chap2 - Chapter 2 Foundations I: States and Ensembles 2.1...

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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 infinite-dimensional function spaces, since it will ensure the convergence of certain eigenfunction expansions e.g., Fourier analysis. But mostly well be content to work with finite-dimensional 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 self-adjoint 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 self-adjoint 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 self-adjoint if A and B are....
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chap2 - Chapter 2 Foundations I: States and Ensembles 2.1...

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