851HW5_09 (1) - PHYS851 Quantum Mechanics I Fall 2009...

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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) (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 )....
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851HW5_09 (1) - PHYS851 Quantum Mechanics I Fall 2009...

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