Preface

One of the primary aims of this tutorial is to demystify the so-called Many Worlds interpretation, or Everett interpretation, of quantum mechanics. We will certainly not offer any sort of formal proof, or decisive experimental evidence. Instead, we will attempt to demonstrate, through a series of simulated thought experiments, what the Everett interpretation is saying, why it is plausible, and how we can begin to wrap our heads around it.

“Many Worlds” is the best of names and it is the worst of names. It is a name catchy enough to grab anyone’s attention, and provides fertile ground for books, movies, and the imagination. But is is a name that spurs overly imaginative accounts that obscure what the interpretation actually says about our universe. Popular writers often try to sell books by mystifying, instead of demystifying.

Thus the goal here is to demystify. Don’t worry though, the real deal is still astonishing and even a bit unsettling.

On the other hand, overly simplified explanations also muddy the waters. Loose analogies can provide an easy entry to grasping concepts, but analogies can only go so far. Trying to extrapolate conclusions based only on an analogy is often doomed to fail.

In this tutorial, we will provide factual explanations based on a solid foundation. To keep things accessible, we try to reduce the foundation to the minimum necessary and avoid unnecessary jargon.

The level of these explanations is aimed at a technically-minded person, but not necessarily with a substantial physics background. Familiarity with looking at graphs is essential, but virtually no math is presented. (Understanding of the few equations shown is not critical.) A pre-existing familiarity with intro physics concepts like momentum, mass, energy, wavelength, and frequency would be helpful. Of course, readers who have taken, say, an upper-level undergraduate quantum mechanics course will gain a deeper insight, likely finding these arguments interesting and illuminating.

Though little to no math is presented here, the results are all obtained using straightforward, standard quantum mechanics.

For the experts

For those in the know, the only ingredients used here are the 1D Schrödinger equation and two-level systems (i.e. qubits). The dynamics of these quantum systems are simulated by discretizing the line into an array of positions, constructing a Hamiltonian, and numerically calculating the unitary time evolution operator from the matrix exponential of the Hamiltonian, with a time step of your choice. Any dynamics can then be obtained by applying the time evolution operator repeatedly to an initial state. It turns out that this method works great for a single particle, but starts to run into the limits of computing power for two particles. We don’t even try to do three particles. Fortunately, there are some workarounds. When looking at two particles, we often can use the trick of transforming the coordinates of the two particles to the center of mass and relative coordinates. It is then sometimes possible to separate the two-particle problem into two one-particle problems. Another simplification occurs if we can safely assume that at any given time, at most two particles are interacting. In this way, for example, we can simulate three particles where at first A and B interact while C moves independently, and later B and C interact while A moves independently (though possibly entangled with B). At some point, I plan to add a section on the technical details including simulation code to allow anyone to try it out for themselves. For now, the code is adapted from the method shown in this example Matlab code.

To be clear, the underlying ideas in this tutorial are not my own. The initial Many Worlds idea was put forth by Hugh Everett III, and expanded on by Bryce DeWitt and others. The picture of how classical reality emerges from the underlying quantum theory was pioneered by people like Dieter Zeh, Wojciech Zurek, David Deutsch, and Sean Carroll.

I like to think that my contribution to this area, presented here, is to illustrate these ideas from the perspective of a humble experimental physicist. The theories developed by the luminaries above are, quite appropriately, expressed in formal mathematical language. This is the language in which ideas can be precisely stated and conclusively proven. But as an experimentalist, I want to know, “How do these theoretical ideas translate into something I can do in my lab?” Or at least something I can picture in my mind as physically real. To this end, I have conceived of a series of physically-motivated examples, simulated their behavior, and visualized the results in what is (hopefully) an illuminating fashion.

As a preview of the type of argument we will see here, a bit of background about quantum mechanics. As it is normally formulated, the foundation of quantum theory has two parts. First, there is a description of how matter is described by waves, and how we can use certain equations to predict how those waves move and change over time. It’s all quite beautiful. Unfortunately, according to the standard theory, we can never see these waves. In order for us to learn anything about these waves we have to perform “measurements” on them. When we do so, the act of measurement instantly and irreversibly alters the wave. There is a complicated set of rules that have to be assumed as axioms of the theory to explain how we actually get information about a quantum wave.

There is a school of thought that suspects that the two parts of the quantum foundation (wave behavior and measurements) are actually one and the same. Perhaps what we call “measurement” can actually be described by the wave behavior alone. Then we could get rid of these additional assumptions that have to be added onto the wave theory. It is plausible that this could be the case and still not be well understood yet, because it turns out that all but the most simple quantum systems are very difficult to understand, and even to simulate on a computer. Over the last few decades, there has been progress in showing how this might come to pass, from an abstract theoretical perspective. Here, we try to distill these ideas into simple, yet physically plausible examples.

We begin by introducing some essential phenomena associated with waves (both regular classical waves and the quantum waves that describe particles, which we will see in many respects are quite similar.) Having laid this groundwork, we will then see a key way in which quantum waves differ from classical waves, leading to the all-important idea of quantum entanglement of multiple particles. We focus on quantum waves in the form of “wave packets” — waves that are spatially localized within a limited range and travel around in some ways similar to our conventional picture of a classical particle. We will see how to visualize wave packets for one or more particles using different plotting methods, and get a feel for how quantum particles move and interact.

We then obtain the first key result — an understanding of the famous double slit experiment. Most importantly, we will see why the interference effect goes away when one tries to learn which path the particle has taken.

We will then start to tackle some of the perplexing phenomena arising from quantum measurements. First, we take a look at the idea of complementary observables – where the measurement of one quantity precludes the deterministic measurement of a second quantity. This leads to a look at what happens when we measure complementary observables on each particle of an entangled pair (so-called Bell state measurements). Similarly, we can use this framework to understand the famous Quantum Eraser experiments.

Through these examples, we will see that the wave behavior alone is enough to reproduce the expected effects, without invoking any assumptions about measurements. The caveats are 1. that the “measurements” we are showing here are still microscopic interactions, not the macroscopic effect you would see with your eye; and 2. that the “measurements” do not yield a single result. Instead, they yield multiple results simultaneously. This is, to say the least, not what we are accustomed to. So we will have to think about some questions. Should we expect different rules for microscopic and macroscopic systems? This question will be discussed in the section on Amplification and decoherence. Is it possible that the standard description of quantum waves is wrong or incomplete? Or is our intuition wrong, and multiple outcomes really do occur even macroscopically, as if they were existing in many separate “worlds?” This brings us to the Many Worlds idea, described in the final sections When Worlds Branch and Born’s Rule.

Acknowledgements

Before diving in, let me acknowledge some of my colleagues in the CWRU department of physics that have helped me while I have been developing this tutorial. Harsh Mathur has been indispensable, both as a sounding board and in suggesting new ideas and methods. In particular, the trick of separating two-particle wave packet scattering problems into center-of-mass coordinates allowed a tremendous simplification of what I as trying to do by simulating two-particle wave functions directly. And I would like to thank Kurt Hinterbichler (as well as Harsh Mathur) for patiently helping me understand relativistic quantum mechanics, which gave rise to the sections on Klein-Gordon waves. Of course, any errors or misrepresentations here are my own fault.