Assignment 15: Problems of Quantum Theory

For submission: Mon. Dec. 1, Tues. Dec. 2, Wed. Dec. 3.

1. Consider a wave packet used in de Broglie's theory to represent a particle. How is the particle's momentum affected if we make the spatial extent of the wave packet bigger or smaller? How does this difference relate to the "Heisenberg Uncertainty Principle"?

2. What is the difference between interpreting the uncertainty of Heisenberg's principle as ignorance as opposed to indeterminateness?

3. What is the "Schroedinger evolution" of a matter wave? What is "the collapse of the wavepacket"?

4. In the standard analysis of the Schroedinger cat thought experiment, what leads to the definite survival or definite death of the cat?

For discussion in the recitation.

A. Quantum theory is an indeterministic theory. That means that a complete specification of the present state of some atomic system does not fix its future. Here's how we apply this idea to radioactive decay. If you have a single atom of Neptunium NP ²³¹ ₉₃, there is a one in two chance that it will decay over the next 53 minutes. According to standard quantum theory, that is all you can know. There is no way to know ahead of time whether the atom will decay. Do you really believe that? Might it be if we had a more complete picture of the compicated, hidden recesses of this atom that we'd see some tiny difference between those atoms that end up decaying and those that do not? Ought we expect some future theory of the insides of atoms to tell us about these sorts of hidden properties? Ought we to demand such a theory before we can say we really understand radioactive decay? Or should we comfortable with the idea that some processes just are indeterministic?

B. To get a sense of how the Heisenberg uncertainty principle applies, consider the problem of balancing a pencil perfectly on its tip. Here is what is needed for success in the balancing operation: you have to align the center of mass of the pencil exactly over the pencil's tip; and, as you take your fingers off the pencil after doing this, you need to leave the pencil perfectly at rest. What does Heisenberg's uncertainty principle tell you about your chances of success?

C. The "measurement problem" remains a lingering difficulty for quantum theory. Yet modern quantum theory remains an extremely successful theory of matter that has given us many fascinating insights into the nature of matter and makes many quantitative predictions that have been borne out by experience. How is this possible?

D. Consdier the Schroedinger cat thought experiment. According to the text book account of quantum measurement, immediately prior to our opening the box, the cat is in a 50-50 superposition of alive and dead states; when we open the box and look at the cat, we trigger a collapse into just one of those states. Most people find that instinctively implausible. However our instincts have mislead us often enough. We all felt instinctively that there is a universal fact over whether two events are simultaneous; or that the sum of the angles of a right angle has to be 180 degrees. Both proved to be false. Should we believe our instincts in this case? If so, why? If not, why not?