Quantum mechanics, postulates

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

The quantities that hold the complete information, the states vectors T and 0(f), can be derived from fhe sysfem operator S2. This, in turn, means that the entire sought informatioiyis also present in or, equivalently, in U(f). The total evolution operator U(f) itself is the major physical content of C(f). Hence, fhe stated quantum-mechanical postulate on completeness implies that everything one could possibly learn about any considered system is also contained in the autocorrelation function C(f). Despite the fact that the same full information is available from T, 0(f), S2,U(f), and C(f), fhe autocorrelation functions are more manageable in practice, since they are observables. As a scalar, the quantity C(f) has a functional form fhaf is defined by its... 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

Quantum mechanics postulates that to every observable property of a system there corresponds a linear Hermitian operator. Mechanical operators having a classical analogue can be constructed when the operator forms for coordinates and momenta are known. 

Available ancient and contemporary textbooks do not explain this situation, see for a recent example reference [64]. However, the current literature presents it as a de facto characteristic feature of quantum mechanics. The usual trend is to classify this oddity within the fuzziness of quantum mechanical postulates. Other authors introduce KE expectation values throughout mathematical manipulations not exempt of difficulty [5]. 

A brief digression about the quantum mechanical postulates may be worthwhile now. Proposed since the formulation of quantum theory, the presence of quantum mechanical postulates in the literature is characterised by quite a large choice of variants. This can be evidenced by perusal of any textbook contained in the limited collection provided in references [3,58,61-72], but looking towards earlier times the situation has not changed since then [4,5]. One can find there, from no postulate description at all [3], up to quite large lists of them [67]. Even the suggestion can be found that quantum mechanical postulates should be substituted by adequate definitions instead [6]. 

So far, the following quantum-mechanical postulates have been introduced ... 

Chapter 10 gives two further quantum-mechanical postulates dealing with spin and the Pauli principle. 

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, ley, 1974, Section 11.4 L. E. Ballentine, Am. J. Phys, 55, 785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single system (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensemble) in this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys, 40,1763 (1972) Rev. Mod. Phys, 42,358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys Rep., 210,223 (1992) Problem 10.3], and this interpretation has been largely rejected. 

Quantum mechanics postulates that the present state of an undisturbed system determines its future state. Consider the special case of a system with a time-independent Hamiltonian H. Suppose it is known that at time to the state function is (x, to)- Derive Eq. (7.100) by substituting the expansion (7.65) with ft = ipi into the time-dependent Schrddinger equation (7.96) multiply the result by integrate over all space, and solve for c . 

Kaplan, I. G. (2002). Is the Pauh exclusive principle an independent quantum mechanical postulate Int. J. Quantum Chem. 89,268-276. 

The main lessons to be kept for the further theoretical and practical investigations of the quantum mechanics postulates and basic applications that are presented in the present chapter pertain to the following ... 

Born approximation Green function Heisenberg picture hydrogenic states interaction picture quantum electric resistance quantum ground level quantum mechanics postulates quantum resonance... 

Based on this commutativity, according to the postulate [P3] of the quantum mechanics, further follows that the Hamiltonian of the system and the translation operator have the same set of eigen-function P, which implies, for the translation operator, the existence of the eigen-values equation (according to the quantum mechanics postulate [PI]) ... 

Directly, without solving the Eq. (3.79), there can be said only that the associated density of probability according to the quantum mechanical postulate [P2], since associated with localization electrons in crystalline field, it should keep the same periodicity, so generating the... 

Chapter 3 Quantum Roots of Crystals and Solids) The quantum mechanics postulates are shortly reviewed for their application in providing the basic crystal Bloch theorem, further specialization to the quantum modeling of crystals in reciprocal space, on various levels of electronic behavior from free to quasi-free, to quasi-binding, to tight-binding models. 

When studying the state of a system, one typically makes various measurements of its properties, such as mass, volume, position, momentum, and energy. Each individual property is called an observable. Because quantum mechanics postulates that the state of a system is given by a wavefunction, how does one determine the value of various observables (say, position or momentum or energy) from a wavefunction ... 

There are, of course, other situations in science. We cannot understand magnetic and electric phenomena on the basis of mechanics without introducing new postulates, which are the generalization of experiments and could not be derived logically from the laws of mechanics. The introduction of new postulates is always connected to the appearance of universal constants (for electrical and magnetic phenomena, i.e., the velocity of light propagation, c). An adequate description of the laws of the microworld requires the introduction of quantum mechanical postulates and the universal constant "h" (Planck s constant). 

These several notions about the quantum world form part of a theoretical basis that is usually referred to as the set of quantum mechanical postulates. These postulates are ideas or statements that in themselves may or may not be true. However, if predictions that follow naturally from these ideas are proven true, then the ideas are accepted as true even if there is no direct proof. 

Examination of the radial functions in Table 10.1 reveals that except for the 1 = 0 functions, all are zero-valued at r = 0 (p = 0). For r > 0, each R , function has n-l-1 points where the function is zero-valued. These points are roots of the polynomials in the R i functions, and they are simply the points where the radial functions change sign. They are nodes in the wave-functions. Figure 10.1 is a plot of several of these functions. Since the quantum mechanical postulates tell us that the square of a wavefunction is the probability density, it is also interesting to notice the forms of R i, which are shown in Figure 10.2. 

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