- #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 single-particle state |n> and the other in state |k> with n not equal to k,

calculate the expectation value of the squared interparticle spacing <(x1-x2)^2>, assuming (a)

the particles are distinguishable, (b) the particles are identical spin-0 bosons, and (c) the particles

are identical spin-1/ 2 fermions in a spin triplet state. Use bra-ket 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 spin-0, 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_1-x_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_1-x_2)^2>##, but choosing an asymetric or symmetric spin changes the space representation which to me seems like it would change

##<(x_1-x_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 spin-0, 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_1-x_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_1-x_2)^2>##, but choosing an asymetric or symmetric spin changes the space representation which to me seems like it would change

##<(x_1-x_2)^2>##, giving me two different possible answers.

As for distinguishable particles, I'm not even sure where to begin...

I appreciate any help!