Coda¶
In this tutorial, we have presented a series of examples that aim to illustrate how we can understand the phenomena of measurements using the theory of quantum waves as described by the Schrödinger equation. This differs from the standard “textbook” understanding, which relies on an additional axiom to impose various phenomena by fiat.
In the Classical waves section, we first looked at some properties of waves in general, such as propagation, reflection, and interference. Later, in the Quantized waves section, we looked at confined waves that can only be composed of a discrete set of frequencies. In both these sections, we were discussing classical waves, such as waves on a stretched string. The point of this section was to review some important concepts used later on, and also to help tease out the similarities and differences between classical and quantum waves. We in particular wanted to take a look at classical Klein-Gordon waves. It is a little-appreciated fact that relativistic quantum waves are essentially Klein-Gordon waves, and non-relativistic quantum waves (those described by the Schrödinger equation) are just those Klein-Gordon-like waves in the non-relativistic limit. As such, some the features of quantum mechanics that are often cited as particularly strange (e.g. the uncertaintly principle, tunneling, or the form of the Schrödinger equation) are equally well exhibited by classical Klein-Gordon waves.
In the Quantum waves section, we saw what is probably the most significant difference between classical and quantum waves — that two particles cannot in general be described by two separate quantum waves. Instead, the quantum wave describing two particles must specify an amplitude for each pair of positions of the two particles, as opposed to a separate amplitude for each particle’s position. We saw how the wave function of two particles moving in one dimension can be visualized using a two-dimensional plot. Specifically, we focused on wave functions in the form of localized wave packets, which appear as blobs that move around on the 2D plots. Much of the tutorial was based on visualizing such states in this way. We were easily able to see the presence of entanglement as non-aligned blobs.
Starting in the Double slit section, we utilized our visualization of two particle wave functions to understand the “which path” measurement in the double slit experiment. This is the first example of our general strategy to boil down the scenario to something as simple as possible, while retaining a physically intuitive picture:
Only look at one-dimensional situations.
Describe things in terms of wave functions, and ideally, Gaussian wave packets. The more typical approach is to abstract away the wave function in terms of qubits and generally abstract vectors in Hilbert space. This results in simpler and/or more powerful mathematics, but loses physical intuition.
Visualize the evolution of two- (or three-) particle wave functions by simulating the 1D Schrödinger equation and displaying 2D (or 3D) plots and videos. We don’t have to take math’s word for it — we can see the results with our own eyes.
As much as possible, do not make use of “external potentials” to act on quantum particles. An external potential of unspecified origin might sweep certain details under the rug, such as whether whatever is generating that potential reacts to, or becomes entangled with, the quantum particle. Only once we have shown that a potential can be plausibly generated by another quantum particle (e.g. a heavy repulsive particle acting as a mirror) do we allow the simplification of replacing that particle by an external potential.
Initially replace “measurements” with microscopic interactions between particles that we refer to as “micromeasurements.” Later in the Amplification and decoherence section, we discuss how to go from “micromeasurement” to “measurement.”
Finally, we can see to what extent the effect of the micromeasurement has captured the behavior usually ascribed to the quantum measurement axiom.
In the Double slit case, we can quickly see why interference disappears when information about the path of the particle is transferred to another particle. The two blobs simply do not overlap any more, even though the two packets of the original particle do still cross each other.
This leads to the idea of Complementary observables, described in the next section. In the double slit experiment (actually a single particle in an interferometer), the two paths are two states with well-defined position. When the particle is split up into both paths, the position is not well defined, but the phase difference between the two packets is well defined. We wish to explore how the measurement of complementary observables comes about in the case of micromeasurements based on the Schrödinger equation alone. Now we need to come up with a way to do a micromeasurement of the phase difference between two packets. We propose a method that relies on overlapping the two packets and measuring the position of the interference fringes using two states of an atom. We spent some time justifying how confined waves could plausibly interact with our wave packets in such a way as to generate the phase micromeasurement. Once we have done that, we then allow ourselves to forget about those details and use the “atom phase sensor” as a general tool to probe phase differences.
