An electron in the hydrogen molecule H can be described using two basis states,|1) and |2), in which the electron is localised near the first or the second hydrogen nucleus,

respectively. The Hamiltonian describing the electron is given by \hat{H}=E_{0}(|1\rangle\langle 1|+| 2\rangle\langle 2|)+\Delta(|1\rangle\langle 2|+| 2\rangle\langle 1|) where E, is the energy of the electron when it is localised close to one of the hydrogen nuclei, and A is the hopping energy which allows an electron to jump between the two nuclei. \text { Show that the allowed energies of the electron are given by } E_{0} \pm \Delta \text {. } ) At time t = 0, the electron is localised in the vicinity of the first hydrogen nucleus,i.e., the state of the electron is |psi(0)) = |1). Find the state of the electron at later times. Are there times when the electron is localised on the second hydrogen nucleus, i.e., in the state |2)? To simplify the calculation, you may set E, = 0. Imagine a hypothetical situation where the first nucleus is unstable and could-decay with some probability. This can be modelled by including an imaginary term in the Hamiltonian, \hat{H}=-i \Gamma|1\rangle\langle 1|+\Delta(|1\rangle\langle 2|+| 2\rangle\langle 1|) e^{t(\hat{A}+\hat{B})} \approx e^{t \hat{A}} e^{t \hat{B}} which is valid for general operators Ä and B and sufficiently small t. where I > 0 is the amplitude for the first nucleus to decay. Explain how localization of the electron depends on r. Justify why ħ/T can be interpreted as the mean decay time.