Expectation values summarize the average outcome of many measurements performed on a quantum state. For a normalized
wave function \(\psi(x)\), the probability density in position space is \(|\psi(x)|^2\), and any position-space average
is computed by weighting with that density.
Position expectation values.
\[
\begin{aligned}
\langle x \rangle &= \int \psi^*(x)\,x\,\psi(x)\,dx \\
\langle x^2 \rangle &= \int \psi^*(x)\,x^2\,\psi(x)\,dx
\end{aligned}
\]
The corresponding uncertainty in position is the standard deviation of the distribution:
\[
\begin{aligned}
\Delta x &= \sqrt{\langle x^2 \rangle - \langle x \rangle^2}
\end{aligned}
\]
Momentum is represented by the operator
\[
\hat p = -\,i\hbar\frac{d}{dx}.
\]
Therefore the momentum expectation values are
Momentum expectation values.
\[
\begin{aligned}
\langle p \rangle &= \int \psi^*(x)\,\hat p\,\psi(x)\,dx \\
\langle p^2 \rangle &= \int \psi^*(x)\,\hat p^2\,\psi(x)\,dx
\end{aligned}
\]
In many introductory and numerical settings, one often studies real-valued wave functions with negligible boundary
values on the chosen interval. In that case, \(\langle p \rangle\) is approximately zero, and a convenient equivalent
form for the second moment is
\[
\begin{aligned}
\langle p^2 \rangle &= \hbar^2 \int \left(\frac{d\psi}{dx}\right)^2 dx
\end{aligned}
\]
The momentum uncertainty is then
\[
\begin{aligned}
\Delta p &= \sqrt{\langle p^2 \rangle - \langle p \rangle^2}
\end{aligned}
\]
Heisenberg uncertainty principle
Once \(\Delta x\) and \(\Delta p\) are known, one can test the Heisenberg relation
\[
\begin{aligned}
\Delta x\,\Delta p \ge \frac{\hbar}{2}.
\end{aligned}
\]
This is not a statement about experimental error or lack of knowledge. It is a structural feature of quantum states.
A wave function that is sharply localized in position necessarily contains a broad spread of momentum components, and a
wave function with narrowly defined momentum must be spatially broad.
Gaussian minimum-uncertainty state
The classic example of a minimum-uncertainty state is the Gaussian wave packet. For a properly chosen Gaussian form,
the product \(\Delta x\,\Delta p\) saturates the Heisenberg bound:
\[
\begin{aligned}
\Delta x\,\Delta p &= \frac{\hbar}{2}.
\end{aligned}
\]
This is why Gaussian states are often used to benchmark numerical uncertainty calculations. If the integration interval
is wide enough and the numerical sampling is fine enough, the computed product should come out very close to
\(\hbar/2\).
Why numerical integration is useful
For many wave functions, symbolic integration is possible only in special cases. A numerical calculator can still
evaluate the relevant moments as long as the wave function is sampled accurately across the region where it is
significant. This is especially useful for user-defined states, finite-window approximations, and comparisons between
different trial wave functions.
In more advanced work, one may also study complex wave functions, coherent states, squeezed states, and phase-carrying
packets with nonzero \(\langle p \rangle\). Those cases require a more general complex-valued treatment, but the core
structure remains the same: compute expectation values, form variances, and compare the resulting uncertainty product
with the quantum lower bound.
| Concept |
Main relation |
Meaning |
| Position mean |
\(\langle x \rangle = \int \psi^* x \psi\,dx\) |
Average position |
| Position spread |
\(\Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}\) |
Uncertainty in position |
| Momentum operator |
\(\hat p = -i\hbar\,d/dx\) |
Quantum momentum operator |
| Momentum spread |
\(\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}\) |
Uncertainty in momentum |
| Real-wavefunction shortcut |
\(\langle p^2 \rangle = \hbar^2 \int (\psi')^2 dx\) |
Useful numerical form for real \(\psi(x)\) |
| Heisenberg bound |
\(\Delta x\,\Delta p \ge \hbar/2\) |
Fundamental quantum limit |