The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
The quantum theoretical description of microscopic systems is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. In matrix mechanics all observables are represented by matrices. All possible eigenvalues of a matrix representing an observable are the possible outcomes of experimental measurement on that observable. This chapter gives the preliminary concepts relative to the use of the matrix method. It presents the various elements of matrix mechanics and discusses the matrix representation of operators of certain systems. The chapter introduces Dirac's notation of state vectors, Hilbert space, projection operator, displacement operators and the generators of translations. The theory of quaternions has found applications in classical mechanics, quantum mechanics and relativity. It has been used in aerospace applications, flight simulators, computer graphics, animation, visualization, fractals and virtual reality.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the fundamental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that represents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.
In quantum mechanics the linear harmonic oscillator representatives of the canonical observables q and p are certain well-known infinite square matrices, qmn and pmn, with denumerable sets of rows and columns. It is pedagogically useful to investigate various problems which arise when one considers the finite matrices Qmn and Pmn which result when qmn and pmn are truncated after the Nth row and column, where N( 2≥2) is some preassigned positive integer. So that the enquiry may be more instructive, it begins with an abstract formulation of the problem of the “truncated harmonic oscillator.” Formally, the latter has the usual Hamiltonian H = 12( Q2+P2 ) , but Q and P are not canonical—their commutator [Q,P] now differing from i by a certain projection operator. The spectrum of H and the (finite) basis { |n〉} generated by H are found by standard methods, though there are now some complications which are absent from the theory of the linear harmonic oscillator. The diagonalization of Q and P presents no difficulties if full use is made of the classical theory of Hermite polynomials. The spectrum of Q is the set of N zeros hs of the Nth Hermite polynomial. Although the wavefunctions 〈 Q′|n〉 are therefore functions of the discrete variable hs, one can formally contemplate them as functions of a continuous real variable h; and some curious intimations of Schrödinger representation appear. The analog of a component of orbital angular momentum is considered, and its spectrum obtained. Finally, the asymptotic limit N → ∞ is briefly discussed, in order thus explicitly to dispel the impression that this limit might somehow represent the “usual” linear harmonic oscillator.