Definition of the Density Matrix for higher spins

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Eni
Posts: 2
Joined: Tue Dec 28, 2021 10:27 pm

Definition of the Density Matrix for higher spins

Post by Eni »

Hello,

I am having problems understanding the definition of the density matrix.
In the Spin Dynamics course we learned, that: rho = |psi><psi|. For a spin 1/2 with the two possible states this can be then [1,0;0,0] or [0,0;0,1]. Since adding/substracting the Identity doesn't change the expectation values or the time evolution, we substract -1 to get the same form as for the Hamiltonians = half the Pauli Matrices. So far so good. The problem comes with a spin 1 particle. Then per definition the states are:
[1,0,0; 0,0,0; 0,0,0]
[0,0,0; 0,1,0; 0,0,0]
[0,0,0; 0,0,0; 0,0,1]
From neither of these do we get easily with adding/substracting the 3x3 identity to the Iz operator [1,0,0; 0,0,0; 0,0,-1].
I am obviously missing something in this logic... can anyone tell me what it is?^^

Cheers, Eni
kuprov
Posts: 77
Joined: Mon Mar 29, 2021 4:26 pm

Re: Definition of the Density Matrix for higher spins

Post by kuprov »

Actually, you do get the right answer for any spin. The spin-half case was:

(+1/2)*|up><up| + (-1/2)*|down><down|

In the spin-1 case you have:

(+1)*|up><up| + (0)*|middle><middle| + (-1)*|down><down|

so exactly the same process.
Eni
Posts: 2
Joined: Tue Dec 28, 2021 10:27 pm

Re: Definition of the Density Matrix for higher spins

Post by Eni »

Ah thank you, Prof. Kuprov.
I think I was missing the multiplication with the expectation values. It is more clear now. :)
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