8.5: An aside about Kinetic Energy

The expectation value of the kinetic energy $\langle \hat{T} \rangle$ is always positive. This can be shown by an integration by parts in which the first term vanishes provided the wavefunction tends to zero at infinity (which it will for a bound state). In 1D:

$$\langle\hat{T}\rangle = \frac{-\hbar^{2}}{2 m} \int \Phi^{*} \frac{d^{2}}{d x^{2}} \Phi d x = \frac{-\hbar^{2}}{2 m} [\Phi^{*} \frac{d}{d x} \Phi]_{-\infty}^{\infty} + \frac{\hbar^{2}}{2 m} \int \frac{d}{d x} \Phi^{*} \frac{d}{d x} \Phi d x = \frac{\hbar^{2}}{2 m} \int\left|\frac{d}{d x} \Phi\right|^{2} d x \nonumber$$

The second term integrand is positive everywhere, so the kinetic energy is always positive.