Every purely attractive one-dimensional potential well, no matter how shallow, has at least one bound state. More precisely, this statement can be formulated as follows. Consider a one-dimensional non-relativistic quantum particle of mass \(m\) inside a purely attractive well potential \(V(x)\): the function \(V(x)\) is such that \(V(x) < 0\) for all \(x\), and \(V(x) \to 0\) as \(x \to \pm \infty\). This system has at least one bound state. That is, the ground state energy \(E_{\text{ground}}\) of the Hamiltonian \[H = -\frac{\hbar^2}{2 m} \frac{d^2}{d x^2} + V(x)\] is strictly negative: \[E_{\text{ground}} < 0 \,.\] Proof: [D. ter Haar, Selected Problems in Quantum Mechanics (Academic, New York, 1964).] According to the variational principle, \[E_{\text{ground}} \le \langle \psi | H | \psi \rangle = \int_{-\infty}^{\infty} dx \, \psi^*(x) H \psi(x)\] for any normalized state \(\psi(x)\). Hence, in order to prove the negativity of \(E_{\text{ground}}\),