In quantum mechanics, the wave function \(\psi(x)\) contains all the information needed to compute measurement
statistics. The probability density for position is
Probability density.
\[
\rho(x)=|\psi(x)|^2=\psi^*(x)\psi(x).
\]
This density tells us where a particle is most likely to be found. However, a single density plot is often not enough.
To summarize the distribution quantitatively, one computes expectation values. The expectation value of position is
Position expectation values.
\[
\langle x\rangle=\int \psi^*(x)\,x\,\psi(x)\,dx
\]
\[
\langle x^2\rangle=\int \psi^*(x)\,x^2\,\psi(x)\,dx.
\]
The first of these gives the average position, while the second helps determine how spread out the position
distribution is. The position uncertainty is the standard deviation
\[
\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}.
\]
Momentum is treated through the momentum operator
Momentum operator.
\[
\hat p=-i\hbar\frac{d}{dx}.
\]
Therefore the momentum expectation values are
Momentum expectation values.
\[
\langle p\rangle=\int \psi^*(x)\,\hat p\,\psi(x)\,dx
\]
\[
\langle p^2\rangle=\int \psi^*(x)\,\hat p^2\,\psi(x)\,dx.
\]
Once those are known, the momentum uncertainty is
\[
\Delta p=\sqrt{\langle p^2\rangle-\langle p\rangle^2}.
\]
The uncertainty product is then
\[
\Delta x\,\Delta p.
\]
According to the Heisenberg uncertainty principle,
\[
\Delta x\,\Delta p\ge \frac{\hbar}{2}.
\]
Example: ground state of the infinite square well
For the 1D infinite square well of width \(L\), the normalized ground-state wave function is
\[
\psi(x)=\sqrt{\frac{2}{L}}\sin\!\left(\frac{\pi x}{L}\right),
\qquad 0
In this case, the position probability density is symmetric about the center of the box, so
\[
\langle x\rangle=\frac{L}{2}.
\]
The momentum expectation value is zero:
\[
\langle p\rangle=0.
\]
This does not mean the momentum is definitely zero. Instead, it means positive and negative momentum contributions
balance in the average. The state still has nonzero kinetic energy and nonzero momentum spread.
For the ground state, one also finds
\[
\langle x^2\rangle=L^2\left(\frac{1}{3}-\frac{1}{2\pi^2}\right)
\]
and
\[
\langle p^2\rangle=\frac{\hbar^2\pi^2}{L^2}.
\]
Therefore
\[
\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}
=L\sqrt{\frac{1}{12}-\frac{1}{2\pi^2}}
\]
and
\[
\Delta p=\sqrt{\langle p^2\rangle}
=\frac{\hbar\pi}{L}.
\]
Multiplying them gives the standard ground-state uncertainty product for the infinite well. This is larger than
\(\hbar/2\), so the Heisenberg principle is satisfied.
Gaussian and oscillator states
Gaussian wave functions are especially important because they often saturate or nearly saturate the uncertainty
relation. For a centered real Gaussian, the expectation value \(\langle x\rangle\) equals the packet center, while
\(\langle p\rangle=0\). The harmonic-oscillator ground state is also Gaussian in position space, which is why it is one
of the most useful exactly solvable minimum-uncertainty states in quantum mechanics.
More advanced problems may involve complex phases, moving packets, or time-dependent superpositions. In those cases,
\(\langle p\rangle\) can become nonzero, and both the density and the expectation values can evolve in time. But the
essential method remains the same: calculate averages from the wave function and its operators.
| Concept |
Main relation |
Meaning |
| Probability density |
\(\rho(x)=|\psi(x)|^2\) |
Measurement likelihood in position space |
| Average position |
\(\langle x\rangle=\int \psi^*x\psi\,dx\) |
Center of the position distribution |
| Average momentum |
\(\langle p\rangle=\int \psi^*\hat p\psi\,dx\) |
Mean momentum value |
| Position spread |
\(\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}\) |
Standard deviation in position |
| Momentum spread |
\(\Delta p=\sqrt{\langle p^2\rangle-\langle p\rangle^2}\) |
Standard deviation in momentum |
| Uncertainty principle |
\(\Delta x\,\Delta p\ge \hbar/2\) |
Fundamental lower bound on simultaneous localization |