851HW4_Solutions09

# 851HW4_Solutions09 - PHYS851 Quantum Mechanics I Fall 2009...

This preview shows pages 1–3. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4: Solutions 1. The 2-Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H = 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? For Δ > 0, the energy of the ground state is − Δ / 2, the degeneracy is 1, and the state vector is | 1 ) . For Δ = 0, the energy of the ground state is 0, the degeneracy is 2 and the degenerate subspace is {| 1 ) , | 2 )} . For Δ < 0, the energy of the ground state si Δ / 2 = −| Δ | / 2, the degeneracy is 1, and the state vector is | 2 ) . (b) Let the full Hamiltonian be H = H + 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). The matrix representation of H in the {| 1 ) , | 2 )} basis is: H = parenleftbigg − Δ 2 Ω * 2 Ω 2 Δ 2 parenrightbigg (1) The characteristic equation is then: det | H − ωI | = − parenleftbigg Δ 2 + ω parenrightbiggparenleftbigg Δ 2 − ω parenrightbigg − | Ω | 2 4 = ω 2 − 1 4 ( Δ 2 + | Ω | 2 ) = 0 (2) the solutions are then ω g = − 1 2 radicalbig Δ 2 + | Ω | 2 (3) ω e = 1 2 radicalbig Δ 2 + | Ω | 2 (4) (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. The eigenvalue equation is ( H − ωI ) | ω ) = 0. Hitting this with ( 1 | and inserting the projector I = | 1 )( 1 | + | 2 )( 2 | , then doing the same for ( 2 | , gives ( ( 1 | H | 1 )− ω ) ( 1 | ω ) + ( 1 | H | 2 )( 2 | ω ) = 0 (5) ( 2 | H | 1 )( 1 | ω ) + ( ( 2 | H | 2 ) − ω ) ( 2 | ω ) = 0 (6) putting in the matrix elements and multiplying by 2 gives − (Δ + 2 ω ) ( 1 | ω ) + Ω * ( 2 | ω ) = 0 (7) Ω ( 1 | ω ) + (Δ − 2 ω ) ( 2 | ω ) = 0 (8) The first equation gives, before normalization, | ω ) = Ω * | 1 ) + (Δ + 2 ω ) | 2 ) . (9) The second gives, | ω ) = (Δ − 2 ω ) | 1 )− Ω | 2 ) . (10) 1 For positive detuning, Δ > 0, according to our answer for (a), we want lim Ω → ( 2 | g ) = 0 and lim Ω → ( 1 | e ) → 0, thus we should use (10) for | ω g ) and (9) for | ω e ) . This gives | g ) = (Δ + radicalbig Δ 2 + | Ω | 2 ) | 1 ) − Ω | 2 ) radicalBig (Δ + radicalbig Δ 2 + | Ω | 2 ) 2 + | Ω | 2 (11) | e ) = Ω * | 1 ) + (Δ + radicalbig Δ 2 + | Ω | 2 ) | 2 ) radicalBig...
View Full Document

## This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at University of Wisconsin.

### Page1 / 7

851HW4_Solutions09 - PHYS851 Quantum Mechanics I Fall 2009...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online