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

Homework Equations

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.

 

[vela]

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

 

[vela]

You're pretty close. You're right that the spin state is symmetric so you need an antisymmetric spatial wavefunction, that the two particles can't both be in the individual lowest-energy states, and that the state is the spatial state x spin state. Your psitotal is also correct.

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.

If you don't actually need to write down the explicit wavefunction, then you're pretty much done. When you add spin to the problem, the states are just the cartesian product of the spatial states and the spin states, i.e. you just write spatial x spin for the state. You just have to make sure the symmetry of the overall state is correct.

 

[paranormal]

I can provide some guidance on how to approach this problem. First, let's break down the problem into smaller parts. We have two spin half particles in a 1D infinite potential well, both with spin up. This means that the spatial part of the wavefunction will be symmetric, while the spin part will be |arrowup,arrowup>.

To find the wavepacket of the ground state, we need to combine the spatial and spin parts. We can write the spatial part as the product of two single particle wavefunctions, psi-ground(1) and psi-ground(2), both in the ground state. This will give us a symmetric wavefunction overall.

Next, we need to consider the spin part. Since both particles have spin up, the total spin will be 1. The angular part will also add up to 1, since each particle has l=1/2. Therefore, we can write the spin part as |1,1>, which is the spin state with total spin 1 and z-component of spin 1.

Now, we can combine the spatial and spin parts to get the overall wavepacket. Since we have two particles, the wavefunction will be a product of two terms, one for each particle:

psi = psi-ground(1) * psi-ground(2) * |1,1>

This is the wavepacket of the ground state for the two spin half particles.

In terms of writing the wavefunction explicitly, we can use the shorthand notation for the spin states, where |1,1> = |arrowup,arrowup>. Therefore, the full wavefunction can be written as:

psi = 1/sqrt(2)(psi-ground(1,2) |arrowup,arrowup>)

where psi-ground(1,2) is the spatial part in the ground state for both particles.

I hope this helps to clarify the problem and guide you in finding a solution. Remember to always break down the problem into smaller parts and use the appropriate notation and concepts to arrive at a solution. Good luck!