Physics edition: The wonders of quantum mechanics and why the probability of everything is 50%

Probability is a concept not really fully grasped by anyone in the world. Its is often a concept we first encounter in middle school maths and as long as we don’t get into it too much, everything seems fine. The classic example of flipping a coin: 50% chance of getting ‘heads’, 50% chance of getting ‘tails’ — two outcomes — simple, right? Well yes, in practice that is what happens most of the time until one really thinks about the nature of probability. There are two main classical approaches to probability: frequentism and epistemic probability.

Frequentism is the approach we use in middle school maths: calculate how many outcomes there are, divide 1 (100%) by the number of outcomes, and get a probability of a specific outcome. Easy, right? This works well for events where the probability is ‘spread evenly’ between each outcome such as flipping a fair coin. If we dive deeper into the nature of probability and more specifically, the nature of frequentism, we discover what we truly mean by: “The probability of getting ‘heads’ is 50%”. This probability value is representing what fraction of the outcomes of flipping a coin will be heads, if we flip the coin an infinite number of times. Because of this, nobody is surprised if after two coin flips, one gets ‘tails’ twice as we think we will get more ‘heads’ than ‘tails’ outcomes in the next few coin flips, each outcome amounting to about 50% of the overall outcomes in the long run. The phrase ‘in the long run’ is really a saving grace for something like flipping a coin, since, if someone gets 6 ‘tails’ outcomes in a row, despite their surprise, they assume they will get much more ‘heads’ outcomes than ‘tails’ outcomes in the next x number of coin flips. In (Everettian) quantum mechanics, which is slowly becoming the more and more convincing version of reality (over classical mechanics), the probability of obtaining 10 ‘heads’ outcomes in a row and the probability of obtaining 5 ‘heads’ outcomes and 5 ‘tails’ outcomes, spread differently over 10 coin flips, is exactly the same: 50%. I will come back to this idea later in the description of Everettian quantum mechanics and probability.

Epistemic probability is the other approach to probability which we use, far more than we think, in day to day life. This approach is far more useful for events where probability is ‘not evenly spread’ between different outcomes. Take, for example, a football match between Chelsea FC (one of the best teams in the world and the champions of Europe at the time of writing of this article) and FC Goa an Indian league team, which barely anyone has even heard of. In this case we would think it far more likely that Chelsea would win the match, most likely with an absolutely obscene scoreline. Now why is it that we don’t assume that the probability of Goa beating Chelsea is 50%, just as with the flip of a coin, there are only two outcomes, right? (There is of course a chance of a draw, however, imagine that in that case the match is settled by penalties to get a definite winner). This is because for events we cannot imagine repeating an infinite amount of times, such as a sports match (because of various factors such as weather, team injuries, player age, etc.), we use epistemic probability. In epistemic probability, one assigns each possible outcome a credence — a personal belief of likelihood of that outcome. This is usually based on either previous events, such as previous encounters between the two teams or on personal knowledge of other factors, such as how ‘good’ the players on either team are. Credences work exactly the same as assigning a probability to an event in frequentism, that is, they have to range from 0% to 100% and the total of the percentages assigned to different credences should add up to 100%. We often use credences to describe past events, such as: “I probably came home at around 19:00, last Friday as that is when I usually come back from work”. However, unlike in frequentism, your credences can change as you gather more/new information. For example, you thought that you likely came home at 19:00 last Friday, but then you remember, you went to see the new Spiderman movie straight after work. You now believe that you came home at around 22:30, assigning a large probability to this new credence and changing the probability assigned to the credence of returning home at 19:00 to 0%. Credences often help us make estimates of probabilities in every day life as such estimates are easy for the human brain to make and comprehend. However, Everettian quantum mechanics renders epistemic probability just as incorrect as frequentism as it states the probability of everything and anything is 50%.

Everettian quantum mechanics is an interpretation of quantum mechanics, often also called the many-worlds interpretation, which states that there is an infinite number of parallel universes due to a phenomenon called decoherence, which causes the branching of the wave function. However, before we can dive into this interpretation and explore the fascinating ideas it has to offer, we must gain, at least a basic understanding of the nature and significance of quantum mechanics itself.

