 #1
MaestroBach
 40
 3
 Homework Statement:

Consider two noninteracting particles of mass m in the harmonic oscillator potential well. For
the case with one particle in the singleparticle state n> and the other in state k> with n not equal to k,
calculate the expectation value of the squared interparticle spacing <(x1x2)^2>, assuming (a)
the particles are distinguishable, (b) the particles are identical spin0 bosons, and (c) the particles
are identical spin1/ 2 fermions in a spin triplet state. Use braket notation as far as you
can, but you will have to do some integrals.
 Relevant Equations:
 N/A
I'm having a hard time understanding how to treat fermions, bosons, and distinguishable particles differently for this problem.
To the best of my understanding, I know that my overall state for bosons must be symmetric, and because they're spin0, this means there's only one coupled spin state available for them, ie ##\ket{S m_s} = \ket{0 0}##, and since their spin is symmetric then their space representation must also be symmetric, giving me ##\psi_{nk} = \psi_n(x_1)\psi_k(x_2) + \psi_n(x_2)\psi_n(x_1)##, from which I can calculate ##<(x_1x_2)^2>##.
However, my confusion comes in when I'm dealing with fermions and distinguishable particles.
For fermions, as far as I understand, the spin could be both asymetric or symmetric, but that would change whether my space representation is asymetric or symmetric. How do I decide which to use? This is especially confusing for me, given that the spin does not play a part in my calculation of ##<(x_1x_2)^2>##, but choosing an asymetric or symmetric spin changes the space representation which to me seems like it would change
##<(x_1x_2)^2>##, giving me two different possible answers.
As for distinguishable particles, I'm not even sure where to begin...
I appreciate any help!
To the best of my understanding, I know that my overall state for bosons must be symmetric, and because they're spin0, this means there's only one coupled spin state available for them, ie ##\ket{S m_s} = \ket{0 0}##, and since their spin is symmetric then their space representation must also be symmetric, giving me ##\psi_{nk} = \psi_n(x_1)\psi_k(x_2) + \psi_n(x_2)\psi_n(x_1)##, from which I can calculate ##<(x_1x_2)^2>##.
However, my confusion comes in when I'm dealing with fermions and distinguishable particles.
For fermions, as far as I understand, the spin could be both asymetric or symmetric, but that would change whether my space representation is asymetric or symmetric. How do I decide which to use? This is especially confusing for me, given that the spin does not play a part in my calculation of ##<(x_1x_2)^2>##, but choosing an asymetric or symmetric spin changes the space representation which to me seems like it would change
##<(x_1x_2)^2>##, giving me two different possible answers.
As for distinguishable particles, I'm not even sure where to begin...
I appreciate any help!