68111116-3-1-Postulates

# 68111116-3-1-Postulates - Foundations of Quantum Mechanics...

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Foundations of Quantum Mechanics* The whole of quantum mechanics can be expressed in terms of a small set of postulates. This chapter introduces the postulates and illustrates how they are used. Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm. Quantum mechanics is a very powerful theory which has led to an accurate description of the micro-physical mechanisms. It is founded on a set of postulates from which the main processes pertaining to its application domain are derived. A challenging issue in physics is therefore to exhibit the underlying principles from which these postulates might emerge. The set of statements we find in the literature as ‘postulates’ or ‘principles’ can be split into three subsets: the main postulates which cannot be derived from more fundamental ones, the secondary postulates which are often presented as ‘postulates’ but can actually be derived from the main ones, and then statements often called ‘principles’ which are well known to be as mere consequences of the postulates. Main postulates (1) State or Wavefunction . Each physical system is described by a state function which determines all can be known about the system. The coordinate realization of this state function, the wavefunction (uppercase psi) plays a central role in quantum mechanics. Two wavefunctions represent the same state if they differ only by a phase factor. The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function. We shall see that the wavefunction of a system will be specified by a set of labels called quantum numbers, and may then be written ca,b, . . . , where a, b, . . . are the quantum numbers. The values of these quantum numbers specify the wavefunction and thus allow the values of various physical observables to be calculated. (2) Equation for the wave function (Schrödinger equation) . The time evolution of the wavefunction of a non-relativistic physical system is given by the time-dependent Schrödinger equation where the Hamiltonian is a linear Hermitian operator, whose expression is constructed from the correspondence principle. The time independent form of the equation can be written as: We shall have a great deal to say about the Schrödinger equation and its solutions in the rest of the text. (3)

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68111116-3-1-Postulates - Foundations of Quantum Mechanics...

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