In non-degenerate first-order perturbation theory, the energy correction to the nth energy level of an unperturbed Hamiltonian $H_0$ is given by:

$" id="MathJax-Element-68-Frame" role="presentation" style="position: relative;" tabindex="0"> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi></mi></msup></mrow></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi></mi></msubsup></mrow></semantics></math> \(E_n^{(1)} = ⟨ ψ_n^{(0)} ∣ H ′ ∣ ψ_n ^{(0)} ⟩ \)$

where $H'$ is the perturbation Hamiltonian, and $" id="MathJax-Element-70-Frame" role="presentation" style="position: relative;" tabindex="0"> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi></mi></msup></mrow></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi></mi></msubsup></mrow></semantics></math> \(ψ_n^{(0)} ψ_0^{(0)}\)$ are the unperturbed wavefunctions. Consider a system where the perturbation $H'$ is proportional to the position operator $x$ . Which of the following systems will have a zero first-order energy correction due to symmetry considerations?