Quantum Vectors

The purpose of this is to extend the concept of vectors beyond the idea of a directional quantity, and to show why the matrix mechanics approach to quantum mechanics (QM) turns out to be the mathematical equivalent of wave mechanics. If you have had a course in QM, this is too elementary for you. This might be helpful to someone starting a QM course or planning to in the near future.

A discrete vector is a list of numbers, not limited to directions in space. In classical mechanics, the orthonormal set of basis (unit) vectors for 3-d quantities such as force are i, j, k: (1,0,0), (0,1,0), and (0,0,1). Any vector in 3-d is composed of a linear combination of these, and in essence is simply a list of three numbers.

Think of an ingredients list in a chili recipe as a vector. One component would be a number representing the amount of beans, for example. This is the way one should view the state vector in quantum mechanics.

(Incidentally, we could argue about whether tomatoes are orthogonal to beans: they both contain water, so in that sense, no. The dot product would be the product of the amounts of water in each + products of any other substances in common. As a practical interpretation, on the other hand, we cannot separate these things into the compounds they are made of, so cooks should think of them as orthogonal: the dot product, bean× tomato = 0. The basis vectors for our chili recipe would be one unit of tomato, one unit beans, etc., etc., written (1,0,0,...), (0,1,0,...), etc. Or one could rotate the system to 0.8 tomato and 0.6 bean, then the next vector could be 0.6 tomato and -0.8 bean, and these would still be orthonormal. In any case, the state vector of the finished chili is a vector which is a linear combination of the basis vectors, with the amount of each ingredient as the appropriate multiplier.)

If you take a spoonful of chili (a measurement), the possible contents of the spoonful are determined by the state vector of the chili. The various components indicate the probabilities of getting the ingredients in the spoonful. In QM, the possible result of a measurement is determined by the state vector. The main difference between QM vectors and chili vectors is that in QM, complex numbers are used for the components.

Dirac's bra-ket (bracket) notation is universally used for QM vectors. <aï is called a bra and ïa> is a ket. They correspond to row and column vectors respectively, but they are defined more abstractly. That is, row and column vectors provide a nice concrete way of viewing them, but this is not the only possible way. The components of <aï are complex conjugates of the components of ïa>. A discrete basis ket ïv_i> may be thought of as a column vector with a one and the rest zeros. The inner product, written <v_iïv_j>, is therefore 1 if i=j and 0 if i¹ j (orthonormal). In the bean× tomato example, <bït> = 0.

The corresponding relation for a continuous case (infinitely many dimensions) is <v_iïv_j> = d (v_i-v_j) (= Dirac delta "function") This seems weird at first; one needs to generalize the concept of a vector. Consider the case of position on the x axis. For each position x_i there corresponds a lx_i> which consists of a list of zeros (infinitely many) except for a spike (Dirac delta "function") at x_i. Furthermore (and this is really astonishing), for any function f(x) there is a ket ïf> such that <x_iïf> = f(x_i). This is written <xïf> = f(x). (We will soon see how this is possible.) So if we think of ïf> as a list of components, each must be infinitesimal in order to compensate for the spike.

To represent f(x) from 0 to 1 as a discrete vector, the list of numbers in ïf> could be f(0), f(.01), f(.02), f(.03),...,f(1), for example. Then the x basis set would be (1,0,0,...0), (0,1,0,0,...0) etc. So why not continue with this: Let a basis = a one and infinitely many zeros, ïf> = infinitely many f(x) values? Perhaps this could be done (with of course other changes in the formalism), but the other way is more convenient in spite of the infinities and the infinitesimals, because it turns out that ò y *y dx = <y ïy >.

Going back to discrete basis kets for the 3-d case, if we multiply column times row lv><vl, we get a matrix. For example,

and if we do the same thing with 0 1 0 we get all zeros except 1 in the middle. Obviously, S lv_i><v_il = I, the identity matrix, all zeros except for ones on the main diagonal.

In the continuous case (infinitely many components) we get all deltas on the diagonal. In wave mechanics, if y (x) is the wave function satisfying Schroedinger's equation, normalization requires ò y *y dx = 1.
If y (x) = <xly >, then y *(x) = <y lx>, so ò y *y dx becomes
ò <y lx><xly >dx, and when you see lx><xl in an integral, you have all those (infinitely many) d 's pulling out all those y components.
So ò y *y dx = 1 is identical to <y ly > = 1.

In what follows, the discussion is very naughty. Things like d (x-x_i) = 1/dx at x = x_i. I know that this is not proper, so don't tell me about it. And if you are in a physics class, do not tell the prof about this. She will be horrified. (Ok if you want to fix it, change the d's to D 's and take the limit as D x approaches zero later, and this will appease and please the math gods, and we will have bountiful harvests of equations in the next millennium.)

An x basis vector for the continuous case consists of (dx)^-½ at one location and the rest zeros. The entire set of x basis vectors consists of infinitely many of these, each having one spike. So there is a spike (dx)^-½ for each x location. For any function f(x) we may define ïf> in which each of the infinitely many components is f(x)(dx)^½. Now you can see explicitly that <x_iïf> = f(x_i) for all x_i. So therefore <xlf> = f(x) and
<glf> = S _inf g*(x)f(x)dx = ò g*(x)f(x)dx.

Then go back to the main quantum page. Or if you haven't seen the quantum mechanics of a crap game yet, you might want to go there.

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