1. Basic Formalism of Quantum Mechanics
a. How are Quantum States represented? States of a physical system are represented by vectors in a vector space A vector is a quantity that has both magnitude (length) and direction. A vector can therefore
be visualized as an arrow.
b. How are observable properties (measurable physical properties)
represented in Quantum Mechanics?
Represented in the algorithm by linear operators on the vector spaces associated with those systems.
c. What are some examples of observables in Quantum Mechanics? Hardness and Softness are examples of observables
d. What is an eigenstate of an operator? a quantum-mechanical state corresponding to an eigenvalue of a wave equation. There’s a rule that connects those operators (and their properties) and those vectors (and their physical states), which runs as follows: If the vector associated with some particular physical state happens to be an eigenvector, with eigenvalue (say) a, of an operator associated with some particular measurable property of the system in question (in such circumstances, the state is said to be an “eigenstate” of the property in question), then that state has the value a of that particular measurable property.
e. What is an eigenvalue (associated with an eigenvector) of an operator? if v is an eigenvector of ^O , then the vector that ^O gives us when operating on v |namely, ^O v |must be some stretched or squished version of v. A squished or stretched version of v is just what you get when you multiply v by some number. This number is the eigenvalue, as illustrated:
f. What is a superposition? superposition claims that while we do not know what the state of any object is, it is actually in all possible states simultaneously, as long as we don’t look to check. It is the measurement itself that causes the object to be limited to a single possibility.
g. How are these things interpreted physically or, at least, how are they
interpreted to allow us to make predictions about what we will measure?
2. Einstein, Podolsky, and Rosen
a. How do Einstein, Podolsky, and Rosen argue that Quantum Mechanics is
“Incomplete”? What does it mean for it to be incomplete? What is the
“criterion of physical reality” that they invoke? How is it used to argue
that Quantum Mechanics is incomplete?
EPR assumed (and this is the only assumption that enters into their argument other than the assumption that the predictions of quantum mechanics about the results of experiments are correct; and the name of this other assumption is locality: that things could in principle be set up in such a way as to guarantee that the measurement of the color of electron 1 produces no physical disturbance whatsoever in electron 2.
Einstein, Podolsky, and Rosen (who have since then come to be known as “EPR”) produced an argument, which was supposed to open the way to that escape, that (if the predictions of quantum mechanics about the outcomes of experiments are correct) the quantummechanical description of the world is necessarily incomplete. Here’s what they meant by “completeness”: a description of the world is complete, for them, just in case nothing that’s true about the world, nothing that’s an “element of the reality” of the world, gets left out of that description.
Criterion for Physical reality: “If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”
b. Why does a simple “hidden variable theory” which does not involve
action at a distance not seem like a viable alternative to the collapse
interpretations above? How did we show that? A Simple “Hidden Variable Theory,” such as Bell’s inequalities fails to act
3. Collapse Interpretations of Quantum Mechanics
a. When a system is in a superposition of eigenstates of some operator, how
do we determine the probability of a certain measurement outcome?
b. How does a system evolve when no measurement is being carried out on a
system? If evolution of a system always obeyed that sort of evolution,
what would the problem be? When no measurements are going on, the states of all physical systems invariably evolve in accordance with the dynamical equations of motion. The problem, as is seen in the example Albert gives regarding Martha’s brain, is that if a system always continues this way it cannot handle the concept of superposition and we end up with the superposition of Martha thinking the pointer is pointing hard and pointing soft, and thus there is not either or.
i. What feature of the time advance (or time evolution) operator
gives rise to the fact that one ought to expect a measuring
instrument to be in a superposition of readings after the
measurement? How does that feature give rise to that fact?
c. What is “collapse” of the state-vector? Why would one posit that it
happens? A counter-intuitive property of quantum systems is that each state can be expressed as a linear combination of other states (this is known as a “superposition” of the other states). Given a sufficiently large number of states, every other state can be expressed as a superposition of the original states. A well-known illustration of superposition is Schrödinger’s cat. When a measurement of a property is carried out, the system “collapses” to one of the state with a defined value for that property, and the measurement corresponding to that particular state is observed. A system in a superposition of states 1, 3, 5, and 6 might collapse to state 3. The probability of collapsing to a given state is determined by the wave function of the system before the collapse.
d. Do collapse interpretations suggest that the world is deterministic? Collapse interpretations suggest that the world is indeterministic.
e. Do collapse interpretations suggest that there is action at a distance? Why
do they suggest that, if they do?
f. When might the collapse occur?
i. Why would one be attracted to the view that it is consciousness
that collapses the state-vector? Why would one be attracted to
that view? The consciousness view supposes that collapses occur only after something is conscious of a measurement and that there are two types of physical systems in the world purely physical systems, which always evolve on dynamical equations of motion, and conscious systems. Problem: depends on consciousness which has not yet been adequately defined.
ii. Why would one be attracted to the view that it is interaction with a
macroscopic body that collapses the state-vector? Why would one
be unattracted to that view? The same problem as consciousness. Too ambigious/ difficult to define what is macroscopic.
iii. Why is it not just a straightforward matter to determine by
experiment when the collapse happens so as to chose between
these two interpretations? Experiements either fail to disntuigsh the 2 thoeries, or, and this is the main problem, it is two difficult and there are too many factors for experiments to empirically distinguish which theory is correct.
g. What is the GRW approach to Quantum Mechanics?
Ghirardi–Rimini–Weber theory is the Ghirardi-Rimini- Weber is a collapse theory that asserts wave function collapse is sopntanerous in an attempt to avoid the measurement problem in quantum mechanics.
i. How does it suggest that the pointers of measurement devices end
up with definite (or “almost definite) positions?
ii. What are some issues with the approach?
That’s what’s been overlooked in the GRW proposal. What the GRW theory requires in order to produce an outcome isn’t merely that the recording in the measuring apparatus be macroscopic (in any or all of the senses just described), but rather that the recording
process involve macroscopic changes in the position of something. And the trouble is that no changes of that latter sort are involved in the kinds of measurements we’ve just been talking about.
The measurement problem in quantum mechanics is the problem of how (or whether) wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer. The wavefunction in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states, but actual measurements always find the physical system in a definite state.
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