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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern