Take the piecewise Hamiltonian for and for

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

Take the piecewise Hamiltonian H^(t)=ℏωσ^x\hat H(t)=\hbar\omega\hat\sigma_x for 0≤t<τ0\le t<\tau and H^(t)=ℏωσ^z\hat H(t)=\hbar\omega\hat\sigma_z for τ≤t<2τ\tau\le t<2\tau. Someone writes down the second-order term of the time-ordered exponential U^(2τ,0)\hat U(2\tau,0) as follows:"The Dyson series to second order is U^(2)=12(−iℏ)2(∫02τH^(t′) dt′)2\hat U^{(2)}=\frac12\left(-\frac{i}{\hbar}\right)^2\left(\int_0^{2\tau}\hat H(t')\,dt'\right)^2, because squaring the total time integral of H^\hat H already accounts for both time orderings, just as (a+b)2(a+b)^2 automatically contains all cross terms when multiplying out an ordinary product." ((a)) Find the error in this reasoning and state, in words, what has been dropped. ((b)) Evaluate the quoted (faulty) expression and the correctly time-ordered second-order term U^(2)=(−iℏ)2∫02τdt1∫0t1dt2 H^(t1)H^(t2)\hat U^{(2)}=\left(-\frac{i}{\hbar}\right)^2\int_0^{2\tau}dt_1\int_0^{t_1}dt_2\,\hat H(t_1)\hat H(t_2) for the stated H^(t)\hat H(t), and identify exactly which term is missing from the faulty version.

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