Quantum Graphs : Physics of Quantum Singularity (2-3)

Settings of Quantum Graphs 3

It is very difficult to solve this problem for the case of general N, and not many facts are known. We have to analyze each N separately starting from small numbers. N = 1 simply gives the edge of half line, and appears fairly trivial, so we move on to look at the case of N = 2 in some detail. A moment's reflection would reveal that the problem of singular node with N = 2 lines is identical to the problem of single line with one point-like defect, which we place at x = 0 which forms a singularity. Two lines now becomes x > 0 and x < 0 parts. Therefore, the problem can be refphrased as follows; What is the most general point interaction in one-dimensional single particle quantum mechanics? The boundary vector in this new interpretation becomes

、

(2.7)

in which wave function and its derivatives are evaluated at the right and left of the singularity, x = 0-, 0+. Unitary matrix now belong to U(2) which is specifiable by four parameters. Here, we use five parameters , , with one constraint and write U down explicitely as

(2.8)

We also adopt the notation H_U for the Hailtonian of the system having singularity specified by U. The representayion (2.8) clearly shows that the set of entire point interaction allowed in quantum mechanics is specified by a parameter space which forms a manifold of four-diminasional hyper sphare S_4 which is isometric to the group U(4).

At this point, readers who have gone through the usual quantum mechanics course will have to say "what?" "I've seen something called Dirac's delta function potential, but never heard of any other types of point interaction. How come there are infinite other alternatives to delta function?"

The secret is of course in our consideration in the previous page where the concept of "self-adjoint extension" is discussed. In usual textbooks, we start from the premise thatthere is a given set of "good" wave function from which we choose the solution to the Schroedinger equation, and that good wave function has to satisfy such natural condition as the continuity. "Nature does not jump, you know." some might say. But the fact is that there ain't any such requirement in the basic axioms of quantum mechanics, and we have to find out a proper set of wave function to make the probability conservation hold even in the presence of singularity.

After N = 2 case, we can think of "Y-type junction" that corresponds to N = 3 case. Then X-type junction which is N = 4. These are expectted to become a mainstray of quantum device, but currently the full mathematical analysis is still lacking. We should start from somewhere and N = 2 is the only case which has received full lattenstion and full solution up to now, which will be the main subject in the following pages.