5.1: The State of a System

The state of a system means the collection of all properties that that system may have. For example, consider an electron. If you wanted to specify the state of that electron as completely as possible, you’d have to specify where it is and its momentum, as well as how well determined its momentum is. You would also have to specify the state of its angular momentum. Does its z spin have a definite or an indeterminate value? If it has a definite value, what is it? If it has an indeterminate value, is it a half and half chance that, if measured, you’ll get +1/2 or -1/2, or is it more likely to be one or the other? If your system includes more than just one particle, you have to include all the information about other particles, as well as any information that arises as a result of the interaction between the particles. For instance, if this electron is moving in some potential, for instance because it’s part of an atom, what (effectively) is the electron’s potential energy?

We are going to introduce an abstract mathematical notation that will indicate “the state of a particle”. The notation itself won’t necessarily have all of the information above. However, what it will give us is a way to talk about the state of a particle. Because the state of a particle potentially includes a lot of information, it will be necessary to use a more abstract notation than you’re used to for mathematical objects. However, remember that even the seemingly-concrete math that you’re comfortable with itself is just constructed from abstract mathematical representations of reality.

Consider algebra. Suppose you have a variable, that may or may not be known. You use the name \(x\) to represent the state of that variable. Now, if we’re dealing with algebra, and we’re dealing with only real numbers, then it’s possible to represent the full information about the state of this variable with just a single number. For instance, suppose you are given the following algebraic equation:

\(\ 2 x+5=9\)

You could use the rules of manipulating algebraic equations to determine that \(\ x = 2\). At that point, you know everything there is to know about the state of this variable. However, you could still represent it with the letter \(\ x\) if you wished to. Even if you don’t, however, and if this equation is supposed to represent something from the real world (say, ages of children in a word problem), even the 2 is a mathematical representation of something in the real world.

Let’s make it more abstract. Suppose I tell you that \(\ 2 x+y=b\), and that \(\ y\) and \(\ b\) are known. However, I haven’t given you a number to fully specify \(\ y\), nor have I given you one for \(\ b\). I then ask you what \(\ x\) is. You could solve this and tell me that

\(\ x=\frac{b-y}{2}\)

Now you would say that \(\ x\) is “known”, even though you can’t reduce it to the concrete representation of real numbers. However, you have given me a representation of \(\ x\) in terms of other things, including this letter \(\ y\) and this other letter \(\ b\). Those two are stand-ins, abstract mathematical representations of “some number that we’ve decided to call \(\ y\)” and “some number that we’ve decided to call \(\ b\)”.

We will use a similar abstract notation to represent “the state of a quantum particle”, or, perhaps, just “the angular momentum state of a quantum particle” (if we don’t care about things like position and momentum). The rules of quantum mechanics will give us mathematical operations we can perform on this representation, and then other things we can do to extract useful information out if it (such as the energy of a particle, or the probability that its \(\ z\) spin will be positive if measured).