Author: Sean Carroll
Publisher: Dutton 2019
Genre: Science (Advances in Theoretical Physics)
Electron cloud of probability (pg 19) The cloud is called the wave function. The amplitude of a particle at x is psi(x). The probability of getting that outcome when we perform a measurement is |psi(x)|^2. This is known as the Born rule. (pg 20)
The wave function is the sum total of reality, and ideas such as position and velocity of electron are merely things we can measure. (Not everyone sees things this way.)
Rules of classical mechanics:
(1) Set up the system by fixing a specific position and velocity for each part.
(2) Evolve the system using Newton’s laws of motion.
Quantum systems are described by the wave function rather than by position and velocities.
Rules of quantum mechanics (part 1):
(1) Set up the system by fixing a specific wave function psi.
(2) Evolve the system using Schrodinger’s equation
Rules of quantum mechanics (part 2):
3. There are certain observable quantities we can measure, such as position, and when we do measure them, we obtain definite results.
(4) The probability of getting any one particular result can be calculated from the wave function. The wave function associates an amplitude with every possible measurement outcome; the probability for any outcome is the square of the amplitude.
(5) Upon measurement, the wave function collapses. However spread out it may have been nn , afterward it is concentrated on the result we obtained.
In modern university curriculum, when physics students are first exposed to quantum mechanics, they are taught some version of these five rules. The ideology associated with this presentation is sometimes called the Copenhagen interpretation. We can just refer to it as “standard textbook quantum mechanics”.
The idea that these rules represent how reality actually works is, needless to say, outrageous.
To put things more pointedly: Why do quantum systems evolve smoothly and deterministically according to the Schrodinger’s equation as long as we aren’t looking at them, but then dramatically collapse when we do look?
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Chapter 2: The courageous formulation
The attitude inculcated into young students by modern quantum mechanics textbooks has been compactly summarized by David Mermin as “Shut up and calculate”.
In statistical mechanics the probability distribution is an epistemic notation—describing the state of our knowledge— rather than an ontological one—describing some objective feature of reality. Epistemology is the study of the knowledge; ontology is the study of what is real.
The minimalist approach has two aspects.
First, we take the wave function seriously as a direct representation of reality. We treat it as ontological, not epistemic.
Second, the wave function always evolves smoothly in accordance with the Schrodinger equation.
In other words, let us erase all of those extra rules about measurement in quantum recipe entirely. This proposal is called “austere quantum mechanics” (AQM) for short.
But there is an immediate problem with it. It sure seems like wave function collapse. When we make measurements of a quantum system with a spread-out wave function, we get a specific answer.
We can use the phrase “quantum state” as a synonym for “wave function”, in direct parallel with calling a set of positions and velocities a “classical state”.
According to AQM, atoms are not empty space. No electron is zipping around. Atoms are described by wave functions that stretch throughout the extent of the atom. How do we get from there to the annoying reality that wave functions appear to collapse when we make measurements?
A quantum state describes systems as superpositions of different measurement outcomes. The electron will, in general start out in a superposition of different measurements. In order to do a measurement you have to use a tool. The tool starts out in some wave function that might look complicated. As soon as you use the tool, the tool to look at the electron, the tool itself is now in superposition of all possible measurement outcomes it might have observed.
However, now there is one wave function that describes the entire system of the electron and the tool. Such a superposition includes every possibility, however, most outcomes are assigned zero weight in the quantum state. This is the quantum phenomenon known as entanglement. The cloud of probability vanishes into nothingness for most possible combinations of electron location and the tool image.
Before the measurement happens, there is one electron and one observer (tool is replaced with an observer). After they interact, rather than thinking of that one observer having evolved into a superposition of possible states, we can think of them as having evolved into multiple possible observers. The right way to describe things after the measurement in this view, is not as one person with multiple ideas where the electron was seen, but as multiple worlds, each of which contains a single person with very definite idea about where the electron was seen. AQM is commonly known as Everett or Multi-Worlds quantum mechanics after Hugh Everett who proposed it in 1957.
Every version of quantum mechanics features two things. (1) a wave function, and (2) the Schrodinger equation, which governs how wave functions evolve in time. The Everrett formulation simply insists that there is nothing else. No wave function collapses.
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Chapter 3: Why would anybody think this?
What the world is made of (nineteenth century)
Particles and fields
What the world is made of (twentieth century and beyond)
A quantum wave function
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Chapter 4: What Cannot be known, because it does not exist
A spinning electron is a tiny magnet with south and north magnetic poles. One way to measure spin is to send the electron through a magnetic field, which will deflect the electron by a bit depending on how its spin is oriented. (As a technicality, the magnetic field has to be focused in the right way—spread out on one side, pinch tightly on the other—for this to work.) We expect the electron to deflect up if its spin axis was aligned with the external field, deflected down if its spin axis was aligned in the opposite direction, and deflected in some intermediate angle if its spin is somewhere in between.
