Physics Friday #12: Interpreting Quantum Mechanics
Preamble: "God does not play dice"
Education Level: High School (Y9/10)
Topic: Quantum Mechanics (Physics)
Developing the Schrödinger Equation
Quantum mechanics had it's origin in the nature of light, and then over the course of 50 years from 1900 - 1950, the entire field of physics was overturned as we realised that waves weren't just limited to light, but everything.
This came to it's head in 1926 with the creation of the Schrödinger wave equation, which dictated how particle 'waves' evolve in time and space.
Now you've probably heard of the wavefunction. Effectively, it's a probability wave, where the amplitude of the wave corresponds to the most likely location you'd find the particle.
The wavefunction doesn't just involve a probability distribution in space, but also in other quantities.
For example, if you put an electron in a small box, you can imbue it with an energy.
But because of quantum mechanics, energy is quantised - there is an energy of X joules, 2X joules, 3X joules, etc. If you put an electron of 2.5X into the system it can't just work like that.
Which is why the electron forms a weighted combination of different states corresponding to specific multiples of our X value. And this superposition just so happens to be the average energy, which is considered the classical energy of the electron.
For example an electron with a superposition of 2X, 50% of the time, and being in state 3X 50% of the time. This averages out to 2.5X.
Collapsing the Wavefunction
But how do you find a particle? Or measure it's energy? Well, via what's known as the wavefunction collapse. When we take a look at the particle as a wave, it suddenly snaps to a specific value and then evolves from there.
This wavefunction collapse can occur for any observable property of the particle. If you measure the energy of our 2.5X particle in our above state, it's a 50/50 chance that you'll catch it in either state.
And once the coin flip occurs, the particle's energy will suddenly jump to 3X or 2X and remain at that value.
You may think this violates the conservation of energy, but remember that the act of 'measurement' intrinsically involves interacting with the electron - a very important point.
But wait, what does this collapse mean?
The Schrödinger equation does not explicitly mention this collapse. It simply describes the evolution of an undisturbed wavefunction. Thus, we need to include collapse as a part of the three postulates of QM:
Particle states are described by a wavefunction, a vector belonging to a Hilbert space
The Schrödinger equation dictates the time evolution of these states
Measurement of an observable (i.e. a hermitian operator) collapses the wavefunction to an observable's eigenstate (each eigenstate being associated with a probability of collapse)
But this still doesn't really answer the question. What is measurement? What counts as measuring an observable property of the particle?
Well here's the thing ... we don't have an answer ... it's an open question and the topic of this post.
An interpretation of quantum mechanics is effectively a theory that aims to answer this question: where and how does this measurement occur?
After almost a century since the formulation of standard QM, we have a litany of many interpretations, most of which fall on a spectrum of when exactly it occurs.
On one end, we have ideas where the wavefunction never existed in the first place, or that the wavefunction naturally collapses.
On the other, we have ideas that the wavefunction collapses at a point very far in the process, or even that it never collapses at all.
I'll talk about 6 of these interpretations, although some of these theories of collapse are more categories of theories.
Think of the Ocean (Pilot Wave)
In the 1920s, de Broglie developed an interpretation of quantum mechanics that posited that subatomic particles do, in fact, physically exist.
The source of the wavefunction and the probabilistic nature of quantum mechanics is caused by the particles being guided by a series of "pilot waves" - which push and move the particles around and imbue them with the motion and energy we observe.
The randomness comes from the fact that the waves themselves depend on the positions of all particles. These guiding waves are dictated by a special guiding equation.
dear lord that's complicated
Image Credit: Wikipedia
This guiding equation, when applied to particles just so happen to result in our neat and clean Schrödinger equaiton.
So what happened to this theory?
The biggest problem with this theory is that it's non-local, meaning that the evolution of the guiding wave requires knowledge of all of the particles in the universe.
This of course, violates special relativity.
Another problem is that it lost the authors' support, or that the authors lost support. de Broglie rejected the theory in 1927 and David Bohm, the other author, was distanced from the other scientists for being outwardly socialist during the early red scares.
