Quantum mechanics. It’s weird, complex, and highly esoteric. Physicists have used quantum theory to make incredibly successful predictions – calculating everything from the energy levels of the hydrogen atom to the anomalous magnetic moment of the electron, to incredible precision. Yet despite all this, the implications of quantum mechanics remain mysterious.

I think the main barrier to understanding what quantum mechanics has to say about reality is that there has never been a clear consensus on what quantum mechanics is fundamentally *about* – and very few physicists have bothered to really answer that question.

Again, this is in part due to the influence of the philosophy of *logical positivism*, which also so affected the debate about the nature of time. Whatever reality underlies physics is not something that can be empirically observed, so physicists have mostly concerned themselves with predicting measurement results, without really seeking to go further with the more philosophical questions.

So by something of a historical accident, the main concepts of quantum theory were developed without any clear specification of what they really meant, or how they related back to the reality that we observe. And the interpretations of the theory have multiplied every since.

### A Brief History of Quantum Mechanics

In the same year, 1905, that Albert Einstein produced his special theory of relativity, he also made a huge contribution to the other pillar of modern physics, quantum mechanics. His discussion of the photoelectric effect showed that the energy of electromagnetic radiation is absorbed and emitted in discrete amounts – *quanta* of energy – where the amount of energy in each discrete unit is proportional to the frequency of the radiation.

This suggested that a wave of electromagnetic radiation could somehow also behave as if was made of particles. This insight led another physicist, Louis de Broglie, to propose a radical hypothesis, the other half of wave-particle duality: that particles could somehow also behave as if they were waves. And indeed, it was discovered that they can. Electrons and other particles diffract and form interference patterns like waves, with wavelengths inversely proportional to their momentum. In the weirdness of quantum mechanics, this behaviour persists even at the level of individual particles.

The duality between particles and waves is what prompted Erwin Schrodinger to formulate what is probably the most fundamental equation of quantum mechanics – Schrodinger’s equation. This is where the modern form of quantum mechanics began, and the mathematics of the theory began to eclipse its physical interpretation.

In crafting his theory, Schrodinger looked for an equation that would have the correct *mathematical behaviour* before anything else – he wanted a wave equation that would produce the relationships between *energy and frequency* on one hand, and *wavelength and momentum* on the other, that were seen in the experimental indications of wave-particle duality.

And indeed, he did find such an equation, and used it to correctly predict the energy levels of the hydrogen atom. But the fundamental object in his equation was now a complex-valued mathematical entity called a ** wavefunction**, and what it represented physically was something of the mystery, even to Schrodinger.

(This, by the way, is where imaginary numbers made their first appearance in quantum physics. They show up in Schrodinger’s equation not because of any physical reality they represent, but because they are a convenient way to get the periodic wave behaviour that he wanted. Imaginary numbers entered into quantum mechanics a second time in a similar way, when Paul Dirac was trying to find mathematical objects that would obey certain relations for *his* equation.)

The wavefunction was retroactively interpreted by most physicists to represent a ** probability amplitude**, a complex quantity with a

*phase*, which allows it to produce wave-like interference effects, and a

*magnitude*, related to the probability that a quantum system may be found in a certain state. The fact that the wavefunction is viewed as the fundamental object of physics is one of the main things that makes quantum mechanics so difficult to understand.

### Superposition

There is a serious conceptual problem with considering the wavefunction as a *probability amplitude* and simultaneously considering it to be a *complete description* of the physical system. This is the ** measurement problem**, and it involves the quantum mechanical phenomenon of

**.**

*superposition*I apologise for the next few paragraphs. I could not figure out how to describe this phenomenon concisely without using a few arbitrary variables. (For a more in-depth explanation, with better examples, David Albert’s book *Quantum Mechanics and Experience* is a decent resource.)

A quantum system in a superposition of two states, lets say A and B, is such that, *when* you make a measurement that can determine whether it is in A or B, there is some probability of the system *being found* in state A and some probability of it *being found* in state B. (This is just an operational definition: I’m afraid it really tells you nothing about what a superposition is like.)

Let’s say that the probability of it being found in each state is 50%. The strange thing about quantum mechanics is that such a system does not behave as a system that has a 50% probability of *being* in A and a 50% probability of *being* in B. We would expect it to behave half the time as if it were in state A and half the time as if it were in state B, but in many cases it does not do that. In fact, a system in superposition of states A and B can exhibit behaviour that neither states A nor B themselves can.

(A well-known example of such extraordinary behaviour is the ** double-slit experiment**, where a particle ends up in a superposition of going through two slits in a screen, creating an interference pattern that cannot be created by particles known to be going through one of the slits.)

