# Expectation value of Sz operator on the 0 Eigenstate in Spin 1 system

I'm trying to evaluate for the expectation value of the $$\hat{S}_z$$ operator on the $$|0>_z$$ eigenstate, which I'm assuming can be expressed $$<\hat{S}>=<0|\hat{S}|0>$$. I know the general eigenvalue equation tells me that $$\hat{S} _z|0> = \hbar 0 |0>$$, but this doesn't make sense when evaluating in matrix form, the output vector is $$\overrightarrow{0}$$:

$$\hbar\ [[1,0,0],[0,0,0],[0,0,-1]]\ \times[0,1,0]\ = \hbar\ [0,0,0]$$

Am I missing something or is there some reason that the expression doesn't translate to matrix form? And what is the final result for $$<\hat{S}>$$?

• You are not missing a thing: 0 [0,1,0]=[0,0,0]. So, what is the expectation value of 0? – Cosmas Zachos Nov 15 at 23:22