# Wave packet of: 2 spin half non interacting indentical particles, wavepacket=?

sharomi

## Homework Statement

The two particles are confined to a 1D infinite potential well both have spin up.
the spin part of the wavepacket is thus: |arrowup,arrowup>
I need to write the wavepacket of the ground state

## The Attempt at a Solution

1. spatial part:
1/sqrt(2)(psi-ground(1,2)-psi-ground(2,1)
2.spin part:
|Z+>|Z+>
i know that total spin will be 1 as well as the angular part since they both add up but I'm not sure how to write this.
3. wavefunction=spatialXspin

i think i understand that the phyics here is that measurments of H will give me twice the ground state of each particles, of L^2 of the eigenvalue of l=1/2+1/2=1 and of spin S=1.
The spatial part has to be Anstisymetric as well since it's fermions and the spin part if symmetric, but as far as writing an actual solution I'm kinda lost in all the algebra. some help would be appretiated.

Staff Emeritus
Homework Helper
It's kind of hard to give advice since it's not clear what you know and don't know, etc. Do you know what the eigenfunctions of the two-particle system look like?

sharomi
emm...They would have to be eigenfunctions of both the Hamiltonian (for the ground state) and the Angular momentum operator (for j=total angular momentum=1). Also measurments of spin should give S=1.

If there was no spin involved i would just write:
psitotal=1/sqrt(2)[ |0,1> - |1,0> ]
where the two states corresspond to the case of the system where one particle is in the lowest energy state and the other occuping the next level (since they can't take the same level).

Now with two spin 1/2 particles i'm not sure what the process for the solution should look like. maybe i also need to assume that they can't take the same energy level - meaning that i can use the above state function (psitotal) for the orbital part. they already state in the question how spin of the two particle system looks like, i.e: |up,up>. so maybe the answer is: psitotal x |up,up>
i don't know if that's correct or perhaps a gross oversimplification of what i need to do in this excercise..

if you didn't understand my answer it's probably because i didn't understand the problem correctly, so just walk me through the solution as you understand it and i'll try to figure out what i was missing.. thanks

Staff Emeritus
What you seem confused about is what the spatial states are. For example, you keep talking about an angular part, separate from the spin, but this is a 1-D problem. What angles are there? The spatial eigenfunctions are of the form $\varphi_n(x_1)\varphi_m(x_2)$ where $\varphi_n(x)$ is the solution for a single particle in a box.