However, that is not what happens.
The experiment was first performed in 1922 by Otto Stern and Walter Gerlach (hence this type of experiments are known as Stern-Gerlach experiments and the magnet is called the Stern-Gerlach magnet). Electrons either go up or down. Nothing in between. The measured spin is quantized.
For an electron, there are two possible outcomes for spin. Either electron is spinning CCW (spin up) or CW (spin down). But this is classical thinking. Suppose we place a second rotated Stern-Gerlach magnet to detect the spin up electrons only. Then they split into spin left and spin right electrons. That is, you can chose any axis you like by rotating your magnet appropriately, and the spin will be quantized wrt to that axis. (Uncertainty principle. page 83)
Systems with two possible measurement outcomes are so common and useful in quantum mechanics that they are given a cute name: qubits. The idea is that a classical bit has just two possible values, say 0 and 1. A qubit (quantum bit) is a system that has two possible measurement outcomes, say spin-up and spin-down along some specific axis. The state of a generic cubit is a superposition of both possibilities, each weighted by a complex number, the amplitude of each alternative.
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Chapter 5: Entangled up in Blue
What really bugged Einstein about quantum mechanics is its apparent non-locality. He codified his objections in EPR paper. This well formulated objection helped aluminate one of the most profound features of the quantum world: the phenomenon of entanglement.
Entanglement arises because there is only one wave function for the entire universe, not separate wave function for each piece of it. EPR believes in the principle of locality—the physical quantities describing nature are defined at specific points in spacetime, not spread out all over the space, and they interact directly only with other quantities nearby, not at a distance. Said in another way, given the speed of light restriction of special relativity, locality would seem to imply that nothing we can do to a particle at one location can instantaneously affect measurements we might perform on another very far away.
Bell’s inequality proves that quantum mechanics is non-local.
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Chapter 6: Splitting the universe
The simple process—macroscopic objects become entangled with the environment, which we cannot keep track of—is decoherence. Decoherence causes the wave function to split, or branch, into multiple worlds. Any observer branches into multiple copies along with the rest of the universe. After branching, each copy of the original observer finds themselves in a world with some particular measurement outcome. To them the wave function seems to have collapsed. The collapse is only apparent, due to the decoherence splitting the wave function.
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Chapter 7: Order and randomness
Deriving the Born rule.
In AQM, the question is “what is the chance I will end up being experimenter on the spin-up branch of the wave function?”. What does “probability” mean in this situation?
If we toss a fair coin infinitely many times, then we expect the total proportion of heads to come closer to 50 percent. So we can define the probability of getting heads as as the fraction of times we actually get heads, if the coin were tossed an infinitely many times. This notion of what we mean by probability is called “frequentism”, as it defines probability as the relative frequency of an occurence in a very large number of trials. Frequentism fits well with the TQM (Textbook Quantum Mechanics) and the Born rule. You could send a very large number of electrons through a magnetic field to measure their spin. (The SG experiment is a favorite one to reproduce in undergraduate lab course for Physics.)
AQM is a different story. At every measurement, the wave function branches into a world with spin-up and one with spin down result. If we record our results, labeling spin-up as “0” and spin-down as “1”. After fifty measurements, there will be a world where the record looks like:
1010101111101…
Suppose there are 24 0’s and 26 1’s. There will also be a world with fifty 1’s and a world with fifty 0’s. In AQM, there is a 100 percent probability that each possibility is realized in some particular world.
The above sequence is generated by a quantum number generator. According to AQM, when I generated that random number, the universe split into 2^50 copies (branches). What was I to think if I am in a world where I get all 1’s. I may think the random number generator is broken. It makes much sense to talk about the frequentism.
Fortunately an alternate approach of probability exists. It is called epistemic probability.
Consider the question “What is the probability that Chicago Bulls will win the 2020 championship? We can repeat this infinitely many times. The 2020 championship happens only once. What we can do is assign credences—degrees of belief—to the various propositions under consideration. Like any probability credences must range between 0 percent and 100 percent, and your total credences must add up to 100 percent. Your credence is something that can change as you gather more information. Statisticians have formalized this procedure under the label Bayesian inference. Bayes derived an equation showing how we should update our credences when we obtain new information. So there is a good notion of probability that applies even when something is only going to happen once. It is a subjective notion rather than an objective notion.
There is an answer that is tempting but wrong: that we don’t know “which world we will end up in”.
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