PHY4221Fall05_Notes1

PHY4221Fall05_Notes1 - PHY4221 Quantum Mechanics I Fall...

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PHY4221 Quantum Mechanics I Fall 2004 1. Fundamental Principles of Quantum Mechanics The underlying structure of the quantum theory is based upon the representation of states of a system which are to be represented by vectors in some suitably chosen vector space. The measurement of the attributes of these states ( dynamcical variables or observables ) are described in terms of operations on these vectors. These operations are assumed to be linear. The operation of an operator on a vector is intended to describe a physical operation on the system. The relation between the mathematical operation and the physical measurement is assumed to be the following: The result of a measurement of a dynamical variable must be an eigenvalue of the linear operator representing that dynamical variable. The state in which the dynamical variable has that value is represented by the corresponding eigenvector. 1.1 Postulates of Quantum Mechanics The above statement is, of course, an imprecise one, and raises many questions. For instance, what is the vector space used to describe a given physical system? How are the operators representing given dynamical variables to be chosen? How does one describe the time evolution of a dynamical system? These and many other questions must be answered if we are to formulate a complete working theory. Here we shall attempt a description of the Dirac formalism of quantum mechanics, which rests on four postulates: Postulate No.1 : The state of a given physical system at any given instant t 0 is defined by the data of a ket | ψ ( t 0 ) i , termed quantum state and belonging to a linear complex vector space called Hilbert space H . 1. Since states of a system to be represented by vectors, the vector space must be determined by the totality of attributes of the system. This is, of course, a function of our understanding, and must be modified from time to time in 1
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the light of discovery. This point may be readily illustrated by reference to the phenomenon of electron spin. Postulate No.2 : Any measurable physical property A is described by a Her- mitian operator A , termed observable and acting in H . 1. It is thus possible to associate with the position of a particle the position operator x j ( j = 1 , 2 , 3 ) and with its momentum the momentum operator p j . The space of the states H is totally characterized by the knowledge of all the algebraic relations between the members of the complete set of linear operators acting in H . This set is such that none of its members can commute with all the others but, rather, only with multiples of the identity operator. Commutation relations [ x j , x k ] , [ p j , p k ] and [ x j , p k ] , for example, totally characterize the space of the states of a particle without internal variables (spin, electric charge, etc.).
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