# Is the ground state energy always larger for the system with higher potential energy?

Say we have two Hamiltonians $$\hat{H}_1$$ and $$\hat{H}_2$$ that differ only in their potential energies and $$V_2(x) > V_1(x)$$ for all $$x$$. Is the energy of the ground state of system 2 necessarily larger than that of system 1?

## 1 Answer

Yes, because $$E_1~=~\langle \psi_1 |\hat{H}_1 |\psi_1\rangle~\stackrel{\begin{array}{c}\text{def. of }|\psi_1\rangle\text{ being}\cr \text{ground state for }\hat{H}_1\end{array}}{\leq}~\langle \psi_2 |\hat{H}_1 |\psi_2\rangle~\stackrel{\begin{array}{c}\hat{H}_2-\hat{H}_1\geq 0\cr \text{semipos. op.}\end{array}}{\leq}~\langle \psi_2 |\hat{H}_2 |\psi_2\rangle~=~E_2,$$ where $$|\psi_i\rangle$$ denote a ground state for system $$i\in\{1,2\}.$$

• This argument doesn't show that $E_2 > E_1$ though. – Lorenz Mayer Oct 25 '18 at 11:21
• It seems to me that your argument does prove that $E_2 > E_1$, since $H_2 - H_1$ is strictly positive. – Lorenz Mayer Oct 25 '18 at 17:03