds to kinetic energy of the electron of mass m = .51MeV as K = mc^2(γ – 1) for γ = 1/√(1 – (v/c)^2) which is about mv^2/2 using binomial theorem. This means a velocity of ≈ .15c, and this relativistic correction has some physics. This is a part of how atomic theory has structure analogous to conformal gravity.

  5. This latest lecture on quantum mechanics makes it plain how limited that this quantum mechanical based calculation tool is in describing the full range of electron behavior. There may be other more effective ways in understanding what electrons can do in a multibody situation.

    For instants, there is a theory of electron behavior that is closely tied to the theory of general relativity. This involves the close connection between superconductivity and black holes.

    Sean Hartnoll two part lecture is an introduction to the possibility of this connection as shown here:

    Part 1

    Part 2

    I believe that this dualism between superconductivity and black holes is the way to explain how electroweak unification can arise.

    The first step on this path is centered on Hole superconductivity. Professor Jorge E. Hirsch, is an Argentine American professor of physics at the University of California, San Diego. He writes as follows:

    Towards an understanding of hole superconductivity

    This is a process where holes join together to form cooper pairs. In simple terms, this many body phenomena initiates a multi-particle cluster formation process where a positively charged core of hole pairs is surrounded by a cloud of negatively charged electrons.

    The types of materials that can form this superconducting seed are wide ranging and include both elements and chemical compounds. Hydrides, carbon, copper, lithium, hydrogen, noble gases, and water are among the materials that support the formation of this clustered superconducting material seed.

    This material forms a nanoparticle sized cluster that is comprised of a positively charged core that in turn is comprised of a lattice of positively charged cooper hole pairs surrounded by a core of very low lying electrons that form a non-orbiting stationary electron spin wave locked in place by the Meissner effect. The electron spin wave is kept in place by a balance between magnetically based electron/hole spin attraction and the magnetic repulsion of the Meissner effect

  6. continued due to length limits:

    In the next step, any light that irradiates this spin wave is captured within the spin wave layer for a period of time that is long enough for the electrons that have been captured in the spin layer to disassociate into polaritons through the entanglement of light and matter.

    A wide range of EMF photons can serve to decompose the electron into its component parts. Heat, RF, visible light, UV light, X-rays and gamma rays can all serve to take the electron to a more fundamental polariton state. In the next step, the spin wave acts as an EMF mixing cavity where the light is constrained, slowed and essentially captured within the spin wave layer for a period of time sufficient for the electrons that have been captured in the spin layer to disassociate into polaritons through the entanglement of light and matter.

    This strong light and matter integration effectively remove charge from the electrons in the spin wave and an EMF based event horizon is formed. What results in an ever expanding polariton Bose Einstein condensate that is dual to a black hole.

    Consistent with the hole theory of superconductivity, to balance the charge of the positive proton core, more free electrons are attracted to the positively charged nanoparticle cluster and subsequently converted to more polaritons. These additional infilling electrons are now converted to more polaritons in a chain reaction that continues only constrained by the availability of free electrons without limits until a magnetically based electroweak singularity is eventually reached.

    Since the nanoparticle is a superconductor and also a superfluid, no energy is lost by the superconducting cluster through dispersion. As the cluster accumulates more spin, a supersolid lattice condensate of polaritons forms as verified by observation. When the amount of available free electrons is large, this spin accretion process continues into the approximately 100 micro sized ranges until a tipping point is reached where instabilities in the polariton lattice occur and the cluster explodes in a Bosenova. This polariton condensate based Bosenova explosion has also been seen and verified in experiment.

    1. I watched Hartnoll’s lectures last summer. It fell a bit short of what I was hoping for. As I recall this involved how Hawking radiation can generate charged particles outside a black hole in a way that cancels out the electric field. Remember, superconductivity breaks the symmetry of the U(1) QED field. This means we can have j = σE with divergent conductivity σ or as its reciprocal zero resistance so that E = 0, but j is finite. Then a condition can occur where there is a condensate of Hawking charged particles that cancels out the charge of the BH. At least this is how I recall this phenomenology. It is a sort of superconductivity.

