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02 - Physics 6210/Spring 2007/Lecture 2 Lecture 2 Relevant...

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Physics 6210/Spring 2007/Lecture 2 Lecture 2 Relevant sections in text: § 1.2 Quantum theory of spin 1/2 We now try to give a quantum mechanical description of electron spin which matches the experimental facts described previously. Let us begin by stating very briefly the rules of quantum mechanics. We shall show what they mean as we go along. But it is best to know the big picture at the outset. Rule 1 Observables are represented by self-adjoint operators on a (complex) Hilbert space H . Rule 2 States are represented by unit vectors in H . The expectation value A of the observable A in the state | ψ is given by the diagonal matrix element A = ψ | A | ψ . Rule 3 Time evolution is a continuous unitary transformation on H . We will now use Rules 1-2 to create a model of a spin 1/2 particle. We will not need Rule 3 for a while (until Chapter 2). We suppose that a spin 1/2 system is completely described by its spin observable S , which defines a vector in 3-d Euclidean space. As such, S is really a collection of 3 observables, which we label as usual by S x , S y , S z , each of which is to be a (self-adjoint) linear operator on a (Hilbert) vector space. We have seen that the possible outcomes of a measurement of any component of S is ± ¯ h/ 2. As we will see, because the set of possible outcomes of a measurement of one these observables has two values, we should build our Hilbert space of state vectors to be two-dimensional. A two dimensional Hilbert space* is a complex vector space with a Hermitian scalar product. Let us explain what all this means.
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