sb3-hw02-2016

# sb3-hw02-2016 - JHU 580.429 SB3 HW2 The L operator is the...

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JHU 580.429 SB3 HW2 The L operator is the Laplace transform, L [ f ( t )] = ˜ f ( s ) = R 0 dte - st f ( t ) . The ? operator is con- volution, f ? g ( t ) = R t 0 dt 0 f ( t - t 0 ) g ( t 0 ) . 1. Standard Laplace transform proofs. (a) Prove that L [ f ? g ( t )] = ˜ f ( s ) ˜ g ( s ) . (b) Prove that L [ ˙ f ( t )] = s ˜ f ( s ) - f ( 0 ) . 2. Whence comes that 2 π ? Suppose we consider eigenfunctions of the time derivative operator that are periodic on the interval t = - T / 2 to T / 2. The eigenfunction with eigenvalue i ω is defined A ω φ ω ( t ) , where A ω is a normalization constant and φ ω ( t ) is the eigenfunction at time t . The phase of the eigenfunctions are fixed by requiring that φ ω ( 0 ) = 1. (a) Provide the function φ ω ( t ) . (b) The periodicity requirement is that φ ω ( - T / 2 ) = φ ω (+ T / 2 ) . What values of ω are permitted? (c) What is the spacing Δ ω between permitted values? This quantization of permitted frequencies is the same phenomenon as the quantization of energy levels for particles in confining potentials, for example a particle in a box or an electron confined by the positive charge of a nucleus.
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