A Confusion So Common
Brad Delong expresses a type of confusion that is so common that it has its own literature. Specifically, he's worried that he doesn't understand what practitioners mean when they write out things using some of the tools of quantum mechanics. In particular, some of the quick and dirty algebraic manipulations that practicing physical scientists throw around when using that most mysterious of objects, the wave function.
It's always a good idea to go back and look at what's going on under the hood. First, remember the first rule: to the best of our understanding, the fundamental particles are all both wave and particle. Photon, electron, all the others, to any degree that we can measure, all are both tiny little particles. And they are waves.
So, anything we do to describe these particles must carry the same fundamental duality. A wave function that describes such an object must carry both particle and wave information, simultaneously, if it is to do its mathematical job. Otherwise, it's not up to the task.
So what then is the mathematical object we write as |A)? |A), our potential wave function, is a complex function. That is, it is a function of complex numbers. As such, the object (A| is the complex conjugate to |A). If these were real numbers, (A| would be the inverse of |A).
Which leads to the next object. (A|A) is a single, real number. Often, depending on normalization convention, as implied by the inverse or complex conjugate, (A|A) = 1.
If |A) were a relatively simple function, that by itself would be enough. But because of the first rule, it's a little more dramatic than that. (A|A) means then something more complicated than simply multiplying |A) by its complex conjugate. What it means more fully is, multiply |A) by its complex conjugate, then integrate the result. If A is a function of space and time, we integrate over space and time to get 1.
If A is a function of momentum and frequency, then we integrate over momentum and frequency. But the operations involved are the same. Multiply, and then integrate.
Of course, the first rule means that this isn't the end. |A) is also a matrix. And (A| is then the conjugate transpose of |A). In which case, (A|A) means multiply the matrix A by its conjugate transpose, which gives a matrix, and then take the trace of that resulting matrix. The trace is then a single number, usually 1 due to normalization.
This goes even further. |A) is also a field, and an operator. But that comes later.
First, let's talk about H. H, the Hamiltonian, is, for the particles we know of at least, a special function (operator, matrix, field) of its own. In particular, H|A), which means to take the operator H and act upon the function A, gives the energy of the system as E|A).
More specifically, if (A|H|A) means operate H on A, multiply the resulting matrix by the complex conjugate of A, and then integrate, then the result is E, the average energy of our particle. Or, in matrix language, multiply the matrix H by the matrix A, multiply the result by the conjugate transpose of A, and then take the trace. The result is E, the energy of the particle. (A|H|A) = E.
H, the Hamiltonian, is the operator which measures the energy of system. Or, alternatively, if we perform an experiment on a particle and measure its energy in a given experimental setup, then H is the theoretic function that we seek which, when operating on a test function, gives the same E as our experiment did. In which case, we speak of H as defining the system. There are other details about H.
One of them is that H also generates the dynamic information of a system, not just its average energy. That object looks like exp(iHt), where exp is the exponential, i is the imaginary number (i.e. square root of -1), and t is time. Then exp(iHt)|A) is the dynamic represenation of |A); alternatively, exp(iHt) acting on |A) generates |A(t)), the propagation of A into the future (or the past).
Either way, the algebra involved always looks like some version of (A|H|A), the multiplication of two matrices, followed by multiplication by the complex conjugate and taking the trace.
Now, let's go back to H|A) = E|A). H is an operator. E is a diagonal matrix of scalar, real numbers.
Or, to put it another, equivalent way, |A) is the matrix which diagonalizes the Hamiltonian. Thus, the wave function is an operator in and of itself. This is where a detailed linear algebra book, one that goes all the way through orthogonality, similarity and unitary transformations, and so on, begins. This is also where practitioners can get funny looks when people ask "what is the wave function?" In practical terms, the wave function here is "all of space", or more particularly any of a broad class of functions which measure (or span) space in a particular way. This is a particular generalization of the way in which position means "any real number" in the equations of classical physics. To ask after the "nature" of a wave function is to ask after the "nature" of numbers. They're the same thing, just written and collected in slightly different ways as needed for the use.
This property of A has some interesting side effects: A can have a simple, easy to write down structure for small scale systems. But that structure can be drastically different at larger scale. So much so that "two-level model" is either a curse or a blessing depending on area of work. Or the time of day, phase of the moon, color of the wine...
All of this is really back to the first rule. Which is that we have to keep track of both particle and wave nature simultaneously. Specifically, we have to deal with functions like A(r,k). r here is position, k is wave number (momentum with certain conditions). All of the notation is a reminder that we must always be careful about when, and in what order, we do something like B(r,k)A(r,k), a multiplication that could be over r, or k, or both, followed by an integration. Or a sum. If you write it out in detail, with full notation, it's tedious, painful, and you're guaranteed to lose track the farther into the work that you go.
Eventually, if you try and do everything in full detail at every stage of derivation, you are guaranteed to screw it up. So first Heisenberg, then Dirac, came up with different shorthand methods. Which just confuses things, because any of the notations can be written as any of the others. And, more unfortunately, Schrodinger's detailed methods involve so many elements that the shorthand has become the common method of representation even when their use confuses everyone involved, expert and non-expert alike.
This is the point where, if you've heard of it, the "shut up and calculate" school of thought stops, more or less. And, for all intents and purposes, that's sufficient. Assuming I haven't just made your confusion worse, the thumbnail description above gives the nuts and bolts elements. For many problems, there's not really any need to go any further.
But there are problems for which this explanation isn't enough. Feynman, Dyson, Bohm, all of them useful and, for some very significant problems, essential to go any farther. Who knows yet whether or how Many-Worlds will lead further, but it's one of the current cases where folks have tackled the basics again. There's always something there to think about anew.
And get confused over. Duality all the way up and down.