Pilot wave theory, in a sense, is so strict on the physicality of particles that it ends up sort-of wrapping around and becoming a many-worlds theory instead, to quote David Deustch:
Pilot-wave theories are parallel-universe theories in a state of chronic denial.
This arises from the problem of branching, a tacked-on attempt to reconcile the nature of the theory. That since the wavefunction was a physical thing, and the pilot wave and particles kept self-interacting, it sort of creates branching realities caused by distant communication with other particles.
Those silly numbers are hiding from us! (Hidden Variables)
The EPR (Einstein-Podolski-Rosen) paradox is another famous problem in QM, caused by entanglement.
Take two electrons and force them to collide with eachother, bounce off, and travel far into the distance. We know that after the interaction, these electrons propagate with free-particle wavefunctions. And we can fire them at eachother such that we don't know their momentums initially - i.e. they entangled.
Now wait for the electrons to travel very far away from eachother, and then measure one of the electrons momenta. In order to maintain conservation of energy, we instantly know what the momenta of the other electron is.
What we also know is that because of this measurement, and that the electron is entangled with the other, that we have just collapsed the other wavefunction instantaneously from a distance.
This is a problem, due to special relativity, we cannot transfer information faster than the speed of light. So clearly our QM is broken.
Hidden variable theories aim to solve the EPR paradox as well as just generally trying to interpret quantum mechanics. Effectively, there are a series of unobservable entities that dictate how wavefunctions collapse.
The wavefunction in the EPR paradox has a hidden variable stating the electrons' momenta so that we aren't violating causality, for example.
Fortunately, but unfortunately, this theory makes a testable prediction via Bell's theorem, which utilises entanglement to determine if these hidden variables work locally.
The experiments conducted show that only a non-local hidden variable theory is possible. One example of this just so happens to be our previous pilot-wave theory!
Observing isn't needed (Spontaneous Collapse)
We could be thinking of this wrong. Perhaps the wavefunction is real, and it is non-deterministic. But that at some point, it collapses on it's own.
There are several ways to do this, but at it's core, these are how the theories go:
There is an extra non-linear term in the Schrödinger equation, that is insignificant at the small scales
This non-linearity causes the wavefunction to be unstable, and prefers it to collapse to observable eigenvalues
With increasing complexity, this term becomes much more important, as more entanglement = more instability
The rate of decay increases as you entangle the system. And if a system is large enough, it's likely to collapse into a classical environment
Effectively, they say that the wavefunction will collapse on their own. And the reason we don't see it on larger scales, or see a collapse when measuring the system, is that the act of interaction (entanglement) causes the wavefunction to be more likely to collapse.
Of course, the theory has trouble reconciling with relativity. As entanglement works over large distances. Models can be made to try and say that entanglement over these distances increases instability for example, but we're still waiting on developments.
Lastly, we have the problem of tails. The wavefunction of a particle exists for all of physical space. At these far out distances, it is very possible for particles to get entangled with distant objects. Meaning that a wavefunction may end up collapsing further than we think.
The easy way out (Copenhagen)
The Copenhagen interpretation was developed in the 20s to attempt to come up with some placeholder answer to what collapse is. It is our middle-of-the-road theory which states that observation of an observable causes collapse.
Observation is defined as the act of applying an observable operator (like the energy operator) to the wavefunction by an external source to gain information on that operator's outcome.
The problem is that this is a meaningless statement. Because anytime a system entangles itself with something greater, it technically does this 'observation'.
Take the double split experiment.
Image Credit: Discovery Channel
What defines the moment of observation? Is it when:
The particles interact with the measuring laser
The measuring laser interacts with the larger observation device
An electronic signal is sent from the device to a computer
The light from the computer interacts with the conscious observer
We can't pinpoint the specific cut-off between the quantum world and the classical.
After all, we know that lasers can entangle themselves with atoms. And that electronic signals are nothing but moving electrons.
The point of the theory is that it's a placeholder. The definitions are ill-defined because we're kinda waiting for another theory to help us.
It's all in your head! (Consciousness)
The immediate answer to the Copenhagen interpretation could be that the collapse occurs at the end of the specified chain. When a conscious observer interacts with the entangled system.
