By J.H. Jeans, M.A., F.R.S
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Extra resources for An Elementary Treatise on Theoretical Mechanics
By a proper choice of linear combinations of them, however, we can always obtain a set of orthogonal (and normalised) eigenfunctions. The spectrum of eigenvalues of the energy may be either discrete or continuous. e. one in which neither the system nor any part of it moves off to infinity. For, with eigenfunctions of a discrete spectrum, the integral J | Ψ | dq, taken overall space, is finite. This certainly means that the squared modulus | Ψ\ decreases quite rapidly, becoming zero at infinity.
24 The Basic Concepts of Quantum Mechanics §6 §6. The passage to the limiting case of classical mechanics Quantum mechanics contains classical mechanics in the form of a certain limiting case. The question arises as to how this passage to the limit is made. In quantum mechanics an electron is described by a wave function which determines the various values of its coordinates; of this function we so far know only that it is the solution of a certain linear partial differential equation. In classical mechanics, on the other hand, an electron is regarded as a material particle, moving in a path which is completely determined by the equations of motion.
CHAPTER 2 CONSERVATION LAWS IN QUANTUM MECHANICS §8. The Hamiltonian operator The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behaviour at all subsequent instants is determined (only, of course, to the degree of completeness which is generally admissible in quantum mechanics). The mathematical expression of this fact is that the value of the derivative dW/dt of the wave function with respect to time at any given instant must be determined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear.