Assignment 1

(Due 10/02/2019)

1.1 Positron, a fundamental anti-particle predicted by Dirac, has a mass that is same as that of electron ($m_e$) but it has an opposite charge ($+e$). Positrons are emitted by Flouride-18 and used in commercial imaging for patients. Positronium is a hydrogen atom analogue where the proton is substituted by a positron. Choose a coordinate system of your choice and a) Write down the non-relativistic Hamiltonian for positronium, b) Will this system be stable? If so, derive a formula for its energy levels, c) In the limit of infinite mass of positron, what is the formula for energy levels of the system, d) Why is there a difference in the energies for the ground-states of case b) and c). (10 pts)

1.2 For a hydrogen atom with only 3 hydrogenic basis functions — 1s, 2s, 2p $_x$ — write down the 1-electron, 2-electron, and 3-electron basis functions which are obtained via direct products. Assuming that the hydrogenic basis functions are orthogonal, show that the 2-electron Hilbert-space basis functions are also orthogonal. Use Dirac notation. (5 pts)

1.3 If the unknown nuclear-electron wavefunction is expressed as: $$\Psi(\mathbf{r},\mathbf{R}) = \sum_{k=1,2;i=1,2,3,4} c_{ki}\Psi_k(\mathbf{r};\mathbf{R})\chi_{ki} (\mathbf R) $$ where $\Psi_k(\mathbf{r};\mathbf{R})$ is the $k^\text{th}$ eigenfunction of $\hat{H}^{el}$ and $\chi_{ki} (\mathbf R)$ is the $i^\text{th}$ eigenfunction of nuclear Schrodinger equation for electronic surface $k$ within BO approximation. For the cases of (i) Exact solution and (ii) Adiabatic approximation, what is the matrix equation dimensionality that one needs to solve to obtain the $c_{ki}$. For block diagonal matrices, provide the dimensions of individual blocks. (5 pts)

1.4 The 1-D quartic oscillator Hamiltonian (in a.u.) $$ \hat{H}^{QO} = -\frac{d}{2\mu dx^2} + px^{4}$$ provides a reasonable description of the bending modes and out-of-plane deformations of a number of molecules, however its exact eigenfunctions are unknown. An approximate solution is to device trial wavefunctions ($\tilde \Psi$)that are linear combinations of harmonic oscillator functions ($\psi_i$) $$ \vert \tilde \Psi \rangle = \sum_{i=0}^{N}c_i\vert \psi_i \rangle \quad \langle\psi_i\vert\psi_j \rangle = \delta_{ij}$$ Using linear variational approach, find the approximate energies of the states using the lowest energetic 1, 3 and 5 harmonic oscillator functions. Discuss the variations in the approximate ground and excited energies of the oscillators as the number of functions in the above expansion is increased. To diagonalize the matrices you can make use of any mathematical software or online tools. Hint: only the following QO Hamiltonian matrix elements, expressed in units of $\left(\frac{p}{32\mu^2}\right)^{\frac{1}{3}}$, in the harmonic basis, are non-zero $$H^{QO}_{i,i} = 3i^2 + 5i + \frac{5}{2} $$ $$H^{QO}_{i,i-2} = 2(i-1)\sqrt{i(i-1)}$$ $$H^{QO}_{i,i-4} = \frac{1}{2}\sqrt{i(i-1)(i-2)(i-3)} ;\text{. (10pts)}$$

1.5 In the class, using the BO basis, we had derived an expression for the first-order non-adiabatic coupling matrix element $T’_{aa’}$. Within same BO basis, show that $T’_{aa}$, the adiabatic matrix element, is vanishing. (5 pts)