To determine the eigenvalues of the Hamiltonian of hydrogen we constructed the operators \(\hat{T}\) and \(\hat{S}\) from angular momentum and the Laplace-Runge-Lenz operator. The actual form of these operators in a specific basis was not so important, just that they satisfied mutually commuting \(\mathfrak{s u}(2)\) algebras. In this problem, we'll learn a bit more about these operators and how they act on energy eigenstates of hydrogen.

(a) First, what is the action of the operators \(\hat{T}_{i}\) and \(\hat{S}_{i}\) on the ground state of hydrogen?

(b) As we did with other \(\mathfrak{s u}(2)\) algebras, we can construct the raising and lowering operators

\[\begin{array}{ll}\hat{T}_{+}=\hat{T}_{x}+i \hat{T}_{y}, & \hat{T}_{-}=\hat{T}_{x}-i \hat{T}_{y}, \tag{9.176}\\\hat{S}_{+}=\hat{S}_{x}+i \hat{S}_{y}, & \hat{S}_{-}=\hat{S}_{x}-i \hat{S}_{y} .\end{array}\]

Consider the action of these operators on a general energy eigenstate of hydrogen, the \(|n, \ell, mangle\). What state or states labeled by radial, total angular momentum and \(z\)-component of angular momentum are produced from acting the raising and lowering operators?

(c) What is the action of the operators \(\hat{T}_{z}\) and \(\hat{S}_{z}\) on an energy eigenstate \(|n, \ell, mangle\) ? Can you identify the linear combination that produces the z-component of the angular momentum operator, \(\hat{L}_{z}\) ?

(d) Now, what do these \(\hat{T}_{i}\) and \(\hat{S}_{i}\) operators look like in position space?