Almost half a century, and with it nearly two entire generations of physicists, has now elapsed since those years in the middle twenties during which quantum mechanics finally crystallized into its presently accepted and universally used form The most fundamental and productive version of quantum mechanics has always been the one introduced by Erwin Schrödinger [1] in the first months of 1926. It was slightly preceded by Heisenberg's matrix mechanics, a discipline which was quickly refined and promptly presented in Born and Jordan's book, Elementare Quantenmechanik [2]. Not long thereafter both Dirac [3] and von Neumann [4] produced their two quite unlike operator versions, essentially completing the conceptual foundations of quantum mechanics
The symbolic and operational techniques have been indispensable in providing a vocabulary for the teaching, discussion, and application of quantum mechanics. Nevertheless when the moment arrives that matrix elements have to be calculated and results obtained, it is the Schrödinger equation which is eventually introduced and has to be solved
We must recall that the substance of Schrödinger's papers, indeed their very title, was the assertion that quantization was an eigenvalue problem. By the selection of a suitable diflerential equation and the imposition of satisfactory boundary conditions we were inexorably led to the quantization which was at once the basis and the most singular and inexplicable feature of Bohr's older quantum mechanics. It is especially to be noted that the term ``eigenvalue problem'' refers to the acquisition of suitable solutions of the boundary value problem of the differential equation, and not to the mere diagonalizalion of a matrix, as the phrase is so often understood nowadays.
However, the specification of the boundary conditions in a way which would be adequate for axiomatic considerations was a concern of some delicacy. Differing viewpoints emerged almost at once, and have persisted in one form or another up to the present day. Schrödinger himself simply imposed the requirements of continuity, single-valuedness, and finiteness. He carefully left the function which actually solved his differential equation devoid of meaning, in spite of an avowed desire to interpret it as a charge density. Eventually, and primarily through the efforts of Max Born, the square of the wave function came to have a probabilistic interpretation, whose natural aftermath was the use of quadratic integrability as a boundary condition.
Unfortunately the requirement of square integrability is not always decisive. There are several instances, one of the most notable of which arises from the ground state of the hydrogen atom, in which all the solutions of the wave equation are found to be square integrable. For s states in the radial equation of the hydrogen atom, there are always two linearly independent solutions, square integrable at the origin. Strictly speaking, we mean that they are integrable in any finite left-hand interval which includes the origin, which is a singular point in the radial wave equation. This means that at whatever energy there can always be found a linear combination which is also integrable in the outward direction, to whatever distance. Thus, according to a criterion of square integrability, there would be no quantization, which is contrary to observation. In actuality the quantizing principle which is applied is to demand that the wave function be finite at the origin.
There is also a considerable literature devoted to the requirements which angular momentum wave functions ought to satisfy. Again finiteness is the quantizing principle which is used in practice, although periodicity suffices for some of the angular coordinates. Here too, it is found that all the solutions would be square integrable, although sometimes some use can be made of the integrability of the derivative of the wave function, or that the unwanted solutions imply unpalatable currents.
Finiteness at the boundary points does not serve as a universally applicable requirement either, as it is powerless to decide the quantization of some of the levels in the Dirac equation for the hydrogen atom.
Continuum wave functions present a normalization problem of a somewhat different nature. Although it is hardly possible to give a probabilistic interpretation to a wave function which extends over the whole of infinite space with a nearly constant amplitude, it is entirely natural either to speak of relative probabilities, or to work with current densities. The characteristic aspect of continuum wave functions is that they are oscillatory rather than exponential. Consequently their normalization integral grows linearly with the volume over which the wave function extends so that it is sensible to resort to the fiction that their orthonormalily is expressed by a Dirac delta function.
As long as our entire interest lies with bound-state problems, there do not arise too many difficulties about the use of square integrability as a boundary condition, nor is there much opportunity for error in manipulating all operators as though they were finite matrices. Nor is there even much difficulty when operators are defined in a finite closed space, such as the configuration spaces of the quantum mechanical tops or rotors. The exceptions which occur have to do with potentials which are unbounded below, especially with some of the more strongly singular potentials.
Investigators who have had the most practical encounters with continuum wave functions, such as the ones which arise in solid state theory or in scattering theory, have always been able to treat them in a very pragmatic manner, which consists mostly in separating bounded solutions from unbounded ones, and among the bounded solutions classifying them as to their direction of propagation The first initiatives were taken by Born, who was particularly anxious to understand the quantum aspects of scattenng theory. Under his influence, Oppenheimer [5] made some of the first extensions of the Schrödinger equation to aperiodic systems, such as arose in the theory of collisions or passage through barriers.
In one way or another it has usually been possible to set up boundary conditions which would produce a satisfactory resolution to the quantization of any system of practical interest. Strongly singular potentials constitute practically the only significant exception. Inasmuch as there are no physical situations in which they occur in their mathematically pure form, there has never been any experimental evidence which would clearly confirm or contradict speculations as to whether or not quantization would occur, or how to achieve it if it did.
Historically, if there had not always existed quantizing principles sufficient for the task at hand, there would certainly have been a concerted effort to resolve the uncertainties which might have existed. Yet it is fair to say that there has never been a single, generally understood, and consistently applied criterion, leaving the procedure to he followed in an unfamiliar case quite cloudy.
We might say that square integrability sufficed for bound-state problems whose potential was bounded below, that an occasional finiteness argument was required for moderately singular potentials, that strongly singular potentials never became an issue, and that continuum wave functions never needed quantizing. Rather, the difliculty with continuum wave functions consisted in the incorporation of their mathematical properties into a theory reminiscent of the theory of finite matrices, which the use of the delta function accomplished to most people's satisfaction.
It would seem that there has been another historical process at work, pertaining to the mathematical sophistication of the physicists and others who needed to use quantum mechanics, and perhaps also to the sophistication of the mathematics with which quantum mechanics itself was phrased. Schrödinger was able to formulate quantization as an eigenvalue problem, precisely because he was familiar with the development which Hilbert had given to eigenvalue problems, and which had been worked out quite precisely for second-order differential equations by Hermann Weyl. The only complication which continuum wave functions caused in an eigenfunction expansion was the necessity to employ a Stieltjes integral in place of a sum on many occasions. Von Neumann met the same difficulty in formulating his theory of operators on Hilbert space, which he likewise resolved wilh a Slieltjes integral over projection operators. Unfortunately use of the Stieltjes integral never became a standard part of the ``mathematical methods of physics.''
The reason was most probably the fact that the ``eigendifferentials'' which occurred in the integral were somewhat hazy concepts, and did not approach a limit having a clearly defined conceptual significance, which was natural. Otherwise it would never have been necessary to resort to the Stieltjes, rather than a Riemannn, integral in the first place. Since the result was to introduce a language which appeared to physicists to involve some rather elaborate circumlocutions which they found bothersome, the mathematical presentations, although quite accurate, never enjoyed much popularity.