A particle is moving in one dimension between x a and x b and is described in

A particle is moving in one dimension between x a and

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A particle is moving in one dimension between x = a and x= b , and is described in this region by a wavefunction ψ x ( ) = A / x , where A is a normalization constant. (a) Evaluate the normalization constant to ensure that the square of the wavefunction describes the probability distribution, P ( x ) of the particle. (b) Using the probability distribution, evaluate the mean value of the position of the particle. You should obtain x = ab b a ( ) ln b a ( ) 6. Modeling electrons in a molecule with the particle in a box. (a) A very simple model of an electron in a chemical bond is to imagine that it is approximately an electron in a box of a typical bondlength, such as 2 Å. Using the value, compute the energy difference between the ground (n=1) and excited (n=2) levels, in eV and kcal/mol. (b) Similarly one can model the 4 delocalized π electrons in a conjugated molecule, such as butadiene, C 4 H 6 , as occupying the particle in a box levels. Using the Aufbau principle, indicate which levels are occupied and then describe what levels are involved in the lowest energy excitation that is possible. Estimate the energy of this excitation (in eV) by making a reasonable definition of the length of the box. The particle in a box model has been used to understand the chain length dependence of the colors of dye molecules (though more accurate methods are now available!).
• Summer '16
• Alistair Sinclair

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