Time Dependent Quantum Mechanical Approach Case Studies Of Ammonia Molecule

The aim of this thesis work is to implement time dependent Quantum mechanicalrnapproaches to study inversion phenomena of ammonia Molecule. This approachesrnwill enable us visualize the time evolution of the orientation of the molecule inside thernsymmetric double well potentials. To study this approach diffusion Quantum MonternCarlo(DQMC) method, which propagates the time dependent Schr¨odinger equationrnin imaginary time, had been used.rnSince solving the schr¨odinger equation exactly with double well potential is not possible,rnnumerical approach had been used. In order to get each energy eigenstates,rnthe time independent Schr¨odinger equation with symmetric double well potential hadrnbeen descritised and changed in to eigenvalue problem. The QL and QR algorithmrnhad been used to solve the eigenvalue problem giving us energy eigenstates. Thernwave functions corresponding to the eigen energy states had also been obtained. Inrnaddition to this, the quantum mechanical energy splitting effect had been studied forrndifferent symmetrical double well potentials.rnThe diffusion function will be obtained as a linear combination of the product ofrnwave function of the time independent eigenfunctions which decay exponentially inrnreal time. The probability density, which shows where the Nitrogen atom is mostrnprobably to be found, being the square of the diffusion function. The full cycle of thernammonia‘s orientation inside the symmetric double well had been done for differentrntime steps.rnFinally comparison between theoretical and computational results had been done forrnviiirndifferent time steps. The simulations shows that similar results had been found inrnboth theoretical and computational approaches

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