Question

Problem 5: 8 points. Two spin-half fermions are in some external potential with single-particle energy levels E? and normalized wavefunctions ?? (?? is the ground state etc.). Their only interaction is through their spins, in the form H??? = ?S? · S? where S?,? are the spins of the fermions and ? a real constant. a) The fermions are in the normalized state ? = ??(r?)??(r?)? where r?,? are the coordinates of the fermions and ? is a state of their spins. Determine ? by expressing in terms of the basis states |uu?, |ud?, |du?, |dd? where, as usual |uu? is the state with both spins having z-component +?/2 etc. b) Find the energy of the state of part (a) (it is an energy eigenstate). c) The fermions are now in the normalized state ?' = (??(r?)??(r?) ? ??(r?)??(r?))?' with ?' a new state of their spins which is an eigenstate of the z-component of the total spin S? = S?? + S??. Determine the possible states ?' and give the value of S? for each. d) Find the energy of the states of part (c) and their degeneracy.

          Problem 5: 8 points. Two spin-half fermions are in some external potential with single-particle energy levels E? and normalized wavefunctions ?? (?? is the ground state etc.). Their only interaction is through their spins, in the form H??? = ?S? · S? where S?,? are the spins of the fermions and ? a real constant.
a) The fermions are in the normalized state ? = ??(r?)??(r?)? where r?,? are the coordinates of the fermions and ? is a state of their spins. Determine ? by expressing in terms of the basis states |uu?, |ud?, |du?, |dd? where, as usual |uu? is the state with both spins having z-component +?/2 etc.
b) Find the energy of the state of part (a) (it is an energy eigenstate).
c) The fermions are now in the normalized state ?' = (??(r?)??(r?) ? ??(r?)??(r?))?' with ?' a new state of their spins which is an eigenstate of the z-component of the total spin S? = S?? + S??. Determine the possible states ?' and give the value of S? for each.
d) Find the energy of the states of part (c) and their degeneracy.

Problem 5: 8 points. Two spin-half fermions are in some external potential with single-particle energy levels E? and normalized wavefunctions ?? (?? is the ground state etc.). Their only interaction is through their spins, in the form H??? = ?S? · S? where S?,? are the spins of the fermions and ? a real constant.
a) The fermions are in the normalized state ? = ??(r?)??(r?)? where r?,? are the coordinates of the fermions and ? is a state of their spins. Determine ? by expressing in terms of the basis states |uu?, |ud?, |du?, |dd? where, as usual |uu? is the state with both spins having z-component +?/2 etc.
b) Find the energy of the state of part (a) (it is an energy eigenstate).
c) The fermions are now in the normalized state ?' = (??(r?)??(r?) ? ??(r?)??(r?))?' with ?' a new state of their spins which is an eigenstate of the z-component of the total spin S? = S?? + S??. Determine the possible states ?' and give the value of S? for each.
d) Find the energy of the states of part (c) and their degeneracy.

Added by Joshua V.

Question

Please give Ace some feedback