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hw07 - Introduction to Quantum and Statistical Mechanics...

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Unformatted text preview: Introduction to Quantum and Statistical Mechanics Homework 7 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) (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) (c) Write down the promoter and demoter expressions in v and d/dv , and then in i and d/di . (6 pts) (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) (e) Find P ˆ of the ground state? Of the first excited state? Use the promoter and demoter to simplify your math (6 pts) (f) What is ) ( ˆ t P when ψ (v,0) has an equal probability of the ground the first excited state? (10 pts) What is P ˆ ∆ at t = 0 ? What is ∆ v ∆ i at t = 0 ? (10 pts) Prob. 7.2 For a simple harmonic oscillator where the initial wave function is a superposition of the two...
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