The Feynman Lectures on Physics, Volume III by unknow

Author:unknow
Language: eng
Format: epub
Published: 2014-05-19T16:00:00+00:00

12–66 The projection matrix for spin one6

We would like now to use our knowledge of the hydrogen atom to do something special. We discussed in Chapter 5 that a particle of spin one which was in one of the base states (, , or ) with respect to a Stern-Gerlach apparatus of a particular orientation—say an apparatus—would have a certain amplitude to be in each of the three states with respect to a apparatus with a different orientation in space. There are nine such amplitudes which make up the projection matrix. In Section 5–77 we gave without proof the terms of this matrix for various orientations of with respect to . Now we will show you one way they can be derived.

In the hydrogen atom we have found a spin-one system which is made up of two spin one-half particles. We have already worked out in Chapter 6 how to transform the spin one-half amplitudes. We can use this information to calculate the transformation for spin one. This is the way it works: We have a system—a hydrogen atom with the energy —which has spin one. Suppose we run it through a Stern-Gerlach filter , so that we know it is in one of the base states with respect to , say . What is the amplitude that it will be in one of the base states, say , with respect to the apparatus? If we call the coordinate system of the apparatus the system, the state is what we have been calling the state . But suppose another guy took his -axis along the axis of . He will be referring his states to what we will call the frame. His “up” and “down” states for the electron and proton would be different from ours. His “plus-plus” state—which we can write , referring to the “prime” frame—is the state of the spin-one particle. What we want is which is just another way of writing the amplitude .

We can find the amplitude in the following way. In our frame the electron in the state has its spin “up”. That means that it has some amplitude of being “up” in his frame, and some amplitude of being “down” in that frame. Similarly, the proton in the state has spin “up” in our frame and the amplitudes and of having spin “up” or spin “down” in the “prime” frame. Since we are talking about two distinct particles, the amplitude that both particles will be “up” together in his frame is the product of the two amplitudes, We have put the subscripts e and p on the amplitudes to make it clear what we were doing. But they are both just the transformation amplitudes for a spin one-half particle, so they are really identical numbers. They are, in fact, just the amplitude we have called in Chapter 6, and which we listed in the tables at the end of that chapter.

Table 12-4 Spin one-half amplitudes

This chapter Chapter 6

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.