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

  1. Mar 28, 2010 #1
    1. The problem statement, all variables and given/known data
    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

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Mar 28, 2010 #2

    vela

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    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?
     
  4. Mar 28, 2010 #3
    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 proccess 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
     
  5. Mar 28, 2010 #4

    vela

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    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 [itex]\varphi_n(x_1)\varphi_m(x_2)[/itex] where [itex]\varphi_n(x)[/itex] 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.
     
    Last edited: Mar 28, 2010
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