Quantum mechanics begins with a very well-known subatomic particle: the electron. In school we are all taught the Bohr model of the atom, consisting of a nucleus made of protons and neutrons, and electron energy levels with electrons orbiting the nucleus in these energy level ‘orbits’, much like a planet orbiting a star. This is a model which works well for classical mechanics and satisfies all the requirements for a basic understanding of atomic physics. Nevertheless, there is another, more accurate model of the atom, called the Schrodinger model. In this model, the atom consist of a nucleus, still made of protons and neutrons, which is orbited by electrons with fixed energies. However, this time, instead of orbiting the atom in regular patterns (much like a planet orbits a star) the electrons ‘blend’ together into a probability cloud called the wave function. This wave function is the mathematical explanation behind electrons’ wave-like properties. In middle school physics, we are taught the electron is a particle and light is a wave, however, this is not entirely true. The electron can display both particle-like and wave-like properties, just as light can display both wave-like and particle-like properties (this is best seen in the double slit experiment). This is the underlying concept of quantum mechanics — a theory to show how a single thing can be both a particle and a wave and can be in two places at the same time. Yes, in two places at the same time. The concept of being in two or more places at the same time, in quantum mechanics, is called a superposition. This superposition is only valid until a measurement of the system is taken. This is well described by a probability cloud. Imagine again the atom, with a cloud of electrons around the nucleus. This cloud has darker and brighter spots, positions where one is more and less likely to find an electron. However, one doesn’t know where the electron is as it is in all these places at the same time, before a measurement is taken, after which, the electron ‘snaps’ into one of the possible places. This cloud of probability is defined by the wave function (which is governed by the Schrodinger equation), which can give one the probability of finding an electron in a certain position, and it is dependent on the total energy of the system. Now in a simpler version of quantum mechanics (textbook quantum mechanics), the wave function changes with time until a measurement is taken, after which the wave function collapses to a certain point, ‘snapping’ the electron into place and giving it particle-like qualities. This interpretation of quantum mechanics is limited as it only works on the subatomic and atomic levels. This is because, in textbook quantum mechanics, one can only talk about the wave function of a single system in a measurement taking scenario, rather than the wave function of the universe, as the wave function of the universe cannot collapse every time the position of a single particle is measured. Nevertheless, the textbook interpretation of quantum mechanics is useful for understanding the underlying principle of quantum mechanics.

Everettian quantum mechanics takes a very different approach, considering the wave function of the universe, rather than of a single system. This approach is far more applicable to larger objects and systems, even as large as the universe, assigning a single wave function to all the particles in the universe. If Everettian quantum mechanics considers the wave function of the universe, rather than a single system, then what happens when the position of a single particle is measured? The wave function can’t collapse, right? Indeed, the wave function does not collapse, but rather, as Everett proposed, it branches. Branches into a different universe. According to Everettian quantum mechanics, every time a quantum measurement is taken, the wave function of the universe branches, to create another universe for every single measurement outcome. For example, if an electron is in a superposition of position 1 and position 2 (with the wave function stating equal probabilities of finding the electron in either of the positions), textbook quantum mechanics states that with a measurement taken, the wave function of the electron will collapse to create one of the possible two measurement outcomes, with a 50% chance of collapse to position 1 and a 50% chance of collapse to position 2. Whereas, Everettian quantum mechanics states that for the same scenario, when a measurement is taken, there is a 100% probability of the wave function of the universe branching into two different universes, one universe where the electron is in position 1, and another universe where the electron is in position 2. At this point, you are likely asking: what do you mean by taking a measurement? In quantum mechanics, a measurement is any interaction a particle has which has the potential to carry information. For example, consider an electron orbiting a nucleus, if one decides to measure the position of the electron using an electron detector, a photon has to interact with the electron and then with the detector to give the observer the position which the electron was detected in, this is considered a measurement. Likewise, if one has no intention of measuring the position of the electron, but the electron interacts with a photon or any other particle outside of the system, this is also considered a measurement, as the other particle has the potential to give information about the position of the electron. The key thing about any quantum measurement is that it causes decoherence, which causes the branching of the wave function into new universes. Decoherence and measurement are somewhat interchangeable phenomena in quantum mechanics, the idea is, decoherence is caused by a particle from outside the system in question, interacting with a particle in the system and then interacting with the environment. This interaction with the environment changes the total energy of the system to an unknown quantity (which humankind cannot even begin to calculate as one needs to factor in the energy of every single particle in the universe). Which then causes the wave function to branch into different universes, giving the particle within the system in question, a definite position, with a universe for every single possible position. Now imagine the scale on which this happens, particles interact with each other all the time. The whole reason we can even see things is because of the interaction of photons with electrons. Each of these interactions causes decoherence and hence the branching of the wave function.

If one considers the universe to be infinite, and hence having an infinite number of particles within it, one must then expect infinite interactions between particles and hence an infinite number of wave function branches. This then means that any possible event will occur in some universe. Since there is an infinite number of parallel universes, any event would have occurred in half of them and not occurred in the other half. Which means, the probability of any event occurring is exactly 50%.

References

Carroll, Sean. 2019. Something Deeply Hidden. N.p.: Penguin Random House LLC.

Tegmark, Max. 2014. Our Mathematical Universe. N.p.: Penguin Random House LLC.