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erok81
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Homework Statement
This is a two part problem. I think I am pretty close, but I'm quite understanding it. Which sucks because this is probably an extremely easy question.
I have five particles in a box with S spin. I am to find the ground state of the five particles if they have spin 0,½,1,3/2.
Homework Equations
My energy in box (infinite potential well) is given by:
[tex]E=\frac{n^2 \pi^2 \hbar^2}{2mL^2}[/tex]
The Attempt at a Solution
For S=0,1 all particles are Bosons and therefore can all fit in the same level. Therefore my ground state energy is given by:
[tex]E= \frac{5 \cdot 1^2 \pi^2 \hbar^2}{2mL^2}~=~ \frac{5 \pi^2 \hbar^2}{2mL^2}[/tex]
For my S=½ I can fit 2n2 where the factor of two arises because the allowed states are doubles since I can fit ± 1/2 spins per level. Since I have five particles, I will occupy up to n=2. Therefore my ground state energy is given by:
[tex]E= \frac{2(n_{1}^{2}+n_{2}^{2}) \pi^2 \hbar^2}{2mL^2}~=~ \frac{5 n^2 \pi^2 \hbar^2}{mL^2}[/tex]
And finally for my S=3/2 I can fit a factor of 4 per level (±3/2,±1/2). Using the same justification as in S=½ I arrive at the ground energy of:
[tex]E= \frac{10 n^2 \pi^2 \hbar^2}{2mL^2}[/tex]
Since the first level will contain four particles and n=2 will have one.
First. Is this correct so far?
The next part is where I am having the most trouble.
I have to find the excited state for all spins in the first half.
I know in order to do this I only need to move one particle to the next energy level. So for S=1,0 I think I have it. They are all in the ground state and only one needs to move up - four stay in the ground state and one moves up. Therefore I have.
[tex]E=\frac{4 \cdot 1^2 \pi^2 \hbar^2}{2mL^2}+\frac{2^2 \pi^2 \hbar^2}{2mL^2}~=~ \frac{4 n^2 \pi^2 \hbar^2}{mL^2}[/tex]
Is this looking okay so far?
That's about as far as I am comfortable with. The next spin I tried was S=½.
Here I have two particles in n=1 and three particles in n=2. Similar to the above case I only need to move one particle up. So I would have two in n=1, two in n=2, and one in n=3 to raise it to the first excited state. I know that much, but when I try to write it out, I am stuck. I think it's that factor of two that throws me off. I don't know if I should include it in each level.
So how does the work look so far? Am I even close?
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