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

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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
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(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. (e) Let us assume that the original Hamiltonian is explicitly time-dependent, but that G ( t ) is chosen so that H is time-independent. Write an expression for | ψ ( t ) ) in terms of the eigenvalues and eigenstates of H , and the initial state | ψ (0) ) . We start from | ˙ ψ ) = i planckover2pi1 H | ψ ) (8) since H is time-independent, this has the solution | ψ ( t ) ) = e iH t/ planckover2pi1 | ψ (0) ) = summationdisplay n | ω n ) e n t ( ω n | ψ (0) ) (9) where H | ω n ) = planckover2pi1 ω n | ω n ) (f) Use the relationship between | ψ ( t ) ) and | ψ ( t ) )
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