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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 Schrdinger equation in the v space (i.e., the current i operator is expressed in the derivative of v ) and then in i space. (6 pts) (b) Why is there a transform, similar to the Fourier transform, between v and i ? 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 = , is P Hermitian? Does P commute with the Hamiltonian? (6 pts) (e) Find P of the ground state? Of the first excited state? Use the promoter and demoter to simplify your math (6 pts) (f) What is ) ( t P when (v,0) has an equal probability of the ground the first excited state? (10 pts) What is P 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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