as Calculating Engine
It's not hard to write down the matrix in terms of interactions between the atoms. In other words, in principle, you can invent a kind of coupling among the
on to make the calculation. But the question is, how do you make the succession of three-atom transformations go bup-bup-bup-bup-bup in a row? It turns out to be rather easythe idea is very simple. [
Feynman draws a row of small circles, and points often to various circles in the row through the following discussion.]
You can have a whole lot of spots, such as atoms on which an electron can sit, in a long chain. If you put an electron on one spot, then in a classical world it would have a certain chance of jumping to another spot. In quantum mechanics, you would say it has a certain amplitude to get there. Of course, it's all complex
and fancy business, but what happens is that the Schrdinger function diffuses: the amplitude defined in different places wanders around. Maybe the electron comes down to the end, and maybe it comes back and just wanders around. In other words, there's some amplitude that the electron jumped to here and jumped to there. When you square the answer, it represents a probability that the electron has jumped all the way along.
As you all know, this row of sites is a wire. That's the way electrons go through a wirethey jump from site to site. Assume it's a long wire. I want to arrange the Hamiltonian of the worldthe connections between sitesso that an electron will have zero amplitude to get from one site to the
because of a barrier, and it can only cross the
with the atoms [of the registers] that are keeping track of the answer. [
In response to a question following the lecture, Feynman did write out a typical
in such a Hamiltonian using an atom-transforming matrix M positioned between electron creation and annihilation operators on adjacent sites.]
That is, the idea is to make the coupling so that the electron has no amplitude to go from site to site, unless it disturbs the
atoms by multiplying by the matrix
2, in this case, or by
3 in these other cases. If the electron started at one end, and went right along and came out at the other end, we would know that it had made the succession of operations
5the whole set, just what you wanted.
But wait a minuteelectrons don't go like that! They have certain amplitude to go forward, then they come back, and then they go forward. If the electron goes forward, say, from here to there, and does the operation
2 along the way, then if the electron goes
, it has to do the operation
Bad luck? No!
2 is designed to be a reversible operation. If you do it twice, you don't do anything; it undoes what it did before. It's like a zipper that somebody's trying to pull up, but the person doesn't zip very well, and zips it up and down. Nevertheless, wherever the zipper is at, it's
up correctly to that particular point. Even though the person unzips it partly and zips it up again, it's always right, so that when it's finished at the end, and the Talon fastener is at the top, the
has completed the correct operations.
So if we find the electron at the far end, the calculation is finished and correct. You just wait, and when you see it, quickly take it away and put it in your pocket so it doesn't back up. With an electric field, that's easy.
It turns out that this idea is quite sound. The idea is very interesting to analyze, to see what a computer's limitations are. Although this computer is not one we can build easily, it has got everything defined in it. Everything is written: the Hamiltonian, the details. You can study the limitations of this machine, with regard to speed, with regard to heat, with regard to how many elements you need to do a calculation, and so on. And the results are rather interesting.