ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Preamble: Evaluate the following integrals for $\psi_{J M}$ eigenfunctions of $\mathbf{J}^{2}$ and $\mathbf{J}_{z}$. 

$\int \psi_{22}^{*}\left(\widehat{\mathbf{J}}^{+}\right)^{4} \psi_{2,-2} d \tau$
Answer: 24
Problem: Preamble: Consider the 3-level $\mathbf{H}$ matrix
\[
\mathbf{H}=\hbar \omega\left(\begin{array}{ccc}
10 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -10
\end{array}\right)
\]
Label the eigen-energies and eigen-functions according to the dominant basis state character. The $\widetilde{10}$ state is the one dominated by the zero-order state with $E^{(0)}=10, \tilde{0}$ by $E^{(0)}=0$, and $-\widetilde{10}$ by $E^{(0)}=-10$ (we will work in units where $\hbar \omega = 1$, and can be safely ignored).

Use non-degenerate perturbation theory to derive the energy $E_{\widetilde{10}}$.  Carry out your calculations to second order in the perturbing Hamiltonian, and round to one decimal place.
Answer: 10.1