HSSP Philosophy of Quantum Mechanics 08/07/11 Lecture Notes
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1 HSSP Philosophy of Quantum Mechanics 08/07/11 Lecture Notes Outline: 1. Homework 4 (discuss reading assignment) 2. The Measurement Problem 3. GRW theory Handouts: None Homework: Yes Vocabulary/Equations: You should be able to: Homework 4 In class, it sounded like HW4 was confusing. Here s what I was hoping you d get out of the reading: 0. I wanted to give you a pretty standard example of some writing in philosophy of physics. This paper wasn t about a theory or new idea, but it was a decent overview stating a problem and classifying possible solutions. Question 1. Maudlin demonstrates that the three postulates, which he calls 1A, 1B, and 1C, cannot all be true. Can you see why? Three incompatible claims: (Maudlin s wording; see also end of Lecture 4 notes) 1.A The wave-function of a system is complete, i.e. the wave-function specifies (directly or indirectly) all of the physical properties of a system. 1. B The wave-function always evolves in accord with a linear dynamical equation (e.g. the Schrödinger equation). 1. C Measurements of, e.g., the spin of an electron always (or at least usually) have determinate outcomes, i.e., at the end of the measurement the measuring device is either in a state which indicates spin up (and not spin down) or spin down (and not spin up). If 1.A and 1.B are correct, 1.C must be wrong: If the wavefunction evolves linearly, then its future state is composed of every part of its present state, acted on by the time
2 evolution equation. For example, a state 1/ 2 red? + 1/ 2 blue> becomes 1/ 2 red> f(t) + 1/ 2 blue> f(t) where f(t) is the equation describing the time evolution of the wavefunction. And, if the wavefunction is complete, that s all there is to the future state. Maudlin calls this kind of superposition future state S*. It is incompatible with the statement that measurements have determinate outcomes, because S*, as a postmeasurement future state, does not represent a definite outcome: you can see this intuitively. If I ask what measurement outcome does the state S* correspond to, you can t reply the electron is measured as red or the electron is measured as blue, or neither or both, as we ve previously seen. If it were possible to measure an electron to be in a superposition, what would that look like? What exit of the box would it come out of? Of course, in real experiments, measurements always seem to have definite outcomes. But the symmetry of the superposition state does not allow one outcome to be the real one while the electron stays in a superposition not without saying that the wavefunction is incomplete. (Hidden variables theories precisely say that since measurements have outcomes, and evolution is linear, the wavefunction can t be complete.) If 1.A and 1.C are correct, 1.B must be wrong: If measurements have outcomes and the wavefunction is complete, then how those definite single-outcome states come about can depend only on things already included in the wavefunction (such as the coefficients in front of states in a superposition, which are often taken to correspond to the square root of the probability that the state corresponds to the outcome of a measurement.) But if wavefunctions evolve in such a way that superpositions become eigenstates, that is not a linear evolution based on the wavefunction alone. It could be a linear evolution with hidden variables, or it could be a nonlinear evolution (like collapse), but not linear evolution with a complete wavefunction. If 1.B and 1.C are correct, 1.A must be wrong: As has been said before, if measurements have outcomes and the evolution of the wavefunction is linear, then there must be something (such as hidden variables) beyond the wavefunction that allows preference for one state over the others in a superposition; then the linear evolution can apply to that preferred state and not the others. Question 2: Summarize how Maudlin divides interpretations of quantum mechanics into 3 categories. Answer: He categorizes interpretations according to which of the three claims above gets rejected. So, there are interpretations which reject that the wavefunction is complete (hidden variables theories), interpretations which reject that the evolution of the wavefunction is linear (collapse theories) and interpretations which reject that measurements have outcomes (for example, the Many Worlds interpretation.) Question 3: How does Maudlin categorize the following theories? -Bohm s theory: rejects 1.A, that the wavefunction is complete -GRW (Ghirardi, Rimini, and Weber): rejects 1.B, that the evolution is linear -Many-Worlds (aka Everettian Interpretation): rejects 1.C, that measurements have
3 outcomes -the Orthodox Interpretation: rejects 1.B, that the evolution is linear (because it adds on an explicit collapse postulate saying that in certain circumstances the evolution is not linear) 4. Briefly, why does Maudlin say the measurement problem is not only philosophical, but also physical? Maudlin says: Each of these options [the rejection of one of the claims above] carries with it an obligation, the discharging of which demands the postulation of new physics. The measurement problem is sometimes portrayed as merely philosophical, or of no interest to physics proper. This is quite untrue. What this means is that theories that offer solutions to the measurement problem do it by clarifying what evolution laws to use when. It s not a debate about words, the meaning of the word measurement. It s about physics: what mathematical rules describe the universe under certain circumstances. The