There is a wide varicly of chcmical and physical theories from which wc select as our example theories of matter (atoms, molcculcs, solids, nuclei, and elementary particles). Wc can illustrate the ACP cpistcmology with the familiar ball-and-stick theory of molcculcs. This theory employs sticks and coloured balls with holes drilled in them at prescribed angles. The procedure consists of assembling die balls and sticks into figures in all possible ways. The predictions of the theory include molccular geometry and the number of isomers cxpcctcd for the molcculc in question. We all feel comfortable with this ball-and-stick theory bccausc it operates in the three-dimensional, classical world of our senses and seems "real" to us. While it is a very useful theory, it has a number of significant failures. For example, it fails to predict both the geometry and the number of isomers of benzene. More seriously, it fails to predict the electronic, vibrational, and rotational spectra of molcculcs. This failure to predict spectra is common to all classical theories and has made ncccssary the development of a new theory which includes predictive powers in this area. The nonclassical quantum theory is a theory that predicts more diversely, more quantitatively, but not more simply than the ball-and-stick theory.
Quantum theory has two essentially equivalent versions: one concerned with wave mechanics (derived from Schrocdingcr's work) and one concerned with matrix mechanics (due to Dirac-Hciscnbcrg) which is our choice. The procedure of the matrix mechanics theory of matter (MMTM) is very interesting but is not relevant to the application of the procedure. It employs vector spacc and their bases, operators, matrices, secular equations, eigenvectors, eigenvalues, groups, group algebras, etc., concepts that are well known to mathematicians but essentially unknown to beginning physicists and chemists. In consequence MMTM may seem less "real" to these beginners than the classical ball-and-stick theory. The MMTM procedure can be applied uniformly to atoms, molcculcs, solids, nuclei, and elementary particles. It is clear that the conccpt of stmcture is much simpler and more intuitive in the ball-and-stick theory than in the MMTM theory. The MMTM structure conccpt is that of a set of building blocks (basis vector of a vector spacc) that are assembled under supervision of the Hamiltonian into a physically significant set of structures (eigenvectors to the Hamiltonian).
The numerical and algebraic calculations required in the MMTM procedure can bccomc quite tedious but fortunately many of them have been or can be programmed for personal computers. The calculations of MMTM then bccomc trivial and operational familiarity is quickly acquired. Consequently, the challenging part of MMTM becomes the selection of the vector spacc and the Hamiltonian and then the interpretation of the output.