PHYSICS
851HW4_09

# 851HW4_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4 1. The 2-Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H 0 = Δ 2 ( | 2 )( 2 | − | 1 )( 1 | ) and a coupling term V = Ω * 2 | 1 )( 2 | + Ω 2 | 2 )( 1 | . (a) What is the energy, degeneracy, and state vector of the bare ground state for Δ > 0, Δ = 0, and Δ < 0? (b) Let the full Hamiltonian be H = H 0 + V . Write down the 2x2 Hamiltonian matrix in the {| 1 ) , | 2 )} basis and then compute the ‘dressed-state’ energy levels for the case Ω negationslash = 0. Use ω g for the lowest eigenvalue, and ω e for the highest (in energy). (c) Following the method shown in lecture (i.e. treating positive and negative detunings separately, and matching the limiting values of the dressed and bare eigenstates in the limits | Δ | → ∞ ), determine the normalized dressed-state eigenvectors. Label the state corresponding to ω g as | g ) and the other state as | e ) . Using Dirac notation, express the Full Hamiltonian as an operator in terms of the kets | g ) and | e ) and the corresponding bras, and then again using the kets | 1 ) and | 2 ) and the corresponding bras. (d) Sketch the energy spectrum versus Ω for the case of fixed Δ > 0. What are ω g and ω e at Ω = 0? What are the corresponding dressed states. What are the limiting values of ω g

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