Measurement Problem Summary: The Measurement Problem is the problem of how to resolve the observation that measurements have definite outcomes with the ideas that the wavefunction is a complete description of a quantum mechanical system, and that it evolves linearly. Other ways to frame the problem include: -Which of the three attractive but incompatible claims above should we reject? -The Orthodox interpretation postulates that wavefunctions evolve linearly except when undergoing a measurement. What circumstances count as a measurement? The Orthodox interpretation fails to answer this. What is really wrong with the Orthodox Interpretation? I ve said that the measurement problem is a quite serious objection to the Orthodox interpretation of quantum mechanics. But isn t it true that the Orthodox interpretation agrees with all experiments so far? Yes, it does, but so do all the other interpretations we discuss in this class. They all fit experiment. So how should we choose among them? We talked a bit about what makes a good scientific theory, and I think this is an important point. A good scientific theory (partial list) -fits with the available experimental data -proposes a minimum of extra stuff (Occam s Razor, keep your theory simple and free of invisible things that have no effects) -is explanatory: it explains what we see, and makes experimental predictions -is precise/complete: it explains exactly what happens in all cases -ideally, its experimental predictions are testable; the theory is disprovable. In class, I talked about how the Orthodox interpretation is successful about being true, but not about being precise. It succeeds in making the right predictions because its predictions are so vague.
4 The next three lectures will cover a theory from each of Maudlin s three categories. Today is GRW, which rejects 1.B, that the evolution is linear. GRW The GRW theory (named for its founders, Ghirardi, Rimini, and Weber) is a collapse theory. It rejects the premise that quantum mechanical dynamics are linear (follow the Schrödinger equation). GRW proposes that collapse is not a separate postulate of quantum mechanics, but describes the dynamics when a particle interacts with its environment. Observations motivating GRW: Relatively large things have relatively definite positions. This is what we actually observe. Meaning that Outcomes are always recorded in macroscopic objects. No one ever sees collapse of one particle in isolation. (How could you watch a single particle? Wait a very long time without watching. Then look. If it had collapsed in that time, you d have one prediction about its state. If it hadn t, you d have another.) So, GRW proposes that 1. There is a fixed probability per unit time that a particle will spontaneously, indeterministically collapse. 2. Particles are loci of collapse; if two (or more) particles are in an entangles system and one of them collapses, the whole system collapses. This is not new, but I m restating it because it s important to GRW. GRW provides an estimate for the value of [probability of collapse] per unit time for a particle: on the order of 1/ 10^8 years. That is a very small probability! Unfortunately, this number would be very very hard to experimentally verify. You d have to watch a particle for 10^8 or 10^9 years, or watch a very very large number of particles for a little less time. So, according to GRW, the probability of a system collapsing is the probability of one particle collapsing (1/10^8 years) multiplied by the number of particles. Is this enough to explain that relatively large objects have relatively definite positions? Let s do an order-of-magnitude calculation: A mole of carbon is 12 grams, the weight of almost 5 pennies. Imagine holding a little lump of carbon in your hand that weighs as much as 5 pennies. This is a macroscopic object. A mole is 6 x 10^23 atoms. An atom of carbon has 18 particles, so you re holding on the order of 10^25 particles. So, the probability of collapse of the system is 10^-8 x 10^25 = 10^17 per year, or 10^9 per second. Every second, a billion
5 spontaneous collapses are going on in that lump. Basically, macroscopic systems are collapsing all the time. That s why they have definite properties. Is GRW probabilistic? Yes Is GRW indeterminate? Yes (isolated particles don t have a determinate state) Is GRW nonlocal? Yes. Collapse happens at the locus, but the whole system collapses at the same time, even if it s extended through space (send entangled particles off in opposite directions.) How does GRW solve the measurement problem? It provides an explanation of what collapse is, and when it happens. Collapse is no longer a mysterious thing, it is a new spontaneous law of the universe. How does GRW deal with the two-path experiment? The particle passing through does not collapse because it does not get entangled with the system. That s important--- entanglement does not happen just because 2 particles are close together, or even when they hit each other. It s a matter of the particles states being in superposition together. Is GRW reasonable? What do you think about it? -It doesn t introduce anything really counterintuitive like backwards causation or the efficiency loophole. If just introduces a new fundamental constant (the rate of spontaneous collapse). -If part of what you dislikes about the Orthodox interpretation was chanciness, indeterminism, nonlocality, GRW does not solve these problems. GRW collapse is fuzzier than orthodox collapse: (ran out of time, next time) 1 objection to GRW: Fails to guarantee definite positions? (next time)
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