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Download Celestial Mechanics. Part II by Shlomo Sternberg PDF

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By Shlomo Sternberg

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Sample text

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. In other words, the probability of infinite values of the coordinates is zero; that is, the system executes a finite motion, and is said to be in a bound state. For wave functions of a continuous spectrum, the integral J | Ψ\ dq diverges.

This means that any physical quantity that is conserved can be measured simultaneously with the energy. Among the various stationary states, there may be some which correspond to the same value of the energy (or, as we say, energy level of the system), but differ in the values of some other physical quantities. Such energy levels, to which several different stationary states correspond, are said to be degenerate. Physically, the possibility that degenerate levels can exist is related to the fact that the energy does not in general form by itself a complete set of physical quantities.

The description by means of the wave function, on the other hand, is a particular case of this, corresponding to a density matrix of the form q(x\ x) = Ψ*(χ')Ψ(χ). The following important difference exists between this particular case and the general one. For a state having a wave function (sometimes called a pure state) there is always a complete set of measuring processes such that they lead with certainty to definite results. For states having only a density matrix (called mixed states), on the other hand, there is no complete set of measuring processes whose result can be uniquely predicted.

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