Postulates of Quantum Mechanics and Initial Considerations
In order to begin the actual study of quantum mechanics, we shall now state a set of five postulates (or axioms), which form the basic structure of the theory. These postulates provide a conceptual and mathematical framework that is strikingly different from classical mechanics in many ways. Within it, however, explanations for apparently contradictory results of early experiments such as quantization of energy and angular momentum can be achieved. In addition, it provides a framework for interpretation of future experiments, as well as additional development of the theory itself.
KeywordsQuantum Mechanics State Vector Quantum System Hermitian Operator SchrOdinger Equation
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- 1.See A. Messiah, Quantum Mechanics, Vol. I, North Holland Publ. Co., Amsterdam, 1965, p. 8 for a more thorough discussion of this point. Also, see Quantum Theory and Beyond, edited by T. Bastin, Cambridge University Press, 1971, for a discussion of the difficulties associated with such an interpretation.Google Scholar
- 6.See J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955, for a more detailed discussion of the mathematical techniques associated with these generalizations.Google Scholar
- 7.It is possible to formulate a relativistic theory of quantum mechanics in which all of the variables (including time) appear symmetrically, i.e., they are treated on an equal footing, but such an approach is not appropriate for the discussions in this text. For a more complete discussion see, for example, P. A. M. Dirac, Quantum Mechanics, 2nd ed., Oxford University Press, Oxford, 1958.Google Scholar
- 10.There exist Hermitian operators that do not possess a complete set of eigenvectors. See, for example, A. Messiah, Quantum Mechanics, Vol. 1, John Wiley, New York, 1958, Vol. 1, p. 188. However, such operators do not correspond to observables, and hence will not be of interest to us.Google Scholar
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- 21.It is assumed here that 4,, Y, and a2 all exist, and lie in the Hilbert space. Operators such as r or ih/aq sometimes cause problems in this regard, and must be treated with care. See A. Messiah, Quantum Mechanics, Vol. 1, John Wiley, 1958, p. 169 for a discussion of this point.Google Scholar
- 31.This representation is based upon the analysis by A. Gamba, Nuovo Cimento, 7, 378 (1950).Google Scholar
- 33.See, for example, Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw Hill, New York, 1966, Chapter 7, for a discussion of the early debates that occurred on this subject.Google Scholar
- 46.For a more detailed description of such an analysis see, for example, G. Arfken, Mathematical Methods for Physicists, Academic Press, New York, 1968, Section 2. 2.Google Scholar
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