# 10_580_hw_6 - Physics 580 Handout 10 5 October 2010 Quantum...

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Unformatted text preview: Physics 580 Handout 10 5 October 2010 Quantum Mechanics I webusers.physics.illinois.edu/ ∼ goldbart/ Homework 6 Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) Periodic functions : In this question we shall consider the LVS formed by complex square-integrable functions on the interval [0 , 2 π ] with periodic boundary conditions f (0) = f (2 π ) , f ′ (0) = f ′ (2 π ) , where f ′ denotes df/dx . By square integrable we mean that ∫ 2 π dx | f ( x ) | 2 < ∞ . Answer, with explanations, the following questions. a) Is the function f ( x ) = exp( ix/ 2) in this space? b) Is the function f ( x ) = x (2 π − x ) in this space? c) Write an arbitrary ket | f ⟩ in this space in terms of the basis {| x ⟩} and the components f ( x ). d) Evaluate ⟨ x | f ⟩ in terms of f ( x ). For an operator Ω to be hermitean on a certain infinite-dimensional linear vector: (i) Ω must be formally self-adjoint, i.e. , up to boundary terms it must satisfy ⟨ g | Ω | f ⟩ = ⟨ f | Ω | g ⟩ ∗ ; and (ii) the boundary conditions must be self-adjoint, so that both functions f and g come from the same function space ( i.e. , satisfy the same boundary conditions). Boundary condi- tions are said to be self-adjoint if their application to f , together with the demand that the boundary term vanishes, obliges g to satisfy identical boundary conditions. We now apply these ideas to the operator T ....
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10_580_hw_6 - Physics 580 Handout 10 5 October 2010 Quantum...

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