Two configurations of a small molecule are described by an orthonormal basis

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

The Structure of Quantum Mechanics, Ch Observables, spectra and the spectral theorem

Problem

Two configurations of a small molecule are described by an orthonormal basis {∣1⟩,∣2⟩}\{|1\rangle,|2\rangle\}, and tunnelling between them is described by the Hamiltonian matrix H=(E0−A−AE0)H=\begin{pmatrix}E_0 & -A\\ -A & E_0\end{pmatrix} in this basis, with E0=2.0×10−4 eVE_0=2.0\times10^{-4}\,\text{eV} and A=5.0×10−5 eVA=5.0\times10^{-5}\,\text{eV} real constants, A>0A \gt 0. The molecule starts, at t=0t=0, in configuration ∣1⟩|1\rangle. (a) Find the eigenvalues EsE_s (symmetric) and EaE_a (antisymmetric) of H^\hat H and their normalised eigenvectors, and give the spectral projectors P^s,P^a\hat P_s,\hat P_a as matrices in the {∣1⟩,∣2⟩}\{|1\rangle,|2\rangle\} basis. (b) Use the spectral theorem to write H^=EsP^s+EaP^a\hat H=E_s\hat P_s+E_a\hat P_a and confirm by matrix multiplication that this reproduces HH. (c) Construct U^(t)=e−iH^t/ℏ=e−iEst/ℏP^s+e−iEat/ℏP^a\hat U(t)=e^{-i\hat Ht/\hbar}=e^{-iE_st/\hbar}\hat P_s+e^{-iE_at/\hbar}\hat P_a and show that the probability of finding the molecule in configuration ∣2⟩|2\rangle at time tt is P1→2(t)=sin⁡2(At/ℏ)P_{1\to2}(t)=\sin^2(At/\hbar). (d) Check this independently, writing H^=E01−Aσ^x\hat H=E_0\mathbb1-A\hat\sigma_x and using e−i(a1+bn^⋅σ⃗)t/ℏ=e−iat/ℏ[cos⁡(bt/ℏ)1−isin⁡(bt/ℏ)n^⋅σ⃗]e^{-i(a\mathbb1+b\hat n\cdot\vec\sigma)t/\hbar}=e^{-iat/\hbar}\left[\cos(bt/\hbar)\mathbb1-i\sin(bt/\hbar)\hat n\cdot\vec\sigma\right] for real a,ba,b and unit vector n^\hat n. (e) Find the smallest time t>0t \gt 0 at which P1→2(t)=1P_{1\to2}(t)=1, and give this time in seconds.

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