Estimate the tunneling probability of a proton with thermal energy E ≈ kBT where T = 300K, when it encounters a barrier of height V ≈ 2E and width 1˚A(as is typically the case for proton tunneling in a hydrogen bonded system).
Let's assume a wave function of particle-in-a-ring Φ = (1 /√ 5)"(2Ψ1 + Ψ−1) (1) where Ψ1 = ( 1 √ 2π)"e{iϕ} and Ψ−1 = ( 1 √ 2π)"e(iϕ)
(i) Determine ⟨L⟩ for the particle described by Φ.
(ii) Determine ⟨E⟩ for the particle described by Φ.
(iii) What are the probabilities of finding the particle in Ψ1 and Ψ−1 eigenstates? Can your answer to (iii) explain the answers to part (i) and part (ii)?