Infinite Well Energies

The lowest energy eigenvalue of the infinite potential well is

$$\varepsilon_1=\frac{\hbar^2 \pi^2}{2mL^2} >0.$$ (256)

This energy is above the bottom at $x=0$ of the infinite well potential (which is zero inside the infinite well).

According to our axiom 2c,

Axiom 2c: The possible outcomes of measurements of the energy corresponding to the hermitian linear Hamilton operator $\hat{H}$ are the eigenvalues of $\hat{H}$. Immediately after the measurement, the quantum system is in the eigenstate of $\hat{H}$ corresponding to the eigenvalue that is measured.

the lowest possible energy value that can be measured is $(1/2)\hbar \omega$ for the linear harmonic oscillator. It is called zero-point energy.