# 3.9: The Uncertainty Principle Redux - Estimating Uncertainties from Wavefunctions

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As will be discussed in Section 4.6, the operators \(\hat{x}\) and \(\hat{p}\) are not compatible and there is **no** measurement that can precisely determine the coresponding observables (\(x\) and \(p\)) simultaneously. Hence, there must be an uncertainty relation between them that specifies how uncertain we are about one quantity given a definite precision in the measurement of the other. Presumably, if one can be determined with infinite precision, then there will be an infinite uncertainty in the other. The uncertainty in a general quantity \(A\) is

\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2} \label{3.8.1}\]

where \(\langle A^2 \rangle\) and \(\langle A \rangle\) are the expectation values of \(\hat{A^2}\) and \(\hat{A}\) operators for a specific wavefunction. Extending Equation \ref{3.8.1} to \(x\) and \(p\) results in the following uncertainties

\[ \Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle ^2} \label{3.8.2a}\]

\[ \Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle ^2} \label{3.8.2b}\]

These quantities can be expressed explicitly in terms of the (time-dependent) wavefunction \(\Psi (x, t)\) using the fact that

\[ \begin{align} \langle x \rangle &= \langle \Psi(t)\vert \hat{x}\vert\Psi(t)\rangle \label{3.8.3} \\[4pt] &=\int \Psi^{*}(x,t) x \Psi(x,t)\;dx \nonumber \end{align} \]

and

\[ \begin{align} \langle x^2 \rangle &= \langle \Psi(t)\vert \hat{x}^2 \vert\Psi(t)\rangle \label{3.8.4} \\[4pt] &= \int \Psi^{*}(x,t) x^2 \Psi(x,t)\;dx \nonumber \end{align} \]

The middle terms in both Equations \(\ref{3.8.3}\) and \(\ref{3.8.4}\) are the integrals expressed in Dirac's Bra-ket notation. Similarly using the definition of the linear momentum operator:

\[\hat{p}_x = - i \hbar \dfrac{\partial}{ \partial x}.\]

So

\[ \begin{align} \langle p \rangle &= \langle \Psi(t)\vert \hat{p} \vert\Psi(t)\rangle \label{3.8.5} \\&= \int \Psi^{*}(x,t) - i \hbar {\partial \over \partial x}\Psi(x,t)\,dx \nonumber \end{align} \]

and

\[ \begin{align} \langle p^2 \rangle &= \langle \Psi(t)\vert \hat{p}^2\vert\Psi(t)\rangle \label{3.8.6} \\ &= \int \Psi ^{*} (x, t)\left(-\hbar^2{\partial^2 \over \partial x^2}\right)\Psi(x,t) \;dx \nonumber \end{align} \]

The Heisenberg uncertainty principle can be quantitatively connected to the properties of a wavefunction, i.e., calculated via the expectation values outlined above:

\[\Delta p \Delta x \ge \dfrac {\hbar}{2} \label {3.8.8}\]

This essentially states that the greater certainty that a measurement of \(x\) or \(p\) can be made, the greater will be the *uncertainty* in the other. Hence, as \(Δp\) approaches 0, \(Δx\) must approach \(\infty\), which is the case of the free particle (e.g, with \(V(x)=0\)) where the momentum of a particles can be determined precisely.

The uncertainty principle is a consequence of the wave property of matter. A wave has some finite extent in space and generally is not localized at a point. Consequently there usually is significant uncertainty in the position of a quantum particle in space.

## Contributors

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")