So a system in a superposition of states A and B is *not in state A*, and *not in state B*, because it shows behaviour that neither state individually can show. But it is always found in exactly one of A or B when it is measured, so in some sense it is *not in neither A nor B*, and it *not in both A and B*. A superposition is a very strange, seemingly contradictory state of affairs.

Now, a very important feature of quantum mechanics is that *the equation which governs the wavefunction* (that is, Schrodinger’s equation) *preserves superpositions*. This means: if the equation says that state A evolves into state C, and that state B evolves into state D, then the equation says that a superposition of A and B evolves into a superposition of C and D.

This feature leads directly to the measurement problem.

### The Measurement Problem

The measurement problem comes about when we ask quantum mechanics to describe what happens when we take a measurement to find the state of a quantum system. To do this, we have to treat the measurement apparatus as second quantum system, interacting with the first. (Which, if quantum mechanics is an accurate description of physical reality, is exactly what the measurement apparatus should be.)

Let’s consider the system I described above, in a superposition of states A and B (with 50% probability of being found in each). A measurement apparatus that can accurately determine whether this system is in A or B should have the following properties:

- When it interacts with a system in state A, it evolves to the state “displaying result A”.
- When it interacts with a system in state B, it evolves to the state “displaying result B”.

These properties, combined with the fact that the equations of quantum mechanics preserve superpositions, directly imply that:

- When the measurement apparatus interacts with a system in a superposition of states A and B, it evolves to be in a superposition of “displaying result A” and “displaying result B”.

* And this is in direct contradiction to the experimental results.* Because what

*actually*happens is this:

- When the measurement apparatus interacts with a system in a superposition of states A and B, it has a 50% chance of displaying result A, and a 50% chance of displaying result B. (For the system described above. In general, the chances depend on the superposition.)

These are two very different situations. If the measurement apparatus is in a superposition of displaying two different results, it is not displaying one or the other, or both, or neither. Instead, it is some strange mode of existence that we really do not understand. (The usual illustration of the role of superposition in the measurement problem is, of course, is the story of Schrodinger’s cat.)

It gets worse. According to the operational definition I gave earlier, if the measurement apparatus is in a superposition of two different states, that means that when the apparatus itself is “measured,” it has some probability of being found in either state. So how do we “measure” the measurement apparatus? Well, we simply look at the readout and see which result it displays.

But if we ourselves are physical systems, then according to quantum mechanics, what happens when we look at the measurement apparatus is that *we ourselves* enter into a superposition of “seeing result A” and “seeing result B”. This is even more blatantly in opposition to the evidence. We have direct introspective access to our own experiences, and no experimenter has ever reported experiencing a superposition of mental states. We know what we experience if we know anything at all, so either quantum mechanics is wrong, or we are wrong – possibly about everything.

I should note here that when a macroscopic system like a measurement apparatus enters into a superposition, what almost inevitably happens is a phenomenon called ** decoherence**. After decoherence, the different states of the superposition no longer interfere with each other and can evolve independently. So if a superposition of states A and B experiences decoherence, its subsequent evolution could be described completely in terms of the way that systems definitely in state A or B would evolve. This suppresses some of the strange behaviour characteristic of quantum systems.

But decoherence does not remove the measurement problem. Decoherence does not cause the measurement apparatus to definitely enter one state or the other: the superposition still exists, it just behaves more simply. The measurement apparatus is still in this strange mode of existence where it is not displaying result A, and not displaying result B, and not both, and not neither.

So, if the wavefunction is a complete description of a physical system, and the standard equations of quantum mechanics are correct and apply to the whole universe, then the “measurement” process that is supposed to resolve superpositions – and the resulting process of ** wavefunction collapse** – never actually occurs.

But in order for quantum mechanics to make predictions that match our experience, and for the interpretation of the wavefunction as a probability amplitude to make any sense, wavefunction collapse must occur. __The measurement problem reveals that the standard interpretation of quantum mechanics is either inconsistent, incorrect, or incomplete.__

We can also see the measurement problem as revealing a different conceptual problem with quantum mechanics: the standard interpretation of quantum theory introduces an unacceptably vague and ad-hoc division between *quantum systems* (for which the wavefunction obeys the Schrodinger equation) on one hand, and *classical systems* (or measurements, or observers, inducing collapse in the wavefunction) on the other. There is no non-arbitrary way to precisely define this division, so as long as it persists, quantum mechanics cannot be regarded as a complete theory of physics.

In my post after next, I will explore the various ways we can resolve the measurement problem and interpret the empirical evidence for quantum mechanics in a coherent way. But first, I want to explore some further aspects of quantum weirdness.