Homework 5, Exercise 2 notation

Homework 5, Exercise 2 notation

par Michael Zhang,
Nombre de réponses : 1

Hello,

I don't understand the notation  \sigma_1^z in the equation  \mathcal{H} = \hbar J \sigma_1^z \otimes \sigma_2^z . I assume that they are 2x2 matrices and that the tensor product gives a 4x4 matrix, but I can't figure out their explicit form.

Thanks in advance for your answer !
En réponse à Michael Zhang

Re: Homework 5, Exercise 2 notation

par Nicolas Macris,

Hello,

If you write out the two Pauli matrices in Dirac notation (in the computational basis |0>, |1>) and do the tensor product by expanding you find a sum of four terms. This is the 4x4 matrix written out in Dirac notation (in the computational basis |00>, |01>, |10>, |11>). You can then see that this matrix is the diagonal matrix with on the diagonal (+1, -1, -1, +1).

More about it in next class also.

Nicolas Macris