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chapter6 - Chapter 4 MANY PARTICLE SYSTEMS The postulates...

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Chapter 4 MANY PARTICLE SYSTEMS The postulates of quantum mechanics outlined in previous chapters include no restrictions as to the kind of systems to which they are intended to apply. Thus, although we have considered numerous examples drawn from the quantum mechanics of a single particle, the postulates themselves are intended to apply to all quantum systems, including those containing more than one and possibly very many particles. Thus, the only real obstacle to our immediate application of the postulates to a system of many (possibly interacting) particles is that we have till now avoided the question of what the linear vector space, the state vector, and the operators of a many-particle quantum mechanical system look like. The construction of such a space turns out to be fairly straightforward, but it involves the forming a certain kind of methematical product of di¤erent linear vector spaces, referred to as a direct or tensor product . Indeed, the basic principle underlying the construction of the state spaces of many-particle quantum mechanical systems can be succinctly stated as follows: The state vector j à i of a system of N particles is an element of the direct product space S ( N ) = S (1) - S (2) - ¢ ¢ ¢ - S ( N ) formed from the N single-particle spaces associated with each particle. To understand this principle we need to explore the structure of such direct prod- uct spaces. This exploration forms the focus of the next section, after which we will return to the subject of many particle quantum mechanical systems. 4.1 The Direct Product of Linear Vector Spaces Let S 1 and S 2 be two independent quantum mechanical state spaces, of dimension N 1 and N 2 , respectively (either or both of which may be in…nite). Each space might represent that of a single particle, or they may be more complicated spaces, each associated with a few or many particles, but it is assumed that the degrees of mechanical freedom represented by one space are independent of those represented by the other. We distinguish states in each space by superscripts. Thus, e.g., j à i (1) represents a state in S 1 and j Á i (2) a state of S 2 . To describe the combined system we now de…ne a new vector space S 12 = S 1 - S 2 (4.1) of dimension N 12 = N 1 £ N 2 which we refer to as the direct or tensor product of S 1 and S 2 . Some of the elements of S 12 are referred to as direct or tensor product states , and are
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124 Many Particle Systems formed as a direct product of states from each space. In other words, from each pair of states j à i (1) 2 S 1 and j Á i (2) 2 S 2 we can construct an element j Ã; Á i ´ j à i (1) - j Á i (2) = j à i (1) j Á i (2) 2 S 12 (4.2) of S 12 ; in which, as we have indicated, a simple juxtaposition of elements de…nes the tensor product state when there is no possibility of ambiguous interpretation. By de…nition, then, the state j Ã; Á i represents that state of the combined system in which subsystem 1 is de…nitely in state j à i (1) and subsystem 2 is in state j Á i (2) : The linear vector space S 12
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