It's a nice idea given that it kills two birds with one stone - it helps point to a physical theory on the nature of sentience, but also allows us to solve the measurement problem.
This does come into conflict with our current understanding of sentience. Our placeholder theorem is effectively that conscious experience is an emergent property of a series of interacting electrical signals in our brain.
This placeholder helps explain why humans are more 'sentient' than animals, or very young children, as we have a very active and complex central nervous system.
Of course, it's just a placeholder. We don't have an actual meaningful answer to sentience, and probably won't for a while. So for now it's left to the dark realm of god of the gaps.
Where it comes into conflict with QM is that a series of interacting electrical signals sounds exactly like an entangled system. So there clearly can't be just emergent properties involved otherwise we're just dealing with a spontaneous collapse theory.
There has to be something physically unique about a sentient brain to cause the collapse - effectively you require the existence of a soul. Something which is even further in the dark realms of philosophy.
Another issue is that it doesn't work with special relativity, as it violates the EPR paradox still.
We also need to determine what counts as sentient. Sentience isn't an on and off switch. There are many ways it can be expressed.
We know that some mammals have some form of conscious experience - so then are cats capable of collapsing the wavefunction?
Finally, what about the universe prior to consciousness? Did it just end up in an entangled nightmare until somehow we got an observer to collapse it all? How can something built of entangled particles end up collapsing itself at some given size?
This interpretation is very interesting, however if it turns out to be true, we'll be stuck with our measurement problem for quite a while.
For now, the biggest problem with the interpretation is that it opens the door to many, many quacks like Deepak Chopra. Who think that we can control this collapse with our minds and alter our reality by just thinking it away WoOOoOoWwowoWoOo!
Forever entangled (Many Worlds)
So, assuming that our consciousness theory is not the right answer, then what causes the collapse?
We can keep getting bigger and bigger:
The electrons in the double slit entangles with the laser photons entangles with the measurement device entangles with the electrical signals entangles with the computer entangles with the observer entangles with the room their in entangles with the Earth entangles with the solar system entangles with the galaxy ...
This out-spiralling entanglement continues without bound until the entire universe is in a superposition of states. And every time an interaction occurs we ourselves are being pulled into a new wavefunction.
This entanglement would've happened early, at about the time of cosmic inflation. But every new quantum event comes with a new set of entanglements.
This leads to the name Many Worlds, as we're creating new realities with every event.
Now it's important to note something important: this is not a multiverse theory. Multiverse theory is proposed source for cosmic inflation. Here, there is still one single universe. Much like how an electron in superposition isn't multiple actual electrons. The universe is just being treated as an electron.
This theory sounds far-fetched. Arguably the fact that it's unfalsifiable makes it not a good interpretation of QM. However, it is a lot simpler than the previous consciousness interpretation - it simply removes the need for a measurement process.
This satisfies Occam's razor as well. It doesn't require a mathematical formalism because the point of the theory is that the formalism doesn't exist.
However, not having a formalism makes it quite difficult to prove. It only seems to be correct in the sense that it doesn't necessarily say that measurement cannot happen, just that it's not measurement. It's entanglement.
Interestingly, the theories on the "wavefunction collapses early" side of the spectrum are more likely to be disproven. Primarily a consequence of the fact that they have the opportunity of making testable predictions.
Despite all of these interpretations, it's clear who stands as the best theories: spontaneous collapse and many worlds. They have their strengths, but they have fair grounding. You could argue that consciousness is also a fair contender, but it's a bit too much in the realm of fantasy - attempting to tie one big unanswered question with another.
Spontaneous collapse has proper mathematical formalism while many worlds seems to work well in an Occam's razor sense.
Regardless, that is a surface-level exploration into the many different ways we have attempted to answer the measurement problem. I hope y'all enjoyed this post and god I need to make them less long.
Please can someone fix this inverted colours issue it's like causing all of my colours on these posts to invert too thx
Reference post: https://www.tumblr.com/oliviabutsmart/732200630726377472/for-some-reason-some-reasons-only-some-images-i
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