# Is this called an operator?

Consider the Hamiltonian: $$H=D\bigg(S_z-\frac{1}{3}S(S+1)\bigg)$$ Where $$S_z$$ is the spin-$$z$$ operator (one half the Pauli matrix for a doublet state) and the matrix representation of $$S$$ is the unit-matrix times the spin of the system considered. Now my question is, is $$S$$ classified as an operator, like $$S_z$$ is? If that is so, then I guess the '$$+1$$' (the unit matrix) is classified as an operator as well?
The reason why I ask is that the typical spin states $$|S\ M_S\rangle$$ are eigenvectors of $$S^2$$, but not of $$S$$, so the one above cannot represent actual angular momentum.

The $$S$$ in your hamiltonian is not an operator, that is just a number, but the combination $$S(S+1)\mathbb{1}$$ is an operator and, again, is a trivial operator since it's just the identity rescaled by some quantity $$S(S+1)$$. The eigenvalue of this new rescaled identity operator is just $$S(S+1)$$.
Since the eigenvalue of the operator $$S^2$$ are in fact $$S(S+1)$$ you'd be better rewriting your hamiltonian in the following manner $$\hat H=D\bigg(\hat{S}_z-\frac{1}{3}\hat{S}^2\bigg)$$ and now there's no doubt about the operator nature of the hamiltonian.
• You say that $S$ is just a number, but it is a number multiplied by the unit matrix, i.e. for a triplet state the matrix representation is: $S=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$. Jun 28, 2020 at 15:23
• That's not correct. You have $$S(S+1)\mathbb{1} = \left(\begin{matrix}S(S+1)&0&0\\0&S(S+1)&0\\0&0&S(S+1)\end{matrix}\right)$$ you should not regard $S$ as an operator since it clearly is not, what is an operator is the product of the number $S(S+1)$ with the identity operator. Jun 28, 2020 at 15:27
• Oh. I have always treated it as a matrix with the spin number along the diagonal, and the '$+1$' as the unit matrix (and '$+2$' as the unit matrix times 2) and this always gives me the correct matrices. Jun 28, 2020 at 15:29
• Can you tell me how I am supposed to interpret the $S^2$ in this Hamiltonian term: $\left(\frac{a}{6}\right)\left[S_{x}^{4}+S_{y}^{4}+S_{z}^{4}-\frac{1}{5} S(S+1)\left(3 S^{2}+3 S-1\right)\right]$? Jun 28, 2020 at 15:53