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Unformatted text preview: Chapter 2 Foundations I: States and Ensembles 2.1 Axioms of quantum mechanics For a few lectures I have been talking about quantum this and that, but I have never defined what quantum theory is. It is time to correct that omission. Quantum theory is a mathematical model of the physical world. To char acterize the model, we need to specify how it will represent: states, observ ables, measurements, dynamics. 1. States . A state is a complete description of a physical system. In quantum mechanics, a state is a ray in a Hilbert space . What is a Hilbert space? a) It is a vector space over the complex numbers C . Vectors will be denoted  ) (Diracs ket notation). b) It has an inner product (  ) that maps an ordered pair of vectors to C , defined by the properties (i) Positivity: (  ) > 0 for  ) = 0 (ii) Linearity: (  ( a  1 ) + b  2 ) ) = a (  1 ) + b (  2 ) (iii) Skew symmetry: (  ) = (  ) * c) It is complete in the norm   = (  ) 1 / 2 1 2 CHAPTER 2. FOUNDATIONS I: STATES AND ENSEMBLES (Completeness is an important proviso in infinitedimensional function spaces, since it will ensure the convergence of certain eigenfunction expansions e.g., Fourier analysis. But mostly well be content to work with finitedimensional inner product spaces.) What is a ray? It is an equivalence class of vectors that differ by multi plication by a nonzero complex scalar. We can choose a representative of this class (for any nonvanishing vector) to have unit norm (  ) = 1 . (2.1) We will also say that  ) and e i  ) describe the same physical state, where  e i  = 1. (Note that every ray corresponds to a possible state, so that given two states  ) ,  ) , we can form another as a  ) + b  ) (the superposi tion principle). The relative phase in this superposition is physically significant; we identify a  ) + b  ) with e i ( a  ) + b  ) ) but not with a  ) + e i b  ) . ) 2. Observables . An observable is a property of a physical system that in principle can be measured. In quantum mechanics, an observable is a selfadjoint operator . An operator is a linear map taking vectors to vectors A :  ) A  ) , A ( a  ) + b  ) ) = a A  ) + b B  ) . (2.2) The adjoint of the operator A is defined by (  A ) = ( A  ) , (2.3) for all vectors  ) ,  ) (where here I have denoted A  ) as  A ) ). A is selfadjoint if A = A . If A and B are self adjoint, then so is A + B (because ( A + B ) = A + B ) but ( AB ) = B A , so AB is self adjoint only if A and B commute. Note that AB + BA and i ( AB BA ) are always selfadjoint if A and B are....
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 Fall '11
 JohnPreskill
 mechanics

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