Symmetry requirements for identical paerticles

Problem:

The Hamiltonian for a system of two non-interacting particles is H = H1 + H2.   Particle 1 is in the normalized eigenstate |ψ1> of H1 and particle 2 is in the normalized eigenstate |ψ2> of H2.  Construct the two-particle eigenstates of H.  Does it depend on the nature of particles 1 and 2?

Solution:

Problem:

Suppose you have two particles, one in single particle states state Φ1 and Φ2, respectively.  
Assuming that Φ1 and Φ2 are orthonormal, construct a two-particle wave function assuming that the particles are
(a)  distinguishable,
(b)  bosons, and
(c)  fermions.

Solution:

Problem:

Consider the center of mass system of two interacting identical fermions with spin ½. 
(a)  What is the consequence of the Pauli exclusion principle on the two-particle wave function?
(b)  Let S1 and S2 be the spin operators of the two individual neutrons.
Show that the operators  P± = ½ ± ¼ ± S1∙S22  are the projection operators of the triplet states and the singlet states of the spin wave functions.
(c)  Using the Pauli exclusion principle and the symmetry properties of the spin and relative orbital angular momentum L, find the allowed values of L for any bound triplet state of the two-particle system.
(d)  Again using the Pauli exclusion principle and the symmetry properties of the space coordinate, show that the particles in a triplet state can never scatter through an angle of 90 degrees in their center of mass system.

Solution:

Problem:

Consider a coupled pair of one-dimensional, distinguishable, simple harmonic oscillators with equal masses, equal individual potentials U(x1) = ½Cx12, U(x2) = ½Cx22, C > 0, and a coupling potential Uc(x1,x2) = ½k(x2-x1)2, k > 0.
(a)  Separate the Hamiltonian in center of mass and relative variables R = ½(x1+x2) and r = (x2-x1).
(b)  Show that the total eigenfunction can be written as a product of two functions and determine the energy eigenvalues of the coupled system of distinguishable particles.
(c)  Now assume that the particles are indistinguishable so that they are either Bosons or Fermions.  Using the symmetry properties of the product functions determined in part (b), determine which of the energy levels found in part (b) are associated with each type of particle.
(d)  Show that for Fermions with their spins aligned the probability of finding the two particles at at the same position is zero.
(Note: In this problem, you may use your knowledge of the wave functions and eigenvalues of the simple harmonic oscillator without derivation or proof.)

Solution:

Problem:

Two identical spin-½ fermions are placed in the one-dimensional harmonic potential
U(x) = ½mω2x2,
where m is the mass of the fermion and ω its angular frequency.
(a)  Find the energies of the ground and first excited states of this two-fermion system.  Express the eigenstates corresponding to these two energy levels in terms of harmonic oscillator wave functions and spin states.
(b) Calculate the square of the separation of the two fermions,
<(x1 - x2)2> = <(x12 + x22 - 2x1x2)>
for the lowest energy state of the two-fermion system.
(c) Repeat the calculations for the first excited states.

Hint:
x1 =  (a0/√2)(a1 + a1),   x2 =  (a0/√2)(a2 + a2),   a0 = (ħ/(mω))½.

Solution: