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3.8: The Uncertainty Principle

In the mid 1920's the German physicist Werner Heisenberg showed that if we try to locate an electron within a region Δx; e.g. by scattering light from it, some momentum is transferred to the electron, and it is not possible to determine exactly how much momentum is transferred, even in principle.

As will be discussed in Section 4.6, the operators \(x\) and \(p\) are not compatible and there is no measurement that can precisely determine both \(x\) 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

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

Thus, for \(x\) and \(p\), we have

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

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

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

\[\langle x \rangle = \langle \Psi(t)\vert x\vert\Psi(t)\rangle =\int \Psi^*(x,t) x \Psi(x,t)\;dx \label{3.8.3}\]

and

\[\langle x^2 \rangle = \langle \Psi(t)\vert x^2 \vert\Psi(t)\rangle = \int \Psi^*(x,t) x^2 \Psi(x,t)\;dx \label{3.8.4}\]

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 \(\left(\hat{p}_x = i\hbar \frac{d}{dx}\right)\),

\[\langle p \rangle = \langle \Psi(t)\vert p \vert\Psi(t)\rangle  = \int \Psi^*(x,t){\hbar \over i}{\partial \over \partial x}\Psi(x,t)\,dx \label{3.8.5}\]

and

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

Then, the Heisenberg uncertainty principle states that

\[\Delta x \Delta p \stackrel{>}{\sim} \hbar \label{3.8.7}\]

which essentially states that the greater certainty with which a measurement of \(x\) or \(p\) can be made, the greater will be the uncertainty in the other. Equation \(\ref{3.8.7}\) can be later refined to, which is the preferred representation of the uncertainty principle:

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

The Heisenberg uncertainly principle essentially states that the greater certainty with which a measurement of \(x\) or \(p\) can be made, the greater will be the uncertainty in the other.

You can see from Equation \(\ref{3.8.8}\) that as \(Δp\) approaches 0, \(Δx\) must approach \(\infty\), which is the case of the free particle (e..g, with \(V(x)=0\)).

Exercise \(\PageIndex{1}\)

Compare the minimum uncertainty in the positions of a baseball (mass = 140 gm) and an electron, each with a speed of 91.3 miles per hour, which is characteristic of a reasonable fastball, if the standard deviation in the measurement of the speed is 0.1 mile per hour. Also compare the wavelengths associated with these two particles. Identify the insights that you gain from these comparisons. 

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.

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