3.4 Expectation Values, and Uncertainties
3.4.1 Expectation Values
As discussed before, $|\psi(x)|^2$ can be treat as probability density. Thus the probability of finding the particle at an interval $(a,b)$ is given by a integration
$$
P=\int_a^b|\psi(x)|^2dx.
$$
We may also find the expectation value of quantity with position dependence via
$$
\overline{f(x)}=\int_{\text{all space}}f(x)|\psi(x)|^2dx.
$$
Or more precisely,
$$
\overline{O}=\int_{\text{all space}}\psi^*(x)\hat{O}\psi(x)dx,
$$
if the observable is described by an operator.
3.4.2 Uncertainties
Uncertainty $\Delta O$ can be defined as
$$
(\Delta O)^2=\overline{(O-\overline{O})^2},
$$
which can be simplified to be
$$
\begin{align}
\overline{(O-\overline{O})^2}&=\overline{(O^2+\overline{O}^2-2O\overline{O})}\\
&=\overline{O^2}+\overline{O}^2-2\overline{O}^2\\
&=\overline{O^2}-\overline{O}^2.
\end{align}
$$
Thus to find uncertainties, we normally find $\overline{O}$ and $\overline{O^2}$ first using
$$
\overline{O}=\int_{\text{all space}}\psi^*(x)\hat{O}\psi(x)dx,
$$
$$
\overline{O^2}=\int_{\text{all space}}\psi^*(x)\hat{O}^2\psi(x)dx.
$$
3.4.3 The Uncertainty Principle
In short, the uncertainty principle can be expressed as
$$
\Delta x\Delta p\ge\frac{\hbar}{2}.
$$
Detailed derivation can be found in the Appendix I.