Abstract The laws of classical mechanics must be generalised when applied to atomic systems, the generalisation being that the commutative law of multiplication, as applied to dynamical variables, is to be replaced by certain quantum conditions, which are just sufficient to enable one to evaluate xy - yx when x and y are given. It follows that the dynamical variables cannot be ordinary numbers expressible in the decimal notation (which numbers will be called c-numbers), but may be considered to be numbers of a special kind (which will be called q-numbers), whose nature cannot be exactly specified, but which can be used in the algebraic solution of a dynamical problem in a manner closely analogous to the way the corresponding classical variables are used. The only justification for the names given to dynamical variables lies in the analogy to the classical theory, e. g., if one says that x, y, z are the Cartesian co-ordinates of an electron, one means only that x, y, z are q-numbers which appear in the quantum solution of the problem in an analogous way to the Cartesian co-ordinates of the electron in the classical solution. It may happen that two or more q-numbers are analogous to the same classical quantity (the analogy being, of course, imperfect and in different respects for the different q-numbers), and thus have claims to the same name. This occurs, for instance, when one considers what q-numbers shall be called the frequencies of a multiply periodic system, there being orbital frequencies and transition frequencies, either of which correspond in certain respects to the classical frequencies. In such a case one must decide which of the properties of the classical variable are dynamically the most important, and must choose the q-number which has these properties to be the corresponding quantum variable.
Abstract The laws of classical mechanics must be generalised when applied to atomic systems, the generalisation being that the commutative law of multiplication, as applied to dynamical variables, is to be replaced by certain quantum conditions, which are just sufficient to enable one to evaluate xy - yx when x and y are given. It follows that the dynamical variables cannot be ordinary numbers expressible in the decimal notation (which numbers will be called c-numbers), but may be considered to be numbers of a special kind (which will be called q-numbers), whose nature cannot be exactly specified, but which can be used in the algebraic solution of a dynamical problem in a manner closely analogous to the way the corresponding classical variables are used. The only justification for the names given to dynamical variables lies in the analogy to the classical theory, e. g., if one says that x, y, z are the Cartesian co-ordinates of an electron, one means only that x, y, z are q-numbers which appear in the quantum solution of the problem in an analogous way to the Cartesian co-ordinates of the electron in the classical solution. It may happen that two or more q-numbers are analogous to the same classical quantity (the analogy being, of course, imperfect and in different respects for the different q-numbers), and thus have claims to the same name. This occurs, for instance, when one considers what q-numbers shall be called the frequencies of a multiply periodic system, there being orbital frequencies and transition frequencies, either of which correspond in certain respects to the classical frequencies. In such a case one must decide which of the properties of the classical variable are dynamically the most important, and must choose the q-number which has these properties to be the corresponding quantum variable.
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.
Abstract The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations qrqs – qsqr =0, prps – pspr = 0 } qrqs – psqr = 0 (r ≠ s) or ih(r = s) where the p's and q's are a set of canonical variables and h is a c-number euqal to (2π)-1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [x, y] defined by xy – yz = ih [x, y] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.
Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation principle a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
Abstract Although the classical electrodynamic theory meets with a considerable amount of success in the description of many atomic phenomena, it fails completely on certain fundamental points. It has long been thought that the way out of this difficulty lies in the fact that there is one basic assumption of the classical theory which is false, and that if this assumption were removed and replaced by something more general, the whole of atomic theory would follow quite naturally. Until quite recently, however, one has had no idea of what this assumption could be. A recent paper by Heisenberg* provides the clue to the solution of this question, and forms the basis of a new quantum theory. According to Heisenberg, if x and y are two functions of the co-ordinates and momenta of a dynamical system, then in general xy is not equal to yx. Instead of the commutative law of multiplication, the canonical variables qrpr (r = 1...u) of a system of u degrees of freedom satisfy the quantum conditions, which were given by the author in the form qrqs ― qsqr = 0 prps ― pspr = 0 qrps ― psqr = 0 qrpr ― prqr = ih (r ≠ s) } (1) where i is a root of — 1 and h is a real universal constant, equal to (2π)-1 times the usual Planck’s constant. These equations are just sufficient to enable one to calculate xy — yx when x and y are given functions of the p’s and q’s, and are therefore capable of replacing the classical commutative law of multiplication. They appear to be the simplest assumptions one could make which would give a workable theory.
In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.
This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.