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Electrons in two atom systems (needed to understand solids)
1. Choose Setup: Well Array (Square), Mouse = Set Eigenstate. Move the slider so Well Count = 1. Choose energy, position, values/dimensions, state phasors, expectation values, uncertainties, wavefunction – probabilities+phase.
(a) Note the energies of the bound states and sketch the amplitude of the wavefunctions of the states (the eigenstates).
(b) Compare the probability of observing the electron outside the well when it is in states 1 and 2 (with state 1 being the lowest-energy state).
2. Assume that this potential well represents the potential energy of electrons near the positively charged core of an individual atom of a particular element.
(a) If the atom has one non-core electron what is its energy when the atom is in its normal state?
(b) If there are 2 non-core electrons, what are their energies? (Hint: use Pauli’s exclusion principle as applied to multiple electrons in one dimensional rectangular wells)
(3) If there are 3 non-core electrons what are their energies?
3. Change the Well Count to 2. The potential energy curve is a simple model of the potential energy of a diatomic molecule, the two atoms of which are held together by what is known as covalent bonding. This means there are electrons, in addition to the core electrons around the nucleus of each atom, which move between the two atoms.
(a) How many bound quantum states are there for this system?
(b) What is the energy of the lowest-energy state? Sketch the amplitude of the wave function for this state.
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4. Determine the energies of states 2, 3 and 4 and write about how the shapes of wave functions for these states are related to those for a single well. Which of the states have a higher probability of being between the atoms? Can you speculate as to how this relates to bonding?
5. Assume each atom has one non-core electron.
(a) What are the energies of the two non-core electrons of the molecule?
(b) Compare these energies with the energy of one non-core electron in a single atom and hence of the “covalent bond” of this molecule
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