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hw07sol - Introduction to Quantum and Statistical Mechanics...

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Unformatted text preview: Introduction to Quantum and Statistical Mechanics Homework 7 Solution Prob. 7.1 For a quantum LC circuit: (a) Write down the time-dependent Schrödinger equation in the v ˆ space (i.e., the current i ˆ operator is expressed in the derivative of v ) and then in i ˆ space. (6 pts) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 i E i di d C L i Li v E v Cv v dv d LC ϕ ϕ ϕ ψ ψ ψ =- = +- 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 F F (b) Why is there a transform, similar to the Fourier transform, between v ˆ and i ˆ ? Write down that transform when C → 0. (6 pts) We can see the exact analog between the ( x, k ) and ( v, i ), and therefore the Fourier transform between ( x, k ) can be directly applied to ( v, i ). Cv x Li k → → ; F ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ∫ ∫ ∞ ∞- ∞ ∞-- = = dv jiv v f i F di jiv i F v f exp 2 exp π , where we have used i as current, and 1- = j . (c) Write down the promoter and demoter expressions in v and d/dv , and then in i and d/di . (6 pts) + = - = ⊥ dv d C v C a dv d C v C a o o o o ϖ ϖ ϖ ϖ 2 1 ˆ 2 1 ˆ + = - = ⊥ ⊥ i L di d L i a i L di d L i a o o o o ϖ ϖ ϖ ϖ 2 ˆ 2 ˆ (d) If we define the “power operator” as v i P ˆ ˆ ˆ = , is P ˆ Hermitian? Does P ˆ commute with the Hamiltonian? (6 pts) 1 From the definition of the Hermitian operator, we can see that v i P ˆ ˆ ˆ = is not Hermitian as: ( 29 ∫ ∫ ∞ ∞- ∞ ∞- =...
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hw07sol - Introduction to Quantum and Statistical Mechanics...

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