One of the things that most surprised me while developing these simulations was how complementary observables are correctly represented by the position and phase micromeasurements. The key thing is that the atom phase sensor not only records the phase difference between two wave packets, but it also acts as a beam splitter for an incoming wave packet. This is no accident — the interaction that allows for the micromeasurement must also have this back-action on the system being measured. The result is that when we send a single wave packet at the atom phase detector (and what is the phase difference of a single wave packet?!), it correctly treats the wave as a superposition of in-phase and out-of-phase packets, yielding a superposition of two micromeasurement outcomes correlated with in-phase and out-of-phase packets, respectively.
From here, it is no stretch to understand some of the most vexing aspects of quantum mechanics, such as the correlations between measurements of Bell states, and the delayed choice quantum eraser experiment. The visualizations get a bit more complicated, but they all boil down to keeping track of the motion of blobs in some number of dimensions, when they overlap or not, and which are correlated with which.
Now before we get too excited, we need to step back and remember that so far we are looking at “micromeasurements,” not macroscopic real-world measurements. We take a look at this issue in the section Amplification and decoherence. In the spirit of physical intuition, we propose a thought experiment that looks like a marble run, or Rube Goldberg device for taking the result of a micromeasurement, and amplifying it into a macroscopic result. This gives us the first real occasion to talk about multiple macroscopic outcomes occurring simultaneously, as if they were in different worlds.
Having broken the ice on the Many Worlds topic, we next dive in for a more in-depth discussion of these worlds in the section When Worlds Branch. Here, we broach the touchy subject of our own perception of the many worlds, and the idea that there may indeed by many different versions of you experiencing and doing different things.
And last of all, we discuss the idea of probabilities of perceiving different worlds. In standard quantum mechanics, this is simply postulated as a rule (Born’s rule.) Here we present (following Zurek, Wallace, and Carroll) an illustration of how Born’s rule can be derived from the Schrödinger equation alone. Some people still claim that Born’s rule cannot be understood in the Many Worlds picture. Indeed, it is hard to wrap one’s head around and I will admit that it was the most difficult section to write. This may not be the last word on the topic.
So where does this leave us? Must you now believe in the Many World’s interpretation of quantum mechanics? Do these “worlds” really exist? Like, actually exist? I have no idea.
What I can say is that I think we have seen that the Schrödinger equation alone, without the measurement axiom, is sufficient to explain the observed non-relativistic quantum phenomena, in at least some simple examples. This allows to say the following:
Many Worlds provides a useful framework for understanding how a many-particle entangled wave function evolves according to the Schrödinger equation.
If the measurement axiom is truly redundant, then it should not be included as an axiom of the theory. Instead, it should be thought of as an emergent property that provides a sort of shorthand for understanding measurements.
If we really do want to include collapse (say, because the Many Worlds idea makes us feel all funny and uncomfortable), it is always possible that the Schrödinger equation is wrong or incomplete. Models such as those proposed by Ghirardi, Rimini, and Weber or Penrose aim to amend the Schrödinger equation. These proposals actually make testable predictions that differ from the usual Schrödinger equation. But so far, no differences have been observed. Until an observation is made that conflicts with the standard Schrödinger equation, then it appears redundant to include collapse as a physical process.
The question of what exists is a separate question, and one that I don’t know how to answer. I am not sure what the question even means. At least, it can be useful to believe (if only tentatively) that certain things exist, so as to create a mental model of the way the world works. (A real philosopher could probably frame this more precisely.) Throughout this tutorial, we have often emphasized that the “many worlds” are a useful mental picture for understanding how a wave function evolves according to the Schrödinger equation. There could be other useful mental models consistent with the Schrödinger equation, which some may also find useful. For example, the Consistent Histories interpretation essentially describes one current reality in terms of many histories that led to it. Or the De Broglie-Bohm interpretation posits that the wave function itself is not what we observe, but instead acts as a guide (a “pilot wave”) that tells point-like physical particles how to move.
The question of whether any of these pictures really describe the fundamental nature of reality we leave as an exercise for the reader.