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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 | ψ ) (Dirac’s ket notation). b) It has an inner product ( ψ | ϕ ) that maps an ordered pair of vectors to C , defined by the properties (i) Positivity: ( ψ | ψ ) > 0 for | ψ ) = 0 (ii) Linearity: ( ϕ | ( a | ψ 1 ) + b | ψ 2 ) ) = a ( ϕ | ψ 1 ) + b ( ϕ | ψ 2 ) (iii) Skew symmetry: ( ϕ | ψ ) = ( ψ | ϕ ) * c) It is complete in the norm || ψ || = ( ψ | ψ ) 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 we’ll 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 ( ψ | ψ ) = 1 . (2.1) We will also say that | ψ ) and e iα | ψ ) describe the same physical state, where | e iα | = 1. (Note that every ray corresponds to a possible state, so that given two states | ϕ ) , | ψ ) , we can form another as a | ϕ ) + b | ψ ) (the “superposi- tion principle”). The relative phase in this superposition is physically significant; we identify a | ϕ ) + b | ϕ ) with e iα ( a | ϕ ) + b | ψ ) ) but not with a | ϕ ) + e iα b | ψ ) . ) 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 : | ψ ) → A | ψ ) , A ( a | ψ ) + b | ψ ) ) = a A | ψ ) + b B | ψ ) . (2.2) The adjoint of the operator A is defined by ( ϕ | A ψ ) = ( A † ϕ | ψ ) , (2.3) for all vectors | ϕ ) , | ψ ) (where here I have denoted A | ψ ) as | A ψ ) ). 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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This note was uploaded on 01/24/2012 for the course PHYS 219 taught by Professor Johnpreskill during the Fall '11 term at Caltech.
- Fall '11