The Bohr model, Landau quantization, and “truth” in science

A few years ago, at a big physics conference, I was party to an argument about whether we should be teaching the Bohr model of the atom in lower-level physics classes.  The argument in favor was that the Bohr model is easy to teach and gives a simple way to think about the structure of atoms.  The argument against was that the Bohr model is completely outdated, conceptually inaccurate, and has long been superseded by a more correct theory.  The major statement of the opposition argument was that it doesn’t do anyone much good to learn an idea that’s wrong.

How strongly I disagree with that statement!

I, personally, love the Bohr model.  It’s founded on a cartoonishly simple way of thinking about quantum mechanical effects, but it can give you a surprisingly solid way of thinking about quantum problems for very little effort.  In other words, even when the Bohr model doesn’t give you the exact right answer, it is very good at teaching you how to feel about a quantum system.

The purpose of this post is, more or less, to be a defense of the Bohr model.  After outlining what the Bohr model is, I’ll show how the exact same logic gives a very quick and surprisingly accurate sketch of another major phenomenon in the quantum world: Landau quantization.  Then, at the end, I’ll wax philosophical a bit about why it’s a mistake to try and teach only “true” ideas in science.

The Bohr model of the atom

The essence of the Bohr model approach is to start by thinking about the problem using only classical physics, and figure out what different states look like.  Then, once you’re done, remember that quantum mechanics only allows certain particular ones of those states.

This way of thinking developed very naturally, because at the time the Bohr model was developed (1913) there was no quantum mechanics.  So as people were puzzling about how to describe the hydrogen atom, they started with the eighteenth and nineteenth-century physics that they knew, and then tried to figure out how it might be modified by funny “new” stuff.

To see how this works, start by thinking about the Hydrogen atom using only high school-level physics.  You have a single electron running around a single proton, and the picture that emerges is that the electron should orbit around the proton in the same way that the earth orbits around the sun.  Like this:

You can work out everything about this orbit (by balancing the attractive force between the charges with the centripetal acceleration), and what you’ll find is that orbits with any radius r are possible.  Orbits with small r have a large momentum (high speed) and a deeply negative energy, while orbits with a large radius have a small momentum and small energy.

More specifically, the momentum is and the energy is .  [Here, is the electron charge, is the electron mass, and is the electric constant.]

Once you’ve figured out everything that would happen for classical electrons, you can remember that quantum mechanics only allows certain kinds of trajectories to be stable.  The key idea, which was developed only slowly and painfully during the first few decades of the 20th century, is that moving particles have a wavelength associated with them, called the “de Broglie wavelength” .  A larger momentum p implies a shorter wavelength: , where is Planck’s constant.  [My way of thinking about the wavelength is that fast-moving particles make short, choppy waves in the quantum field, while slow-moving particles make gentle, long-wavelength ripples.]  For a trajectory to be stable, the orbit of the electron needs to have an integer number of wavelengths.  Otherwise, the trajectory gets unsettled by ripples in the quantum field.  This stability is often demonstrated with pictures like this:

The orbit on the left is stable, while the one on the right is not.

My own personal image for the stability/instability of different quantum trajectories comes from all the time I spent playing in the bathtub as a little kid.  Like many kids, I imagine, I used to try and slide back and forth in the tub to get the water sloshing from side to side in big dramatic waves. Like this:

What I found is that making these big “tidal waves” requires you to rock back and forth with just the right frequency.  If you continue to rock with that frequency, then you get one big wave moving back and forth, and you can slide around the tub while staying inside the biggest part of the wave as it shifts from one side to the other.  But if you try to change your frequency, then suddenly you find yourself colliding with the tidal wave and water goes flying everywhere.  This is something like what happens with electrons in the Bohr model.  If they travel around their orbits at just the right speed, then they move together with the ripples in the quantum field.  But moving at other speeds leads to some kind of unstable mess, and not a stable atom.

…I wonder whether I can go back and explain to my parents that all that water on the floor was really just an important part of my training for quantum mechanics.

Anyway, applying the Bohr stability condition to the classical electron trajectories gives the result that the orbit radius can only have the following specific values:

,

where (and meters).  Correspondingly, the energy can only have the values

.