      The onset of superconductivity is a situation where the conduction electrons associated with an ion in a lattice, here being weakly interacting, are bunched. The electric field will of course tend to cause them to be apart, but at low temperature the configuration has lower energy when they pair up. These are the Cooper pairs. They oppose electric and magnetic fields so the QED field symmetry is violated. This electric field repulsion is 2πħc/2e and corresponds to the quantum flux of magnetic field 2πħ/2e ~ 2.07×10^{−15} Wb. That mysterious 2e in the denominator is a feature of this Cooper pairing.

      This is a many body problem, and there is every reason to think the electrons condense in a larger N-body system. There is a lot of structure here, which is similar to the symmetries of entanglements.

    2. Regarding: “The onset of superconductivity is a situation where the conduction electrons associated with an ion in a lattice, here being weakly interacting, are bunched. The electric field will of course tend to cause them to be apart, but at low temperature the configuration has lower energy when they pair up. These are the Cooper pairs.”

      If I may, some of the environmental assumptions implicit in your last reply may not be applicable to these hole based clusters. There is another aspect of the behavior of electrons when they interact in these superconducting pico and micro clustered configurations that might well strain credulity. Taking metallic hydrogen as a prototype of this more general Hole case, the electron shell that surrounds the positive hole core protects that polariton condensate and positive hole core from all temperature and pressure conditions no matter how severe through quantum degeneracy pressure. The electron shell encapsulates and protects both the core and the condensate. For example, in the core of Jupiter, metallic hydrogen survives the multi magabar pressures and extreme temperatures found there.

      This implies that these hole superconductors might well be expected to survive in conditions well above ambient. These clusters are unlikely to fall apart even under the most extreme temperature and/or pressure conditions.

      As an example of this case, John LeClair found that water based hole clusters survived and remained intact under laser induced cavitation erosion of all hard substances up to and including diamond.

  7. Even after more than 100 years of quantum mechanical description of the hydrogen atom, the phenomenological aspects remain unexplained. There is an abundance of trend-setting misinterpretations which are surprising from an epistemological point of view, since the qualitative-analytical understanding of the omissions is very transparent.
    Interesting is the fact that the Arnold Sommerfeld expansion of the Bohr atomic model correctly describes the fine structure of the hydrogen atom without the spin postulate. Sommerfeld introduced relativistic, Kepler elliptical orbits for the electron, which circled radially symmetrically in the Bohr model, referred to spherical coordinates and quantized them independently of one another.
    What (also) everyone could know but many are reluctant to admit ...
    Phenomenologically speaking…
    The calculation of ground state energies is neither quantum mechanically nor quantum electrodynamically justified. Since a significantly decisive part is determined by the ratio of the interacting masses. There is neither QM nor QED based the possibility of introducing the reduced mass mred = mA / (1 + mA / mB) quantum field phenomenologically. The reduced mass is - whether you want it to be true or not - historically derived from "Newtonian celestial mechanics" within the framework of standard physics. In plain language this means that in the sense of atomic interactions, these are neither based on QM nor QED. By the way: Equating an electrical centripetal force with a mass-dependent centrifugal force is not a mathematical problem, but phenomenologically unfounded in the context of physics. Regardless of the masses of the charge carriers, two charges of the same amount do not “suffer” a mass shift, since the gravitational interaction (proton-electron for example) is ~ 40 powers of ten smaller. The suggestive model used is obviously phenomenologically unfounded. The question of how does a mass interact with a charge remains unanswered.

  8. How does the Zitterbewegung interpretation of quantum mechanics fit in with these quantum energy levels. Is the Zitterbewegung concept still valid in the light of the Higgs field role in producing mass? Is Penrose right in saying that Zitterbewegung is synonymous with the Higgs field mass generation mechanism?


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