851HW5_09Solutions - PHYS851 Quantum Mechanics I, Fall 2009...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
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 5 1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven two-level system onto the time-independent 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 | ψ ′ ( t ) ) = U G | ψ ( t ) ) , where | ψ ( t ) ) is the original state-vector, and | ψ ′ ( t ) ) is the state vector in the new ‘frame of reference’. For the case of a time-dependent Hamiltonian, H ( t ) and a time-dependent generator G ( t ), we would like to determine the effective Hamiltonian, H ′ ( t ), which governs the evolution of the state | ψ ′ ( t ) ) . (a) Begin by differentiating both sides of the equation | ψ ′ ( t ) ) = U G ( t ) | ψ ( t ) ) with respect to time. Use Schr¨odinger’s equation to eliminate d dt | ψ ( t ) ) . (Tip: keep in mind that in general [ H ( t ) ,G ( t )] negationslash = 0) | ˙ ψ ′ ) = ˙ U G | ψ ) + U G | ˙ ψ ) (1) = ˙ U G | ψ ) − i planckover2pi1 U G H | ψ ) (2) (b) The effective Hamiltonian in the new ’frame of reference’ must satisfy the equation: i planckover2pi1 d dt | ψ ′ ( t ) ) = H ′ ( t ) | ψ ′ ( 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 | ψ ′ ) = ˙ U G U † G ( U G | ψ ) ) − i planckover2pi1 U G HU † G ( U G | ψ ) ) (3) = − i planckover2pi1 bracketleftBig U G HU † G + i planckover2pi1 ˙ U G U † G bracketrightBig | ψ ′ ) (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 time-dependent? What is H ′ in the case where H and G are both time-independent and [ H,G ] = 0? If G is not time-dependent, 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 12/02/2010 for the course PHYSICS 851 taught by Professor M.moore during the Fall '09 term at Michigan State University.

Page1 / 9

851HW5_09Solutions - PHYS851 Quantum Mechanics I, Fall 2009...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online