4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square well of length L. Construct the state vector and determine the energy for the ground-state of the two-particle system assuming the particles are non-interacting. i. ii. Determine the first excited state energy of the two-particle system and give all its state vectors, again assuming the particles are non-interacting. What is the degeneracy of the first excited state? ji. Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k is positive and small. Discuss quantitatively how the ground and first-excited state energies differ from the case for non-interacting particles. 5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the hydrogenic wave functions adapted to the one-electron atom to determine the dependence of the ground state energy of the atom on ry.
4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square well of length L. Construct the state vector and determine the energy for the ground-state of the two-particle system assuming the particles are non-interacting. i. ii. Determine the first excited state energy of the two-particle system and give all its state vectors, again assuming the particles are non-interacting. What is the degeneracy of the first excited state? ji. Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k is positive and small. Discuss quantitatively how the ground and first-excited state energies differ from the case for non-interacting particles. 5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the hydrogenic wave functions adapted to the one-electron atom to determine the dependence of the ground state energy of the atom on ry.
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Transcribed Image Text:4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square
well of length L.
Construct the state vector and determine the energy for the ground-state of the
two-particle system assuming the particles are non-interacting.
i.
ii.
Determine the first excited state energy of the two-particle system and give all its
state vectors, again assuming the particles are non-interacting. What is the
degeneracy of the first excited state?
ji.
Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k
is positive and small. Discuss quantitatively how the ground and first-excited state
energies differ from the case for non-interacting particles.
5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of
radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of
the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the
hydrogenic wave functions adapted to the one-electron atom to determine the dependence
of the ground state energy of the atom on ry.
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