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

Settings of Quantum Graphs 2

In order4 to have a sound quantum mechanics in the presence of a singular point, the quantum states and operators have to satisfy certain basic conditions that guarantee the probablistic interpretation of observation of quatum system. A central requirement is that an observable has to be a self-adjoint operator in appropriate Hilbert space. When applied to the free hamiltonian (2.1), which is nothing but the Laplacian operator, this self-adjointness becomes the continuity condition of probability current at the site of singularity. For usual problems in quantum mechanics, we seek a solution with a tacit assumption of the existence of a set of wave function with "physically reasonable property" such as "smooth and square integrable". With the assumption of the existence of this set, among which the solution will be drawn, self adjointness of an observable operator becomes identical to its Hermiticity. But in the current problem, the requirement of Laplacian being self-adjoint forms a condition on a set of wave function within which Laplacien is kept Hermitian. This process of finding a proper set of wave function (or more precisely a proper subset of Hilbert space) is called self-adjoint extention. In the current context, it is the requirement of "the continuity of probability current" which is a most fundamental requirement of quantum mechanics, that sets a condition for the condition of proper wave function. Actual form of that condition is easily obtained as the connection condition of wave functions at the singularity x_i = 0 in the form

(2.3)

If we denote the point on the i-th brach right adjacent to thenode by 0i, and define N-dimensional vectors \Phi and \Phi' by

、

(2.4)

we can easily show, by explicit calculation, that the condition of probability density flus (2.3) is written as

(2.5)

which can be further rewritten as . Here, L₀ is an arbitrary constant with the dimension of length which is introduced to have consistent dimensionality for each term in the equality. This equality signifies the transfomability of the inside of absolute value of LHS to that of RHS by a unitary matrix of dimension N. Writing that unitary matrix belonging to U(N) as U, after some reformulation, we have an equation

(2.6)

which is the fundamental equation for us. The meaning of L_0, that emmerged from "nowhere" with the dimensional argument will be discussed later in full detail. End result is that the Schroedinger equation (2.2) yields a definite state only when supplemented by the condition (2.6) that specifies how the values of various braches of wave functions are connected at the node x_i=0. The unitary matrix U appearing in that connection condition has N^2 parameters that define the quantum mechanical property of the node. In other word, at the node that is an object of "set of measure zero", there are infitenite number of different quantum point interactions that are specified by N^2 parameters of U(N) matrix.