A simplified two-state model of a molecule (of the ammonia-inversion type) has

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations

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

Problem

A simplified two-state model of a molecule (of the ammonia-inversion type) has, in the basis {∣1⟩,∣2⟩}\{|1\rangle,|2\rangle\} of two localised configurations, the Hamiltonian matrix H^=(0−A−A0),A=2.0×10−6 eV.\hat H = \begin{pmatrix}0 & -A\\ -A & 0\end{pmatrix},\qquad A = 2.0\times10^{-6}\,\mathrm{eV}. At t=0t=0 the molecule is prepared in the localised state ∣1⟩|1\rangle. ((a)) Diagonalise H^\hat H: find its eigenvalues and normalised eigenvectors in the {∣1⟩,∣2⟩}\{|1\rangle,|2\rangle\} basis. ((b)) Use the spectral decomposition to build U^(t)\hat U(t) and find ∣ψ(t)⟩|\psi(t)\rangle. ((c)) Find the probability of finding the molecule still in ∣1⟩|1\rangle at t=1.0×10−10 st = 1.0\times10^{-10}\,\mathrm s.

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