Now, the Bohr model is not a true representation of the inside of an atom.  The movement of electrons around a nucleus is not nearly so simple as the circular orbits I drew above. The Bohr model also doesn’t tell you anything about how many electrons you can fit on different orbits.  And you certainly couldn’t use the Bohr model to predict subtle effects like the Lamb shift.  But the Bohr model very quickly tells you some important things: how big the atom is, how deep the energy levels are, and how the energy levels are arranged.  And in this case, it happens to get those answers exactly right.

To a certain degree, Bohr was lucky that this line of very approximate reasoning got him the exact right answers.  But I think it is often underappreciated how useful the Bohr model is as a paradigm for approaching quantum problems.  To illustrate this point, let me use the same exact thinking on another problem that wasn’t figured out until decades after the Bohr model.

Landau levels

As it happens, there is another kind of problem where charges run around in circular orbits, which you are also likely to learn about in a first or second-year physics course.  This is the problem of a electrons in a magnetic field.  As you might remember, a magnetic field pushes on moving charges, bending their trajectories into circles.  Like this:

A beam of moving electrons (the purple streak) is pulled into a circular trajectory by a magnetic field.

A magnetic field makes a force that is always perpendicular to the velocity of a moving charge, causing the charges inside the field to run in closed circles.  In the classical world, those circles can have any size, but quantum mechanics should select only some of them to be stable.

So what kind of quantum states can electrons in a magnetic field have?

Following the Bohr model philosophy, we can approach this problem by first working everything out as if quantum mechanics did not exist.  The physics of charges in a magnetic field is a few hundred years old, and fairly simple.  You can use it to figure out that faster charges have bigger orbit radii, according to the relation , where is the strength of the magnetic field.  The kinetic energy of the electron is , which in terms of the radius means .  So, there is a whole range of classical trajectories with different radii.  Those trajectories with larger radius correspond to faster electron speed and larger energy.

Now we can examine this classical picture through the lens of Bohr’s stability criterion, which says that only trajectories with just the right radius can be stable.  In particular, only trajectories whose length is an integer multiple of the de Broglie wavelength can survive as stable orbits (remember the “sloshing in the bathtub” analogy).  Applying this condition gives:

,

where, again, .  If you put this result for into the kinetic energy, you get:

.

These discrete energies are called “Landau levels” (after the Soviet Union’s legendary alpha physicist).  And while the quantization of magnetic trajectories is not quite as widely known as the hydrogen atom, it has been the source of just as many strange scientific observations, and has kept many scientists (myself included) gainfully employed for more than half a century.

Here the Bohr model approach again provides a quick and easy guide to these energy levels.  First, one can see that different energy levels come with a uniform spacing in energy, , where is the “cyclotron frequency.”  Second, the radius of the corresponding trajectories grows with the square root of the energy.  Finally, the smallest possible cyclotron trajectory has a radius , which is called the “magnetic length.”

These are important (and correct) results which can lead you quite far in conceptual thinking about what magnetic field does to electronic states.  And while they were really only appreciated in the second half of the twentieth century (after quantum mechanics had come into full bloom), they could have been mostly derived as early as 1913 using Bohr’s way of thinking.

[As it happens, the formulas above are not exactly correct, as they were in the Bohr model.  The correct result for the energy is

.

So the “Bohr model” type approach gives the exact right answer for the lowest energy level, but is wrong about the spacing between levels by a factor of two.

UPDATE: Here‘s a simple addition you can make to get the answer exactly right.]

What is “truth,” really?

I have no qualifications as a philosopher of science or of anything else.  That much should be emphasized.  But nonetheless I can’t resist making a larger comment here about the idea that certain scientific ideas shouldn’t be taught because they are “not true.”

Science, as I see it, is not really a business of figuring out what’s true.  As a scientist, it is best to take the perspective that no scientific theory, model, or idea is really “true.”  A theory is just a collection of ideas that can stick in the human mind as a useful way of imagining the natural world.

Given enough time, every scientific theory will ultimately be replaced by a more correct one.  And often, the more correct theory feels entirely different philosophically from the one it replaces.  But the ultimate arbiter of what makes good science is not whether the idea an true, but only whether it is useful for predicting the outcome of some future event.  (It is, of course, that predictive power that allows us to build things, fix things, discover things, and generally improve the quality of human life.)

It is undeniable at this point that the Bohr model is decidedly not true.  But, as I hope I have shown, it is also undoubtedly very useful for scientific thinking.  And that alone justifies its presence in scientific curricula.