The first section of this chapter shows that the mathematical representation of amplitude cannot be as simple as a real number, but must be at least as complicated as a two-dimensional arrow. If you're willing to accept this as fact, then you may skip that rather technical and involved section. But in no case should you skip over the second section of this chapter, which makes a simple but subtle and important general point.

Amplitude is represented by an arrow

I'm going to introduce one more type of analyzer: the “front–back analyzer” (also called the “y analyzer”). This will be the last new analyzer, I promise. The left half of this analyzer is just like the left half of a traditional Stern–Gerlach analyzer, with its traditional non-uniform magnetic field. But while the right half of the traditional Stern–Gerlach analyzer contains only plumbing to make sure the atoms come out parallel to the sides of the box, the right half of the front–back analyzer contains also a magnetic field that changes direction slowly from place to place. Along the path towards the upper exit, the magnetic field starts by pointing straight up. A little farther on it tilts a bit to the right. The tilt angle of the field increases gradually until, just before the exit, the field points directly to the right. The path towards the lower exit is similar, except that in this case the field starts out pointing down and gradually tilts until it points directly to the left.