Two bosons in a well. Two particles of mass m are in the spherical potential
well
V(r)={(0,r<=r_(0)),(infty ,r>r_(0)):}
where r_(0) is a positive constant. Assume the two particles do not interact, so
that the Hamiltonian is
H=(p_(a)^(2))/(2m)+(p_(b)^(2))/(2m)+V(r_(a))+V(r_(b)).
Suppose the two particles are identical spin-1 bosons. The symmetry condition
in this case is that the total wavefunction is symmetric under interchange of all
the quantum numbers belonging to the two particles.
In terms of the eigenfunctions psi _(n)(r) for a single particle in a square well, (a)
find the allowed values of the total spin of the two spin 1 bosons in the ground
state, and write down the ground state wave function. (Extra credit: write
down the wave function in terms of the actual square well wave functions). (b)
Find the allowed values of the total spin in the first excited state.
3. Two bosons in a well. Two particles of mass m are in the spherical potential well 0u>u 0o r>ro
where ro is a positive constant. Assume the two particles do not interact, so that the Hamiltonian is
H= 2m
-+V (ra)+V (r) . 2m
Suppose the two particles are identical spin-1 bosons. The symmetry condition in this case is that the total wavefunction is symmetric under interchange of all the quantum numbers belonging to the two particles.
In terms of the eigenfunctions /n(r) for a single particle in a square well, (a) find the allowed values of the total spin of the two spin 1 bosons in the ground state, and write down the ground state wave function. (Extra credit: write down the wave function in terms of the actual square well wave functions). (b) Find the allowed values of the total spin in the first excited state.