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
Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 5 1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven twolevel system onto the timeindependent Rabi model. Here we will investigate how this change of variables can be treated more formally as a unitary transformation. Unitary operators are those which, when acting on (transforming) any state, always preserve the norm of the state. Any Hermitian operator, G can be used to generate a unitary transformation, via the Unitary operator U G = e iG . The Unitary transformation is then defined by  ( t ) ) = U G  ( t ) ) , where  ( t ) ) is the original statevector, and  ( t ) ) is the state vector in the new frame of reference. For the case of a timedependent Hamiltonian, H ( t ) and a timedependent generator G ( t ), we would like to determine the effective Hamiltonian, H ( t ), which governs the evolution of the state  ( t ) ) . (a) Begin by differentiating both sides of the equation  ( t ) ) = U G ( t )  ( t ) ) with respect to time. Use Schrodingers equation to eliminate d dt  ( t ) ) . (Tip: keep in mind that in general [ H ( t ) ,G ( t )] negationslash = 0)  ) = U G  ) + U G  ) (1) = U G  ) i planckover2pi1 U G H  ) (2) (b) The effective Hamiltonian in the new frame of reference must satisfy the equation: i planckover2pi1 d dt  ( t ) ) = H ( t )  ( t ) ) . Use the fact that U G U G = I , and your result from 1a, to give an expression for H ( t ) in terms of H ( t ) and G ( t ). d dt  ) = U G U G ( U G  ) ) i planckover2pi1 U G HU G ( U G  ) ) (3) = i planckover2pi1 bracketleftBig U G HU G + i planckover2pi1 U G U G bracketrightBig  ) (4) Thus we see that H = U G HU G + i planckover2pi1 U G U G (5) (c) What is H ( t ) in the special case where G is not explicitly timedependent? What is H in the case where H and G are both timeindependent and [ H,G ] = 0? If G is not timedependent, then U G = 0, so that H = U G HU G (6) If [ H,G ] = 0, then it follows that [ U G ,H ] = 0, so that H = U G HU G = HU G U G = H (7) 1 (d) By definition, H ( t ) negationslash = H ( t ) is defined as the energy operator. In general, would it be safe to assume that the eigenstates of H ( t ) are the energy eigenstates of the system? No, it would not be a safe assumption, because H is not just a unitary transformation on H , due to the addition of the U G term. Thus H and H will likely not have the same spectrum....
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 Wisconsin.
 Spring '11
 Smith
 Physics, Work

Click to edit the document details