System and State

The notion of a system and its state is a fundamental one in physical science and engineering. Both the words system and state derive from a Proto Indo European base word meaning to stand, and we use them in senses that have varying degrees of precision. In general, a system is a collection of things that "stand together," and the state of that system is the "way they stand together." In physics, a system might be a single elementary particle, a bottle of gas, a star, or the universe, and by its state we mean mean some collection of variables that specify that state. For an electron, for example, its charge, mass, velocity and position.

In classical physics the state variables tend to be measurable numbers, what the quantum pioneers called c-numbers. Things become more abstract in quantum mechanics. The state of a quantum system is specified by something that we call a ray in Hilbert space. It's worth noting, though, that even the classical description is a big step in abstraction from ordinary speech. A planet or a canonball has a complex description in ordinary speech, but if you want to know where it's going next you need some numbers, like velocity and mass, for example.

We are all forced to do some thinking in numbers: the state of my pocketbook is pretty well characterized by the number of dollars in it, for example. Physical sciences and engineering carry numerical thinking to an extreme, but it's not really enough for the world of the quantum. Physicists didn't go gladly to the Hilbert space description - at any rate many of them didn't. They were forced to it by the counter intuitive behavior of quantum phenomena, especially by quantum system's habit of exhibiting both wave and particle like behavior, and by the uncertainty principle.

Consider the spin of an electron. If we measure that spin it always turns out to be either up or down - never something in between. Could anything be simpler? Here's the rub - in between measurements an electron acts like it's partly spinning up and partly spinning down. If this seems strange and bizarre to you, it seems that way to many physicists too, even those who have spent decades thinking about it. I won't go into the ample evidence that this is the case, but suffice it to say that nobody has been able to make it go away, and plenty have tried, including Einstein and other quantum pioneers.

How do we describe something like that? It turns out that a Hilbert space is a good way to do it. I don't have a simple explanation of what a Hilbert space is (see the link for Wikipedia's discussion) but I will try to give a rough intuitive feel for it. Start by imagining a point in space that we will call the origin. Draw an arrow from that point to some other point in space - that arrow is a vector in our three dimensional space. If we put some coordinate axes in our space, with the zeroes of the axes at our origin, we can label our vector with the coordinates of the head of the arrow - each set of coordinates correspondes to a unique vector and vice versa.

Together all the vectors form a vector space. One important property of vectors is that you can add or subtract them, or multiply them by a constant. For example, if we represent the vector with head at x=1, y=2, and z=1 in our coordinate system by |1,2,1> and similarly another vector by |1,0,1> then we can do arithmetic like so:

|1,2,1> + 2|1,0,1> = |1,2,1>+|2,0,2>=|3,2,3>

which is the vector with head at coordinates 3, 2, and 3. If you have studies some vector algebra, you remember that this geometrically corresponds to stretching the second vector by a factor of two, moving its tail to the head of the first, and then drawing a new vector from the origin to the head of the doubled, displaced, second vector.

A Hilbert space is a generalization of a vector space - it can have infinitely many dimensions, for example. We won't need that many for our electron spin state - two will do, but we will need another attribute, because our multiplying constants can be complex numbers. We choose two fundamental vectors |up> (the electrons spin is "completely" up) and |down> (the spin is all down). Then the most general spin state is represented by a up> + b down> where a and b complex numbers. (I won't go into details, but we want to make all vectors of unit length, also).

Part of the point here is that while the measurement can only yield up or down, the general pre-measurement state is more complicated. The additional complexity of the representation is related to the uncertainty in quantum measurements. In general, even a complete knowledge of the state is not enough to determine a measurement, since state vector yields only a probability distribution.

As usual, correction and other improvements are welcomed.